{"continents":[{"blurb":"The one continent where a question can be closed for good. A proof settles it for every future reader, and a proof assistant can now check it line by line. Most of the atlas's machine-checkable problems live here, drawn from formal-conjectures and erdosproblems.","id":"mathematics","method":"Proof, now machine-checkable","name":"Mathematics","wash":"#3d4a7a"},{"blurb":"What can be computed, how fast, and how securely. Its questions are mathematics about machines, and they increasingly get settled by machines as well.","id":"computation","method":"Proof plus construction","name":"Computation","wash":"#5e4a7e"},{"blurb":"The laws under everything else. Physics closes a question when a prediction survives a measurement designed to break it. Its open problems are where the two great theories, quantum mechanics and general relativity, stop agreeing.","id":"physics","method":"Prediction against measurement","name":"Physics","wash":"#7a4a6e"},{"blurb":"The universe as a whole, studied from one vantage point with no way to rerun the experiment. About 95% of its contents are named but not understood.","id":"cosmos","method":"Observation at a distance","name":"Cosmos","wash":"#8e3f47"},{"blurb":"The one planet we can study from inside. Its questions span deep time, the churning interior and a climate we are now changing faster than we can measure.","id":"earth","method":"Observation and modeling of one planet","name":"Earth","wash":"#9a5a33"},{"blurb":"Making and understanding substances. Its open problems decide what materials the next century is built from and how its energy is stored.","id":"chemistry","method":"Synthesis and measurement","name":"Chemistry and matter","wash":"#a67c2e"},{"blurb":"Systems that copy themselves with variation and so were shaped by selection rather than designed. Biology's great solved problems read the code; its open ones ask how the code builds bodies, how it began, and why it wears out.","id":"life","method":"Experiment on evolved systems","name":"Life","wash":"#6f7a34"},{"blurb":"How matter comes to think, remember and speak. The continent now has two kinds of subjects, brains and trained models, and questions about each shed light on the other.","id":"mind","method":"Experiment on minds, natural and artificial","name":"Mind","wash":"#4a7a45"},{"blurb":"Knowledge put to work against disease. Medicine settles its questions with the controlled trial, and its record includes the only human disease ever eradicated.","id":"medicine","method":"The trial","name":"Medicine","wash":"#2f7a6a"},{"blurb":"How people live together, and what they left behind. Society cannot run controlled experiments on nations, so it reads history and finds natural experiments, and some of its questions have exact answers anyway.","id":"society","method":"Natural experiments and history","name":"Society","wash":"#2d6d7e"},{"blurb":"Problems that close when something works. Engineering's verifiers are physical: a clock that keeps time at sea, a plane that stays up, a reactor that makes more energy than it takes. Many of its open problems decide the energy future.","id":"engineering","method":"The working prototype","name":"Engineering","wash":"#2e5a86"},{"blurb":"Where questions go before anyone knows how to settle them. Several continents on this map began here: physics as natural philosophy, psychology as philosophy of mind, logic as the study of argument. What remains are the questions no experiment has yet been able to reach.","id":"philosophy","method":"Argument","name":"Philosophy","wash":"#6e5a45"}],"details":{"abc-conjecture":{"explanation":"It ties addition to multiplication and implies dozens of other results, including an asymptotic Fermat's Last Theorem. Shinichi Mochizuki announced a proof in 2012 through his inter-universal Teichmüller theory. Scholze and Stix identified a step they consider a fatal gap, and most number theorists do not regard the conjecture as settled. The status here is open, with the dispute recorded rather than resolved.","legacy":"wiki-abc","links":[["Wikipedia: abc conjecture","https://en.wikipedia.org/wiki/Abc_conjecture"]],"posed":"1985, Joseph Oesterlé and David Masser","statement":"For coprime a + b = c, the product of the distinct primes dividing abc is rarely much smaller than c.","verifier":"Lean statement in formal-conjectures","y":[1985,null,null,false]},"age-of-earth":{"closed":"1956, Clair Patterson: 4.55 billion years","explanation":"Kelvin's physics seemed to rule out the long ages geology and evolution needed, because radioactivity was unknown. Patterson dated meteorites with uranium–lead isotopes. Fighting lead contamination in his own lab led him to discover how much lead industry had put into the environment, and to a campaign that ended leaded petrol.","links":[["Wikipedia: Age of Earth","https://en.wikipedia.org/wiki/Age_of_Earth"]],"posed":"1862, Lord Kelvin's cooling estimate of 20–400 million years","statement":"How old is the Earth?","y":[1862,1956,null,false]},"aging":{"explanation":"Researchers list molecular 'hallmarks' of aging, but which ones are causes rather than symptoms is disputed. Caloric restriction and several drugs extend lifespan in model organisms. Partial reprogramming with Yamanaka factors reverses some aging markers in mice.","links":[["Wikipedia: Ageing","https://en.wikipedia.org/wiki/Ageing"]],"statement":"What causes aging, and can it be slowed or reversed in humans?","verifier":"A reproducible lifespan extension in mammals with a known mechanism","y":[1825,null,null,true]},"ai-alignment":{"explanation":"Current methods, human feedback and constitutional training, work in practice and come with no guarantees. The problem grows harder as systems become more capable than the people overseeing them.","links":[["Wikipedia: AI alignment","https://en.wikipedia.org/wiki/AI_alignment"]],"statement":"How do we make capable AI systems reliably pursue the goals their users and society intend?","y":[1960,null,null,true]},"alzheimers":{"explanation":"The amyloid hypothesis dominated for decades and drew hundreds of failed trials. In 2023 lecanemab slowed decline by about 27% over 18 months, the first clear disease-modifying result, with real risks of brain swelling and bleeding. Halting the disease, rather than slowing it, has not been achieved.","links":[["Wikipedia: Alzheimer's disease","https://en.wikipedia.org/wiki/Alzheimer%27s_disease"]],"posed":"1906, Alois Alzheimer","statement":"What causes Alzheimer's disease, and can it be halted or prevented?","verifier":"A trial endpoint: halted cognitive decline","y":[1906,null,2023,false]},"ambient-ammonia":{"explanation":"Haber–Bosch uses about 1–2% of world energy and emits over 1% of global CO2. Nitrogenase enzymes do it at room temperature. Electrochemical routes are promising, but many reported results were later traced to contamination, so the field adopted strict isotope-labeling protocols.","links":[["Wikipedia: Nitrogen fixation","https://en.wikipedia.org/wiki/Nitrogen_fixation"]],"statement":"Can nitrogen be reduced to ammonia efficiently at room temperature and pressure?","verifier":"15N isotope-labeled yield under a standard protocol","y":[1960,null,null,true]},"amoc-tipping":{"explanation":"Its collapse would cool northwest Europe sharply and shift tropical rainfall. Direct measurements only began in 2004, and proxy-based early-warning analyses disagree about how close it is to a threshold.","links":[["Wikipedia: Atlantic meridional overturning circulation","https://en.wikipedia.org/wiki/Atlantic_meridional_overturning_circulation"]],"statement":"Will the Atlantic Meridional Overturning Circulation collapse this century, and at what warming?","y":[1961,null,null,false]},"antimicrobial-resistance":{"explanation":"Resistant infections were directly responsible for over a million deaths in 2019. Few new antibiotic classes have been discovered since the 1980s, largely because they earn too little to justify development. Machine-learning screens have found candidates such as halicin and abaucin, and pull incentives such as subscription payments are being tried to fix the economics.","links":[["Wikipedia: Antimicrobial resistance","https://en.wikipedia.org/wiki/Antimicrobial_resistance"]],"statement":"Can new antibiotics and strategies outpace the evolution of drug-resistant bacteria?","y":[1945,null,null,true]},"arc-agi":{"explanation":"The benchmark was designed to measure skill acquisition rather than stored knowledge. It is run as a prize competition with hidden test sets, which makes it a working example of a verifier-backed prize.","links":[["ARC Prize","https://arcprize.org/"]],"posed":"2019, François Chollet","prize":"ARC Prize","statement":"Can a system solve novel abstract-reasoning grid puzzles as efficiently as humans do?","verifier":"Hidden evaluation set","y":[2019,null,null,false]},"arrow-impossibility":{"closed":"1951, Kenneth Arrow's impossibility theorem","explanation":"Arrow proved that no ranked voting rule can satisfy unanimity, independence of irrelevant alternatives and non-dictatorship together. It is a theorem about democracy with a short proof, and it shifted the field from searching for the perfect rule to choosing which condition to give up. Gibbard and Satterthwaite later showed every reasonable rule can be manipulated.","links":[["Wikipedia: Arrow's impossibility theorem","https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem"]],"posed":"1785, the Marquis de Condorcet's voting paradox","statement":"Is there a way to combine individual rankings of three or more options into a group ranking that satisfies a few natural fairness conditions?","y":[1785,1951,null,false]},"arrow-of-time":{"explanation":"The laws of physics are nearly time-symmetric, and the second law explains the direction of time only given a low-entropy past. Why the early universe was so special, the past hypothesis, is a question for cosmology that physics has not answered.","links":[["Wikipedia: Arrow of time","https://en.wikipedia.org/wiki/Arrow_of_time"]],"statement":"Why did the universe begin in a state of such low entropy?","y":[1876,null,null,true]},"artificial-photosynthesis":{"explanation":"Lab devices have exceeded plant efficiencies, but they combine cost, stability and efficiency poorly: the fast ones degrade or use rare metals. It is a race against photovoltaics plus electrolysis, which keeps getting cheaper.","links":[["Wikipedia: Artificial photosynthesis","https://en.wikipedia.org/wiki/Artificial_photosynthesis"]],"statement":"Can sunlight turn water and CO2 into fuel efficiently, cheaply and durably?","y":[1912,null,null,false]},"autonomous-driving":{"closed":"2020 onward, driverless commercial robotaxi service in limited cities","explanation":"Driverless taxis operate commercially in several cities, with published safety records that compare favorably with human drivers in those areas. Driving everywhere, in all weather and without detailed maps, is not achieved.","links":[["Wikipedia: Self-driving car","https://en.wikipedia.org/wiki/Self-driving_car"]],"statement":"Can cars drive themselves safely everywhere people drive?","y":[2004,null,2020,false]},"baryogenesis":{"explanation":"The Big Bang should have made equal amounts, which would have annihilated. About one extra matter particle per billion pairs survived. The Standard Model's CP violation is far too weak to explain that excess, so something beyond it is required.","links":[["Wikipedia: Baryon asymmetry","https://en.wikipedia.org/wiki/Baryon_asymmetry"]],"posed":"1967, Andrei Sakharov's conditions","statement":"Why does the universe contain matter but almost no antimatter?","y":[1967,null,null,false]},"bell-local-realism":{"closed":"2015, loophole-free Bell tests at Delft, NIST and Vienna","explanation":"Bell turned a philosophical argument into an inequality that experiments could test. Aspect, Clauser and Zeilinger's experiments violated it, and in 2015 three groups closed the remaining loopholes at once. The 2022 Nobel Prize recognized the work. Local realism is ruled out; which assumption to give up is still argued.","links":[["Wikipedia: Bell test","https://en.wikipedia.org/wiki/Bell_test"]],"posed":"1935, Einstein, Podolsky and Rosen; 1964, John Bell made it testable","statement":"Can the correlations of quantum mechanics be explained by local hidden variables?","y":[1935,2015,null,false]},"benzene-structure":{"closed":"1865, August Kekulé; 1931, Linus Pauling's resonance and Erich Hückel's orbitals","explanation":"Kekulé's ring of alternating bonds explained the formula but not the unusual stability. Quantum mechanics explained it: the electrons are shared around the whole ring. The idea of aromaticity became a pillar of organic chemistry.","links":[["Wikipedia: Benzene","https://en.wikipedia.org/wiki/Benzene"]],"statement":"How are benzene's six carbon atoms arranged, and why is it so stable?","y":[1825,1931,1865,false]},"bieberbach-conjecture":{"closed":"1985, Louis de Branges","explanation":"Proved for small n one case at a time over seven decades. De Branges's proof was at first met with doubt, partly because he had earlier announced flawed results. A Leningrad seminar verified it, and the key inequality turned out to be a known fact about special functions, found by Askey and Gasper.","links":[["Wikipedia: De Branges's theorem","https://en.wikipedia.org/wiki/De_Branges%27s_theorem"]],"posed":"1916, Ludwig Bieberbach","statement":"The coefficients of a normalized univalent function on the unit disk satisfy $|a_n| \\le n$.","y":[1916,1985,null,false]},"big-bang-vs-steady-state":{"closed":"1965, Penzias and Wilson detect the cosmic microwave background","explanation":"Two radio engineers traced a persistent hiss in their antenna to a uniform glow from every direction. It was the leftover heat of the early universe that Big Bang models predicted, and steady-state theory could not explain it. They first suspected pigeon droppings in the antenna.","links":[["Wikipedia: Cosmic microwave background","https://en.wikipedia.org/wiki/Cosmic_microwave_background"]],"statement":"Did the universe have a hot, dense beginning, or has it always looked the same?","y":[1948,1965,null,false]},"birch-swinnerton-dyer":{"explanation":"It predicts how many independent rational solutions a cubic equation has from an analytic function built out of counting solutions modulo primes. It is proved when the analytic rank is 0 or 1, by Gross–Zagier and Kolyvagin. It is one of the rare Millennium problems whose statement came out of machine computation.","legacy":"millennium-bsd","links":[["Wikipedia: Birch and Swinnerton-Dyer conjecture","https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture"]],"posed":"1965, Bryan Birch and Peter Swinnerton-Dyer, from early computer experiments","prize":"$1,000,000 Clay Millennium Prize","statement":"The rank of an elliptic curve's rational points equals the order of vanishing of its L-function at s = 1.","verifier":"Lean statement in formal-conjectures","y":[1965,null,null,false]},"black-hole-image":{"closed":"2019, Event Horizon Telescope (M87*); 2022, Sagittarius A*","explanation":"A planet-sized array of radio telescopes, synchronized by atomic clocks, resolved the ring of light around M87's black hole. The shadow's size matched general relativity's prediction.","links":[["Wikipedia: Event Horizon Telescope","https://en.wikipedia.org/wiki/Event_Horizon_Telescope"]],"statement":"Can the shadow of a black hole's event horizon be seen?","y":[1973,2019,null,true]},"black-hole-information":{"explanation":"Hawking radiation looks thermal, which would destroy information and break quantum mechanics. Since 2019, calculations using 'islands' and replica wormholes have reproduced the Page curve, which says information does come back out. How it escapes, in terms of a local mechanism, is still debated, so the problem is marked partial.","links":[["Wikipedia: Black hole information paradox","https://en.wikipedia.org/wiki/Black_hole_information_paradox"]],"posed":"1976, Stephen Hawking","statement":"Is information that falls into a black hole destroyed when the black hole evaporates?","y":[1976,null,2019,false]},"boolean-pythagorean-triples":{"closed":"2016, Marijn Heule, Oliver Kullmann and Victor Marek","explanation":"No: the integers 1 to 7,825 cannot be colored that way, while 1 to 7,824 can. A SAT solver proved it and produced a 200-terabyte certificate that independent checkers verified. Graham paid the $100 prize he had offered. It is the pure form of a machine-verified prize.","links":[["Wikipedia: Boolean Pythagorean triples problem","https://en.wikipedia.org/wiki/Boolean_Pythagorean_triples_problem"]],"posed":"1980s, Ronald Graham","prize":"$100 from Ronald Graham, paid","statement":"Can the positive integers be colored red and blue so that no Pythagorean triple is monochromatic?","verifier":"DRAT proof certificate, independently checked","y":[1980,2016,null,false]},"bounded-prime-gaps":{"closed":"2013, Yitang Zhang; bound improved by James Maynard and Polymath8","explanation":"Zhang proved that some gap below 70 million recurs infinitely often, the first finite bound ever. Within a year, the Polymath8 project and Maynard's simpler multidimensional sieve brought the bound down to 246. It is one of the clearest recent examples of a public collaboration improving a result in the open. Under the Elliott–Halberstam conjecture the bound drops to 6.","links":[["Wikipedia: Prime gap","https://en.wikipedia.org/wiki/Prime_gap"],["Wikipedia: Polymath Project","https://en.wikipedia.org/wiki/Polymath_Project"]],"posed":"Implicit in the twin prime question","statement":"Some fixed gap between consecutive primes occurs infinitely often.","y":[1849,2013,null,true]},"broca-localization":{"closed":"1861, Paul Broca; 1874, Carl Wernicke","explanation":"Broca's patient 'Tan' could understand speech but say only one syllable. His autopsy showed damage to the left frontal lobe, which gave the first strong evidence for localized brain function. Modern imaging shows language relies on distributed networks around these areas.","links":[["Wikipedia: Broca's area","https://en.wikipedia.org/wiki/Broca%27s_area"]],"statement":"Is the capacity for language tied to particular regions of the brain?","y":[1825,1861,null,true]},"burnside-problem":{"closed":"1964 Golod–Shafarevich (general); 1968 Novikov–Adian (bounded exponent); restricted version solved by Zelmanov, 1991","explanation":"No: infinite such groups exist, even with a uniform bound on the orders. The restricted version, which asks only about finite quotients, is true, and Efim Zelmanov's proof of it won a Fields Medal in 1994.","legacy":"wiki-bounded-burnside-problem","links":[["Wikipedia: Burnside problem","https://en.wikipedia.org/wiki/Burnside_problem"]],"posed":"1902, William Burnside","statement":"Is a finitely generated group in which every element has finite order necessarily finite?","y":[1902,1964,null,false]},"busy-beaver-5":{"closed":"2024, the bbchallenge collaboration; proof checked in Coq","explanation":"The answer is 47,176,870. A volunteer collaboration decided whether each of roughly 180 million machines halts, and the whole argument was checked in Coq. It combines a mathematical result, a large computer search and a formal proof, the pattern this atlas is built for. BB(6) is at least beyond 2↑↑↑5 and may run into problems like Collatz.","links":[["Wikipedia: Busy beaver","https://en.wikipedia.org/wiki/Busy_beaver"],["bbchallenge.org","https://bbchallenge.org/"]],"posed":"1962, Tibor Radó","statement":"What is the maximum number of steps a halting 5-state, 2-symbol Turing machine can run?","verifier":"Coq proof of BB(5) = 47,176,870","y":[1962,2024,null,false]},"busy-beaver-6":{"explanation":"Some 6-state machines behave like Collatz-type iterations, so deciding them may require solving problems that mathematics has not solved. It is a clean, checkable target where every step of progress is a certificate.","legacy":"wiki-busy-beaver","links":[["bbchallenge.org","https://bbchallenge.org/"]],"statement":"Determine the maximum number of steps of a halting 6-state, 2-symbol Turing machine.","y":[1962,null,2025,false]},"cambrian-explosion":{"explanation":"Candidate causes include rising oxygen, predation arms races, and the evolution of developmental gene networks. Precambrian fossils have made the explosion look less sudden than it once did, but its trigger is still argued.","links":[["Wikipedia: Cambrian explosion","https://en.wikipedia.org/wiki/Cambrian_explosion"]],"statement":"Why did most animal body plans appear within about 20 million years, around 540 million years ago?","y":[1859,null,null,true]},"cancer":{"explanation":"Cancer is hundreds of diseases. Some once-fatal cancers are now usually curable, including childhood leukemia, testicular cancer and Hodgkin lymphoma. Immunotherapy and CAR-T therapy produce durable remissions in some patients. Metastatic solid tumors remain mostly incurable.","links":[["Wikipedia: Cancer research","https://en.wikipedia.org/wiki/Cancer_research"]],"statement":"Can cancer be reliably prevented, detected early and cured?","y":[1971,null,null,true]},"carbon-removal":{"explanation":"Most climate scenarios that meet the Paris targets assume large removals by mid-century. Direct air capture works today at thousands of tonnes a year and several hundred dollars a tonne. Enhanced weathering and other methods are cheaper but harder to measure, so verification is a large part of the problem.","links":[["Wikipedia: Carbon dioxide removal","https://en.wikipedia.org/wiki/Carbon_dioxide_removal"]],"statement":"Can CO2 be removed from the atmosphere permanently at billions of tonnes per year and under $100 a tonne?","y":[1999,null,null,true]},"catalan-conjecture":{"closed":"2002, Preda Mihăilescu","explanation":"Tijdeman showed in 1976 that there are only finitely many solutions, but with a bound too large to search. Mihăilescu finished it with the arithmetic of cyclotomic fields and no computer search at all. The formal-conjectures entry keeps related Diophantine questions open.","legacy":"wiki-catalan","links":[["Wikipedia: Catalan's conjecture","https://en.wikipedia.org/wiki/Catalan%27s_conjecture"]],"posed":"1844, Eugène Catalan","statement":"8 and 9 are the only consecutive perfect powers.","y":[1844,2002,null,false]},"cellular-reprogramming":{"closed":"1962, John Gurdon (frog cloning); 2006, Shinya Yamanaka (four factors)","explanation":"Gurdon showed a frog's intestinal cell nucleus could make a whole frog, so differentiation does not erase genes. Yamanaka found that four transcription factors turn skin cells into pluripotent stem cells. They shared the 2012 Nobel Prize.","links":[["Wikipedia: Induced pluripotent stem cell","https://en.wikipedia.org/wiki/Induced_pluripotent_stem_cell"]],"statement":"Can a specialized adult cell be returned to an embryonic state?","y":[1892,2006,1962,true]},"chronic-pain":{"explanation":"About one in five adults lives with chronic pain. Opioids work poorly long-term and caused an addiction crisis. In 2025 suzetrigine became the first new class of non-opioid pain drug approved in decades, for acute pain. Chronic pain involves changes in the nervous system itself, which makes it harder.","links":[["Wikipedia: Chronic pain","https://en.wikipedia.org/wiki/Chronic_pain"]],"statement":"What causes pain to persist after injury heals, and can it be treated without addictive drugs?","y":[1965,null,2025,true]},"classification-finite-simple-groups":{"closed":"1983 announced; 2004 gaps closed (Aschbacher and Smith)","explanation":"The proof spans tens of thousands of pages by more than a hundred authors over fifty years. A 'second generation' proof, begun by Gorenstein, Lyons and Solomon, is still being written up to make it checkable by one person. It is the strongest argument for formal verification at scale.","links":[["Wikipedia: Classification of finite simple groups","https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups"]],"statement":"Every finite simple group is cyclic, alternating, of Lie type, or one of 26 sporadic groups.","y":[1892,2004,1983,true]},"climate-sensitivity":{"explanation":"The Charney report's range stood for forty years. Combining process understanding, historical warming and paleoclimate narrowed it: the IPCC's 2021 assessment gives a likely range of 2.5 to 4 °C. Cloud feedbacks remain the biggest source of uncertainty, and they are what separates a manageable future from a severe one.","links":[["Wikipedia: Climate sensitivity","https://en.wikipedia.org/wiki/Climate_sensitivity"]],"posed":"1896, Svante Arrhenius; 1979 Charney report: 1.5–4.5 °C","statement":"How much does global temperature eventually rise if atmospheric CO2 doubles?","verifier":"A narrower observed range","y":[1896,null,2021,false]},"collatz-conjecture":{"explanation":"A child can run it and nobody can prove it. Terence Tao showed in 2019 that almost all orbits reach almost bounded values, the strongest result so far. Conway showed that natural generalizations are undecidable, which hints that the specific numbers 3 and 1 matter. A counterexample would be a number a computer can check, which makes this one of the few famous problems where a finite search could still end it.","legacy":"wiki-collatz-conjecture","links":[["Wikipedia: Collatz conjecture","https://en.wikipedia.org/wiki/Collatz_conjecture"]],"posed":"1937, Lothar Collatz","statement":"Halving even numbers and sending odd n to 3n + 1 always reaches 1.","verifier":"Lean statement in formal-conjectures","y":[1937,null,null,false]},"collective-action-climate":{"explanation":"Elinor Ostrom showed that communities often manage local commons well without either privatization or central control, and won the 2009 Nobel for it. Global commons such as the climate are harder: the Montreal Protocol worked, and climate agreements have so far fallen short of their own targets.","links":[["Wikipedia: Tragedy of the commons","https://en.wikipedia.org/wiki/Tragedy_of_the_commons"]],"statement":"How can nations cooperate to protect shared resources when each gains by free-riding?","y":[1968,null,null,true]},"complete-human-genome":{"closed":"2003 draft, Human Genome Project; 2022 gapless, Telomere-to-Telomere consortium","explanation":"The 2003 reference left about 8% unsequenced, mostly repetitive regions. Long-read sequencing let the T2T consortium finish the first truly complete sequence in 2022, nearly twenty years after the first was declared done.","links":[["Wikipedia: Telomere-to-telomere","https://en.wikipedia.org/wiki/Telomere-to-telomere"]],"statement":"What is the full sequence of a human genome?","y":[1990,2022,2003,false]},"connectome":{"closed":"1986, C. elegans (302 neurons); 2024, the adult fruit fly brain (about 140,000 neurons), FlyWire","explanation":"The fly connectome was mapped with a mix of machine segmentation and crowd proofreading. A mouse brain is about 500 times larger. Whether wiring alone predicts behavior is itself an open question: C. elegans has been mapped for decades and is not yet fully understood.","links":[["Wikipedia: Connectome","https://en.wikipedia.org/wiki/Connectome"]],"statement":"Can the complete synaptic wiring of a brain be mapped, and does it explain behavior?","y":[1972,null,2024,true]},"continuum-hypothesis":{"closed":"1940 Kurt Gödel and 1963 Paul Cohen: independent of ZFC","explanation":"Gödel showed it cannot be disproved from the standard axioms, and Cohen, inventing forcing, showed it cannot be proved. The standard axioms leave it open. Whether new axioms should settle it, and which ones, is still argued. Woodin has changed his own proposed answer over the years.","links":[["Wikipedia: Continuum hypothesis","https://en.wikipedia.org/wiki/Continuum_hypothesis"]],"posed":"1878, Georg Cantor; Hilbert's first problem, 1900","statement":"There is no set whose size is strictly between that of the integers and that of the real numbers.","y":[1878,1940,null,false]},"cosmic-inflation":{"explanation":"Inflation explains why the universe is so flat and uniform, and its predictions for the spectrum of density fluctuations have been confirmed. Its signature in primordial gravitational waves, B-mode polarization, has not been seen. BICEP2 announced a detection in 2014 that turned out to be galactic dust.","links":[["Wikipedia: Inflation (cosmology)","https://en.wikipedia.org/wiki/Inflation_(cosmology)"]],"posed":"1980, Alan Guth","statement":"Did the universe undergo a brief period of exponential expansion before the hot Big Bang?","y":[1980,null,null,false]},"dark-energy":{"explanation":"The simplest answer is a cosmological constant, but its observed value is famously far smaller than quantum field theory naively predicts. In 2024 and 2025 DESI's baryon acoustic oscillation data hinted that dark energy may change over time. If confirmed, that would rule out a pure constant.","links":[["Wikipedia: Dark energy","https://en.wikipedia.org/wiki/Dark_energy"]],"posed":"1998, accelerating expansion found by two supernova teams","statement":"What drives the accelerating expansion of the universe?","y":[1998,null,null,false]},"dark-matter":{"explanation":"Its gravity shows up in galaxy rotation, gravitational lensing, the cosmic microwave background and the growth of structure, all consistent with the same amount. Direct detection experiments have ruled out much of the favored parameter space for WIMPs without a signal. Axions, sterile neutrinos and primordial black holes remain candidates, and a minority argue for modified gravity instead.","links":[["Wikipedia: Dark matter","https://en.wikipedia.org/wiki/Dark_matter"]],"posed":"1933, Fritz Zwicky; 1970s, Vera Rubin's rotation curves","statement":"What is the substance that makes up about 85% of the matter in the universe?","verifier":"A direct-detection signal confirmed by independent experiments","y":[1933,null,null,false]},"dinosaur-extinction":{"closed":"1980, Luis and Walter Alvarez propose an impact; 1991, Chicxulub crater identified","explanation":"A thin clay layer worldwide is rich in iridium, which is rare in Earth's crust and common in asteroids. The 180-kilometre crater was found under the Yucatán a decade later. The role of the Deccan Traps volcanism alongside the impact is still argued.","links":[["Wikipedia: Cretaceous–Paleogene extinction event","https://en.wikipedia.org/wiki/Cretaceous%E2%80%93Paleogene_extinction_event"]],"posed":"19th century","statement":"What caused the mass extinction at the end of the Cretaceous?","y":[1850,1980,null,false]},"early-supermassive-black-holes":{"explanation":"Growing from stellar remnants at the usual maximum rate takes too long. JWST has found more massive early black holes and galaxies than models predicted. Direct collapse of gas clouds and heavy seeds are leading candidates.","links":[["Wikipedia: Supermassive black hole","https://en.wikipedia.org/wiki/Supermassive_black_hole"]],"statement":"How did black holes of a billion solar masses form within a few hundred million years of the Big Bang?","y":[2001,null,null,true]},"earthquake-prediction":{"explanation":"Long-term hazard maps work well; short-term prediction has failed repeatedly. The 1985 Parkfield experiment expected an earthquake by 1993, and it came in 2004. Machine learning on laboratory quakes has found precursors that have not yet carried over to real faults.","links":[["Wikipedia: Earthquake prediction","https://en.wikipedia.org/wiki/Earthquake_prediction"]],"statement":"Can the time, place and size of a large earthquake be predicted in advance?","y":[1910,null,null,true]},"egyptian-hieroglyphs":{"closed":"1822, Jean-François Champollion, with the Rosetta Stone","explanation":"Scholars had assumed hieroglyphs were pure symbols. Champollion showed they mix sound signs with meaning signs, using Coptic, the descendant of Egyptian, as a key. Thomas Young had made progress on the names before him.","links":[["Wikipedia: Decipherment of ancient Egyptian scripts","https://en.wikipedia.org/wiki/Decipherment_of_ancient_Egyptian_scripts"]],"statement":"What do Egyptian hieroglyphs say, and how does the script work?","y":[1650,1822,null,true]},"einstein-tile":{"closed":"2023, David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss","explanation":"Penrose needed two tiles. In 2023 an amateur, David Smith, found the 'hat', a 13-sided polykite, and the team proved it tiles only aperiodically. Because the hat's tilings use mirror images, they followed with the 'spectre', which needs no reflections. The discovery was made by hand and exploration software, not by theory.","links":[["Wikipedia: Einstein problem","https://en.wikipedia.org/wiki/Einstein_problem"]],"statement":"There is a single shape that tiles the plane, but only non-periodically.","y":[1966,2023,null,true]},"eukaryote-origin":{"explanation":"Mitochondria are descended from engulfed bacteria, which is now settled after being ridiculed for years. Asgard archaea, discovered in 2015 and first cultured in 2019, appear to be the host lineage. How the nucleus and the other compartments arose is still open.","links":[["Wikipedia: Eukaryogenesis","https://en.wikipedia.org/wiki/Eukaryogenesis"]],"posed":"1967, Lynn Margulis's endosymbiosis theory","statement":"How did complex, nucleus-bearing cells evolve from simpler prokaryotes?","y":[1967,null,2019,false]},"exoplanets":{"closed":"1992, Wolszczan and Frail (pulsar planets); 1995, Mayor and Queloz (51 Pegasi b)","explanation":"More than 5,000 are now confirmed. The first around a Sun-like star was a hot Jupiter that orbits in four days, a kind of planet no theory had predicted. The 2019 Nobel Prize went to Mayor and Queloz.","links":[["Wikipedia: Exoplanet","https://en.wikipedia.org/wiki/Exoplanet"]],"statement":"Do other stars have planets?","y":[1584,1995,1992,true]},"expanding-universe":{"closed":"1929, Edwin Hubble, building on Lemaître (1927) and Slipher's redshifts","explanation":"Galaxies recede at speeds proportional to their distance. Einstein had added a cosmological constant to keep his universe static and later dropped it. It returned in 1998 as dark energy.","links":[["Wikipedia: Expansion of the universe","https://en.wikipedia.org/wiki/Expansion_of_the_universe"]],"statement":"Is the universe static, as Einstein assumed, or changing in size?","y":[1917,1929,null,true]},"fault-tolerant-quantum":{"closed":"2024, Google's Willow chip shows error rates falling as code size grows","explanation":"The threshold theorem says error correction works if physical error rates are low enough. In 2024 a surface code crossed that threshold, with logical errors halving at each step up in code size. A machine large enough to run Shor's algorithm on real key sizes needs roughly a million physical qubits or much better codes.","links":[["Wikipedia: Quantum error correction","https://en.wikipedia.org/wiki/Quantum_error_correction"]],"statement":"Can a quantum computer correct its own errors well enough to run long algorithms?","y":[1995,null,2024,false]},"fermats-last-theorem":{"closed":"1995, Andrew Wiles, with Richard Taylor","explanation":"For 358 years it produced whole fields of mathematics as by-products, from Kummer's ideal numbers onward. The end came indirectly. Frey, Serre and Ribet showed that a counterexample would give an elliptic curve that could not be modular. Wiles then proved enough of the modularity conjecture to rule that curve out. The theorem itself was a corollary of a much larger structure.","links":[["Wikipedia: Fermat's Last Theorem","https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem"]],"posed":"1637, Pierre de Fermat, in a margin","statement":"No positive integers $a, b, c$ satisfy $a^n + b^n = c^n$ for any $n > 2$.","y":[1637,1995,null,false]},"fermi-paradox":{"explanation":"The galaxy is old enough for a spacefaring civilization to have spread across it many times. Proposed answers range from life being rare, to civilizations being short-lived, to our searches being too narrow. Biosignature searches in exoplanet atmospheres are the first measurements that could constrain the answer.","links":[["Wikipedia: Fermi paradox","https://en.wikipedia.org/wiki/Fermi_paradox"]],"posed":"1950, Enrico Fermi: \"Where is everybody?\"","statement":"If intelligent life is common, why do we see no sign of it?","y":[1950,null,null,false]},"four-color-theorem":{"closed":"1976, Kenneth Appel and Wolfgang Haken; formalized in Coq 2005","explanation":"It was the first major theorem proved with essential help from a computer, which checked 1,936 configurations. Many mathematicians refused to accept a proof no human could check. Robertson, Sanders, Seymour and Thomas simplified it in 1996, and Georges Gonthier checked it completely in Coq in 2005, which ended the argument.","links":[["Wikipedia: Four color theorem","https://en.wikipedia.org/wiki/Four_color_theorem"]],"posed":"1852, Francis Guthrie","statement":"Every planar map can be colored with four colors so that neighboring regions differ.","verifier":"Formal proof in Coq (Gonthier, 2005)","y":[1852,1976,null,false]},"free-will":{"explanation":"Compatibilists hold that freedom means acting on one's own reasons without coercion, which determinism does not threaten. Libertarians and hard determinists deny this. Libet's 1983 experiments on readiness potentials were taken to show decisions are made before awareness; later work suggests the signal may be noise that builds toward a threshold rather than a decision.","links":[["Wikipedia: Free will","https://en.wikipedia.org/wiki/Free_will"]],"statement":"Are our choices free, and is freedom compatible with a universe governed by laws?","y":[-300,null,null,true]},"function-of-sleep":{"explanation":"Every animal studied closely sleeps, including fruit flies and jellyfish. Proposed functions include memory consolidation, synaptic downscaling and clearing metabolic waste, and none is established as the core reason.","links":[["Wikipedia: Sleep","https://en.wikipedia.org/wiki/Sleep"]],"statement":"Why do animals sleep, given the risks of being unconscious?","y":[1953,null,null,true]},"fusion-power":{"closed":"December 2022, National Ignition Facility: target gain above 1","explanation":"NIF's laser shot released 3.15 MJ from 2.05 MJ of laser energy delivered to the target, the first scientific ignition, though the lasers drew far more from the wall. Tokamaks and stellarators, public and private, aim for net electricity in the 2030s. The remaining problems are engineering ones: materials that survive neutron bombardment, tritium breeding, and running continuously.","links":[["Wikipedia: Fusion ignition","https://en.wikipedia.org/wiki/Fusion_ignition"],["Wikipedia: Fusion power","https://en.wikipedia.org/wiki/Fusion_power"]],"statement":"Can controlled fusion deliver net electricity to the grid at competitive cost?","verifier":"Net electricity to the grid, metered","y":[1951,null,2022,true]},"general-purpose-robotics":{"explanation":"Moravec's paradox: what is easy for people, such as folding laundry, is hard for robots, and what is hard for people is easy for them. Learned policies trained on large demonstration datasets have made fast progress since 2023. Reliability over thousands of different tasks in cluttered spaces is the open part.","links":[["Wikipedia: Moravec's paradox","https://en.wikipedia.org/wiki/Moravec%27s_paradox"]],"statement":"Can a robot do most physical household or factory tasks as reliably as a person, in unstructured settings?","y":[1966,null,null,true]},"generalization-deep-learning":{"explanation":"Classical learning theory predicts that models with more parameters than data should overfit. Networks fit even random labels and still generalize on real data, as Zhang and colleagues showed in 2017. Implicit regularization, double descent and scaling laws describe the behavior without fully explaining it.","links":[["Wikipedia: Double descent","https://en.wikipedia.org/wiki/Double_descent"]],"statement":"Why do overparameterized neural networks that can memorize their data still generalize to new data?","y":[2016,null,null,false]},"genetic-code":{"closed":"1961–1966, Marshall Nirenberg, Heinrich Matthaei, Har Gobind Khorana and others","explanation":"Nirenberg and Matthaei's poly-U experiment showed UUU codes for phenylalanine, and all 64 codons were assigned within five years. The code is nearly universal across life, which is strong evidence of common descent.","links":[["Wikipedia: Genetic code","https://en.wikipedia.org/wiki/Genetic_code"]],"statement":"How do sequences of DNA bases specify proteins?","y":[1954,1966,1961,false]},"genotype-phenotype":{"explanation":"Most traits depend on thousands of variants of tiny effect interacting with environment. Height, one of the best-studied traits, is now largely explained by common variants, but for most traits prediction is weak. The missing link is development: how a genome builds a body.","links":[["Wikipedia: Genotype–phenotype distinction","https://en.wikipedia.org/wiki/Genotype%E2%80%93phenotype_distinction"]],"statement":"Can an organism's traits be predicted from its genome?","y":[1909,null,null,true]},"geodynamo":{"explanation":"Convection in the liquid iron outer core generates the field, and 1995 simulations by Glatzmaier and Roberts produced spontaneous reversals. Why reversals happen irregularly, and whether one could be predicted, is open. So is how the field was sustained before the solid inner core formed.","links":[["Wikipedia: Dynamo theory","https://en.wikipedia.org/wiki/Dynamo_theory"]],"statement":"How does Earth's core sustain its magnetic field, and what triggers reversals?","y":[1919,null,1995,true]},"geometric-langlands":{"closed":"2024, Dennis Gaitsgory, Sam Raskin and collaborators","explanation":"The proof runs across five papers and roughly 800 pages. It settles the geometric analogue of the Langlands correspondence, in which functions on a curve replace numbers. It does not settle the arithmetic Langlands conjectures, but it gives a precise target and a set of tools for them.","links":[["Wikipedia: Geometric Langlands correspondence","https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence"]],"posed":"1980s, Drinfeld and Laumon","statement":"The categorical geometric Langlands equivalence holds for curves over the complex numbers.","y":[1980,2024,null,false]},"germ-theory":{"closed":"1860s–1880s, Louis Pasteur and Robert Koch","explanation":"Semmelweis showed in 1847 that handwashing cut deaths in childbirth, and he was ignored. Pasteur and Koch established that specific microbes cause specific diseases, and Koch's postulates set the standard of proof. Sanitation, antisepsis and vaccines followed.","links":[["Wikipedia: Germ theory of disease","https://en.wikipedia.org/wiki/Germ_theory_of_disease"]],"statement":"Are diseases caused by bad air, imbalance of humors, or living microorganisms?","y":[1546,1876,null,true]},"gettier-problem":{"explanation":"Gettier gave cases where someone has a justified true belief by luck, and plainly does not know. Sixty years of proposed fourth conditions have each met counterexamples. It is one of the most-cited short papers in philosophy.","links":[["Wikipedia: Gettier problem","https://en.wikipedia.org/wiki/Gettier_problem"]],"posed":"1963, Edmund Gettier, in three pages","statement":"Is knowledge justified true belief, and if not, what is missing?","y":[1963,null,null,false]},"glass-transition":{"explanation":"As a liquid cools into a glass its viscosity rises by many orders of magnitude with little change in structure. Philip Anderson called it the deepest unsolved problem in solid state theory in 1995. Simulations have not settled whether an underlying ideal glass transition exists.","links":[["Wikipedia: Glass transition","https://en.wikipedia.org/wiki/Glass_transition"]],"statement":"Is the glass transition a true thermodynamic phase transition or only a dynamic slowing down?","y":[1948,null,null,true]},"goldbach-conjecture":{"explanation":"It has been checked by computer up to $4 \\times 10^{18}$. The weak version, about odd numbers and three primes, was proved in 2013. The binary version resists because two primes give too little room for the averaging arguments that work with three. Chen Jingrun came closest in 1973: every large even number is a prime plus a number with at most two prime factors.","legacy":"wiki-goldbach-conjecture","links":[["Wikipedia: Goldbach's conjecture","https://en.wikipedia.org/wiki/Goldbach%27s_conjecture"]],"posed":"1742, Christian Goldbach in a letter to Euler","statement":"Every even integer greater than 2 is the sum of two primes.","verifier":"Lean statement in formal-conjectures","y":[1742,null,null,false]},"graph-isomorphism":{"closed":"2015, László Babai: quasipolynomial time","explanation":"It is one of very few natural problems that is in NP but believed to be neither in P nor NP-complete. Babai's quasipolynomial algorithm was the biggest advance in decades. A bug found in 2017 was fixed within days. Whether polynomial time is possible is open.","links":[["Wikipedia: Graph isomorphism problem","https://en.wikipedia.org/wiki/Graph_isomorphism_problem"]],"statement":"Can graph isomorphism be decided in polynomial time?","y":[1972,null,2015,true]},"gravitational-waves":{"closed":"2015 detection, announced 2016, LIGO and Virgo","explanation":"LIGO measured a distortion one thousandth the width of a proton over its 4-kilometre arms, from two black holes merging 1.3 billion light years away. Einstein himself doubted the waves could ever be detected. The 2017 Nobel Prize went to Weiss, Barish and Thorne, and gravitational waves now serve as an astronomical instrument in their own right.","links":[["Wikipedia: First observation of gravitational waves","https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves"]],"posed":"1916, Albert Einstein","statement":"Do ripples in spacetime exist, and can they be measured?","y":[1916,2015,null,false]},"great-divergence":{"explanation":"Candidate explanations include institutions, geography, culture, coal and colonial extraction. Acemoglu, Johnson and Robinson won the 2024 economics Nobel for evidence on institutions from natural experiments. East Asia's catch-up shows convergence is possible, and nobody knows a recipe that reliably reproduces it.","links":[["Wikipedia: Great Divergence","https://en.wikipedia.org/wiki/Great_Divergence"]],"statement":"Why did sustained economic growth begin in northwestern Europe, and why do some countries still not converge?","y":[1776,null,null,true]},"hadwiger-nelson":{"explanation":"The answer is 5, 6 or 7. The lower bound of 5 came in 2018 from Aubrey de Grey, an amateur, who found a unit-distance graph needing five colors. A Polymath project then shrank his graph from 1,581 vertices to about 500. The answer may depend on the axioms of set theory if the coloring is not required to be measurable.","links":[["Wikipedia: Hadwiger–Nelson problem","https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem"]],"posed":"1950, Edward Nelson","statement":"How many colors are needed to color the plane so that no two points at distance 1 share a color?","y":[1950,null,2018,false]},"hard-problem":{"explanation":"Neuroscience can explain which processes correlate with awareness without explaining why they feel like anything. Some philosophers deny the problem exists. There is no verifier here, and the atlas says so: this is a space where explanation is the entire product.","links":[["Wikipedia: Hard problem of consciousness","https://en.wikipedia.org/wiki/Hard_problem_of_consciousness"]],"posed":"1995, David Chalmers (the name); the question is ancient","statement":"Why is there subjective experience at all, rather than information processing in the dark?","y":[1995,null,null,false]},"herculaneum-scrolls":{"closed":"2023–2024, Vesuvius Challenge: first passages read from inside a sealed scroll","explanation":"X-ray scans plus machine-learning ink detection revealed the word 'purple' in 2023, and the 2023 Grand Prize read about 2,000 characters of an unknown philosophical text, probably by Philodemus. It was a public prize with a clear verifier, and hundreds of scrolls remain.","links":[["Vesuvius Challenge","https://scrollprize.org/"],["Wikipedia: Herculaneum papyri","https://en.wikipedia.org/wiki/Herculaneum_papyri"]],"prize":"Vesuvius Challenge prizes","statement":"Can the carbonized scrolls buried by Vesuvius in 79 CE be read without unrolling them?","verifier":"Legible text confirmed by papyrologists","y":[1752,null,2023,false]},"hierarchy-problem":{"explanation":"Quantum corrections should pull the Higgs mass up toward the highest energy scale in the theory unless enormous cancellations occur. Supersymmetry was the favored solution, and the LHC has not found it at the energies where it was most expected. Physicists now take seriously that the answer may be environmental or may need a new principle.","links":[["Wikipedia: Hierarchy problem","https://en.wikipedia.org/wiki/Hierarchy_problem"]],"statement":"Why is the Higgs mass, and so the weak scale, so much smaller than the Planck scale?","y":[1976,null,null,true]},"higgs-boson":{"closed":"2012, ATLAS and CMS at the Large Hadron Collider","explanation":"Found 48 years after it was predicted, at 125 GeV, by two independent detectors. Its measured mass leaves the vacuum near the edge of stability, which is itself a puzzle. The discovery completed the Standard Model's particle list; everything beyond it is now open territory.","links":[["Wikipedia: Higgs boson","https://en.wikipedia.org/wiki/Higgs_boson"]],"posed":"1964, Englert and Brout; Higgs; Guralnik, Hagen and Kibble","statement":"Does the field that gives elementary particles their mass exist?","y":[1964,2012,null,false]},"high-tc-mechanism":{"explanation":"BCS theory explains conventional superconductors through vibrations of the crystal lattice, and it does not account for the cuprates, which superconduct far above what it would allow. Nearly four decades on there is no consensus mechanism. Understanding it is widely seen as the route to room-temperature superconductors.","links":[["Wikipedia: High-temperature superconductivity","https://en.wikipedia.org/wiki/High-temperature_superconductivity"]],"posed":"1986, Bednorz and Müller discover cuprate superconductors","statement":"What pairs the electrons in cuprate and other unconventional superconductors?","y":[1986,null,null,false]},"hilbert-second":{"closed":"1931, Kurt Gödel's second incompleteness theorem","explanation":"Gödel showed that no consistent system strong enough for arithmetic can prove its own consistency, so Hilbert's program as stated cannot succeed. Gentzen proved consistency in 1936 using induction up to the ordinal ε₀, which trades one assumption for another. The problem is marked refuted because the kind of proof Hilbert asked for is impossible.","links":[["Wikipedia: Hilbert's second problem","https://en.wikipedia.org/wiki/Hilbert%27s_second_problem"]],"posed":"1900, David Hilbert","statement":"Prove, by finitary means, that the axioms of arithmetic are consistent.","y":[1900,1931,null,false]},"hilbert-tenth":{"closed":"1970, Yuri Matiyasevich, completing Martin Davis, Hilary Putnam and Julia Robinson","explanation":"No such algorithm exists: every computably enumerable set is Diophantine, so a decision procedure would solve the halting problem. The version over the rationals is still open.","links":[["Wikipedia: Hilbert's tenth problem","https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem"]],"posed":"1900, David Hilbert","statement":"Find an algorithm that decides whether any Diophantine equation has an integer solution.","y":[1900,1970,null,false]},"hiv-cure":{"explanation":"Antiretroviral therapy turned HIV into a manageable condition, and a few people have been cured through stem cell transplants from donors with the CCR5-delta32 mutation, starting with Timothy Ray Brown in 2008. Transplants are far too dangerous to scale. The latent reservoir of infected cells is the obstacle to a general cure.","links":[["Wikipedia: HIV/AIDS research","https://en.wikipedia.org/wiki/HIV/AIDS_research"]],"statement":"Can HIV infection be cured, not just controlled?","y":[1983,null,2008,true]},"hodge-conjecture":{"explanation":"It asks when topology on a complex shape comes from actual subvarieties cut out by polynomials. It is known for divisors, the Lefschetz (1,1) theorem, and fails for integer coefficients, as shown by Atiyah and Hirzebruch. It is the least elementary of the Millennium problems to state, and even the right formalization is work in progress.","links":[["Wikipedia: Hodge conjecture","https://en.wikipedia.org/wiki/Hodge_conjecture"]],"posed":"1950, W. V. D. Hodge","prize":"$1,000,000 Clay Millennium Prize","statement":"On a projective complex manifold, every rational Hodge class is a rational combination of classes of algebraic cycles.","y":[1950,null,null,false]},"hubble-tension":{"explanation":"The cosmic microwave background implies about 67 km/s/Mpc, while the Cepheid–supernova distance ladder gives about 73, a gap well beyond quoted errors. Either one method has an unrecognized systematic error or the standard cosmological model is missing something. JWST measurements have not yet settled which.","links":[["Wikipedia: Hubble's law","https://en.wikipedia.org/wiki/Hubble%27s_law"]],"statement":"Why do early-universe and late-universe measurements of the expansion rate disagree?","verifier":"Independent measurements that converge, or a model that fits both","y":[2016,null,null,true]},"indus-script":{"explanation":"There are about 4,000 inscriptions, most only a few signs long, and no bilingual text. Whether the signs even encode language is disputed.","links":[["Wikipedia: Indus script","https://en.wikipedia.org/wiki/Indus_script"]],"statement":"Do the symbols of the Indus Valley Civilization write a language, and if so which?","y":[1924,null,null,true]},"interpretability":{"explanation":"Features and circuits have been found inside language models, including with sparse dictionary methods since 2023. Explaining a whole frontier model's behavior in these terms is far off. Benchmarks for what counts as an explanation are still being built.","links":[["Wikipedia: Mechanistic interpretability","https://en.wikipedia.org/wiki/Mechanistic_interpretability"]],"statement":"Can the internal computations of a trained neural network be fully understood?","y":[2013,null,2023,true]},"invariant-subspace":{"explanation":"Enflo and Read built counterexamples on other Banach spaces, so the Hilbert space case is what remains. Positive results cover compact operators and many other classes. Claimed proofs have circulated since 2023 without being accepted, and it is listed open here.","legacy":"wiki-invariant-subspace-problem","links":[["Wikipedia: Invariant subspace problem","https://en.wikipedia.org/wiki/Invariant_subspace_problem"]],"statement":"Does every bounded operator on a separable infinite-dimensional Hilbert space have a nontrivial closed invariant subspace?","verifier":"Lean statement in formal-conjectures","y":[1935,null,null,true]},"inverse-galois":{"explanation":"Known for solvable groups (Shafarevich) and for all but one of the sporadic groups; the Mathieu group M23 is the famous holdout. Most methods use rigidity, building extensions from the geometry of covering spaces.","legacy":"wiki-inverse-galois","links":[["Wikipedia: Inverse Galois problem","https://en.wikipedia.org/wiki/Inverse_Galois_problem"]],"posed":"19th century; systematic work from Hilbert and Emmy Noether","statement":"Is every finite group the Galois group of some extension of the rational numbers?","verifier":"Lean statement in formal-conjectures","y":[1850,null,null,false]},"is-ought":{"explanation":"Hume observed that arguments slide from 'is' to 'ought' without justifying the step. Naturalists try to ground ethics in facts about well-being; others hold that values are a separate category. Aligning AI systems has made the question practical, because it requires writing down what systems ought to do.","links":[["Wikipedia: Is–ought problem","https://en.wikipedia.org/wiki/Is%E2%80%93ought_problem"]],"posed":"1739, David Hume","statement":"Can conclusions about what ought to be done be derived from facts about what is?","y":[1739,null,null,false]},"island-of-stability":{"explanation":"Nuclear shell theory predicts a region of relatively long-lived superheavy nuclei. The heaviest elements made so far, up to oganesson with 118 protons, lack enough neutrons to reach it. Nobody yet has a way to make the neutron-rich isotopes the island requires.","links":[["Wikipedia: Island of stability","https://en.wikipedia.org/wiki/Island_of_stability"]],"statement":"Do superheavy elements near 114 protons and 184 neutrons live long enough to study?","y":[1966,null,null,true]},"jacobian-conjecture":{"closed":"July 2026, Levent Alpöge, crediting the discovery to Claude Fable 5","explanation":"The conjecture says a polynomial map that is invertible to first order everywhere must be invertible globally. It falls in every dimension n ≥ 3. Alpöge posted an explicit degree-7 map of three variables with Jacobian determinant −2 that sends three different points to one. Because the colliding points are real, the same map also answers the real question with constant Jacobian. The plane case, which Ludwig Kraus stated in 1884 and which is older than Keller's question, is untouched. A companion note and interactive atlas are linked below.","legacy":"wiki-jacobian-conjecture","links":[["Alpöge's announcement (Jul 19, 2026)","https://x.com/__alpoge__/status/2079028340955197566"],["Terence Tao, A digestion of the Jacobian conjecture counterexample","https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/"],["David Speyer, The new counterexample to the Jacobian conjecture","https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/"],["Danielle Fong and Fable, The State of the Jacobian Conjectures","https://jacobianconjectures.com/note/"],["jacobianconjectures.com, an atlas of the day after","https://jacobianconjectures.com/"],["Wikipedia: Jacobian conjecture","https://en.wikipedia.org/wiki/Jacobian_conjecture"]],"posed":"1939, Ott-Heinrich Keller","statement":"A polynomial map from $\\mathbb{C}^n$ to $\\mathbb{C}^n$ with constant nonzero Jacobian determinant has a polynomial inverse.","verifier":"The counterexample is checkable by hand or by computer algebra","y":[1939,2026,null,false]},"kadison-singer":{"closed":"2013, Adam Marcus, Daniel Spielman and Nikhil Srivastava","explanation":"It started as a question about the foundations of quantum mechanics and was shown equivalent to problems in a dozen fields. The solution came from graph theory, through interlacing polynomials, the same method the authors used to build Ramanujan graphs. A question in operator algebras was settled by a combinatorial method.","links":[["Wikipedia: Kadison–Singer problem","https://en.wikipedia.org/wiki/Kadison%E2%80%93Singer_problem"]],"posed":"1959, Richard Kadison and Isadore Singer","statement":"Does every pure state on the diagonal of B(ℓ²) extend uniquely to a pure state on all of B(ℓ²)?","y":[1959,2013,null,false]},"kahn-kalai":{"closed":"2022, Jinyoung Park and Huy Tuan Pham","explanation":"It says a random structure appears roughly as soon as it is expected to appear. Park and Pham's proof was about six pages. Its shortness surprised the field and came shortly after the related Talagrand conjectures were settled.","links":[["Quanta Magazine: Elegant six-page proof","https://www.quantamagazine.org/elegant-six-page-proof-reveals-the-emergence-of-random-structure-20220425/"]],"posed":"2006, Jeff Kahn and Gil Kalai","statement":"The threshold of any monotone graph property is within a log factor of its expectation threshold.","y":[2006,2022,null,false]},"kakeya-3d":{"closed":"2025, Hong Wang and Joshua Zahl","explanation":"A Kakeya set can have zero volume yet still hold a needle in every direction. The conjecture says such sets cannot be small in dimension. The two-dimensional case was done in 1971. Wang and Zahl's 2025 proof of the three-dimensional case is widely considered one of the decade's biggest results in analysis. The case of dimension four and up remains open.","links":[["Wikipedia: Kakeya set","https://en.wikipedia.org/wiki/Kakeya_set"]],"statement":"Every set in $\\mathbb{R}^3$ containing a unit segment in every direction has Hausdorff dimension 3.","y":[1917,2025,null,true]},"kakeya-general":{"explanation":"Settled for n = 2 (Davies, 1971) and n = 3 (Wang and Zahl, 2025), and over finite fields (Dvir, 2008, with a startlingly short polynomial-method proof). The general Euclidean case is open. It sits under many problems in harmonic analysis, including restriction and Bochner–Riesz.","legacy":"wiki-kakeya","links":[["Wikipedia: Kakeya set","https://en.wikipedia.org/wiki/Kakeya_set"]],"statement":"Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $n$.","verifier":"Lean statement in formal-conjectures","y":[1917,null,2025,true]},"keller-plane":{"explanation":"Ludwig Kraus stated the two-variable case in 1884, with a proof that fails at its last step, and Keller posed the question in all dimensions in 1939. The two-variable case survived the 2026 counterexamples, which all need three variables. Any counterexample in the plane is known to have degree above 100, and plane polynomial automorphisms have a rich structure, described by the Jung–van der Kulk theorem, that has no analogue in dimension 3. After July 2026 the plane case is the whole of what is left of the Jacobian conjecture.","links":[["Wikipedia: Jacobian conjecture","https://en.wikipedia.org/wiki/Jacobian_conjecture"],["The State of the Jacobian Conjectures","https://jacobianconjectures.com/note/"]],"posed":"1939, Ott-Heinrich Keller","statement":"A polynomial map of $\\mathbb{C}^2$ with constant nonzero Jacobian determinant is invertible.","verifier":"Lean statement jacobian_conjecture_two_variables in formal-conjectures","y":[1939,null,null,false]},"kepler-conjecture":{"closed":"1998, Thomas Hales with Samuel Ferguson; formally verified 2014","explanation":"Hales reduced the problem to thousands of inequalities checked by computer. Referees said they were 99% certain it was right and could not be fully certain. Hales answered by leading the Flyspeck project, which completed a formal proof in HOL Light and Isabelle in 2014. It became the standard example of formal verification settling a proof that humans could not referee.","links":[["Wikipedia: Kepler conjecture","https://en.wikipedia.org/wiki/Kepler_conjecture"]],"posed":"1611, Johannes Kepler","statement":"No packing of equal spheres in space is denser than the cannonball packing, π/√18 ≈ 74%.","verifier":"Formal proof: Flyspeck (HOL Light, Isabelle)","y":[1611,1998,null,false]},"langlands-reciprocity":{"explanation":"The general correspondence is known for GL(1) (class field theory) and in many cases for GL(2), plus pieces beyond. Each new case has taken years and has usually settled famous problems along the way. The geometric version was proved in 2024, and the arithmetic version remains the central open problem of the field.","links":[["Wikipedia: Langlands program","https://en.wikipedia.org/wiki/Langlands_program"]],"posed":"1967, Robert Langlands","statement":"Galois representations and automorphic representations correspond, compatibly with L-functions.","y":[1967,null,null,false]},"language-acquisition":{"explanation":"Chomsky argued from the 'poverty of the stimulus' for an innate universal grammar. Large language models learn grammar from text alone, though they need far more data than any child hears. That reopened the debate without settling it.","links":[["Wikipedia: Language acquisition","https://en.wikipedia.org/wiki/Language_acquisition"]],"statement":"How do children learn language from limited input, and how much of the ability is innate?","y":[1959,null,null,true]},"liar-paradox":{"explanation":"Tarski showed in 1933 that a sufficiently strong language cannot define its own truth predicate, which blocks the paradox by moving truth to a metalanguage. Kripke's 1975 theory of truth allows gaps instead. Gödel used a relative of the paradox to prove incompleteness. Formal fixes exist; whether any captures ordinary truth is debated.","links":[["Wikipedia: Liar paradox","https://en.wikipedia.org/wiki/Liar_paradox"]],"posed":"4th century BCE, attributed to Eubulides","statement":"Is the sentence 'This sentence is false' true or false?","y":[-350,null,1933,true]},"linear-a":{"explanation":"The sound values can be partly read from Linear B, which descends from it, but the language itself is unknown and seems unrelated to any known family. The corpus is small, about 1,400 inscriptions, mostly short.","links":[["Wikipedia: Linear A","https://en.wikipedia.org/wiki/Linear_A"]],"statement":"What language does the Minoan Linear A script write?","y":[1900,null,null,false]},"linear-b":{"closed":"1952, Michael Ventris, with John Chadwick; building on Alice Kober's analysis","explanation":"Kober's painstaking grids of inflected word forms made the decipherment possible; she died in 1950. Ventris, an architect, expected Etruscan and found Greek, older than Homer by centuries. The reading predicted words on newly found tablets, which is how decipherments are verified.","links":[["Wikipedia: Linear B","https://en.wikipedia.org/wiki/Linear_B"]],"statement":"What language does the Bronze Age Linear B script of Crete and Greece write?","y":[1900,1952,null,false]},"long-duration-storage":{"explanation":"Lithium-ion batteries have fallen in cost by about 90% since 2010 and dominate storage up to about four hours. Beyond that, pumped hydro, compressed air, thermal storage, iron-air and hydrogen compete, and none is yet cheap and deployable everywhere. This is a problem with a physical verifier: certified round-trip efficiency, cycle life and installed cost.","links":[["Wikipedia: Grid energy storage","https://en.wikipedia.org/wiki/Grid_energy_storage"]],"statement":"Can electricity be stored for days to seasons at a cost that makes a mostly renewable grid reliable?","verifier":"Certified round-trip efficiency and cost per kWh on a test stand","y":[2010,null,null,true]},"longitude":{"closed":"1761–1773, John Harrison's marine chronometer H4","explanation":"Parliament set a prize for a method accurate to half a degree, after naval disasters caused by navigation errors. Harrison, a self-taught clockmaker, built a watch that kept time through storms and temperature swings. The Board of Longitude, favoring the astronomical method of lunar distances, kept changing the test, and Harrison was paid in full only after appealing to King George III and Parliament. It is the founding example of a prize for an open problem, and of what goes wrong when the verifier is not fixed in advance.","links":[["Wikipedia: Longitude prize","https://en.wikipedia.org/wiki/Longitude_prize"],["Wikipedia: John Harrison","https://en.wikipedia.org/wiki/John_Harrison"]],"posed":"1714, the Longitude Act offers £20,000","prize":"£20,000 under the Longitude Act, paid in part","statement":"How can a ship at sea determine its longitude reliably?","y":[1714,1761,null,false]},"machine-consciousness":{"explanation":"Once a question for thought experiments, it now bears on how models are built and treated. Indicator-based assessments derived from neuroscience theories have been proposed, and every such theory is contested.","links":[["Wikipedia: Artificial consciousness","https://en.wikipedia.org/wiki/Artificial_consciousness"]],"statement":"Could a trained model have experiences, and how would anyone tell?","y":[1950,null,null,true]},"malaria-vaccine":{"closed":"2021 RTS,S and 2023 R21 recommended by WHO","explanation":"Parasites are far harder to vaccinate against than viruses. The first two recommended vaccines cut clinical cases substantially in young children, with R21 showing about 75% efficacy in trials. Durable, high protection for everyone remains the goal.","links":[["Wikipedia: Malaria vaccine","https://en.wikipedia.org/wiki/Malaria_vaccine"]],"statement":"Can a vaccine prevent malaria at high efficacy?","y":[1967,null,2021,true]},"map-of-knowledge":{"explanation":"Every map of knowledge chooses what its geometry means and gives up the rest. Garfield's 1964 citation historiographs put time on one axis; later maps of science place fields by citation similarity and lose chronology. Even one clean criterion is hard: minimizing the edge crossings of a drawing is NP-hard, shown by Garey and Johnson in 1983. This atlas has two drawings with different trade-offs: the Map keeps containment and loses time, and the Chart keeps time and field and loses similarity. A layout that keeps all three at every scale is an open problem.","links":[["Wikipedia: Graph drawing","https://en.wikipedia.org/wiki/Graph_drawing"],["Wikipedia: Crossing number (graph theory)","https://en.wikipedia.org/wiki/Crossing_number_(graph_theory)"],["Wikipedia: Citation analysis","https://en.wikipedia.org/wiki/Citation_analysis"]],"statement":"Is there a layout of a knowledge graph that keeps time, dependency and similarity faithful at every zoom level, and can it be computed efficiently?","y":[1964,null,null,true]},"matrix-multiplication-omega":{"explanation":"Strassen broke the $n^3$ barrier, and the bound has crept down since, to about 2.371 by 2024. Each step now wins in the third decimal place. AlphaTensor in 2022 found new small-case algorithms by search, showing that machine search can contribute. A faster algorithm is a checkable object; a lower bound would need a proof.","links":[["Wikipedia: Matrix multiplication algorithm","https://en.wikipedia.org/wiki/Matrix_multiplication_algorithm"]],"posed":"1969, after Volker Strassen's $O(n^{2.81})$ algorithm","statement":"Is $\\omega = 2$, that is, can two $n \\times n$ matrices be multiplied in $n^{2+\\varepsilon}$ operations for every $\\varepsilon > 0$?","y":[1969,null,null,false]},"measurement-problem":{"explanation":"Copenhagen, many-worlds, pilot-wave and collapse theories all reproduce the predictions and disagree about what exists. Collapse models predict tiny deviations that experiments are starting to test. Decoherence explains why interference disappears but not why one outcome occurs.","links":[["Wikipedia: Measurement problem","https://en.wikipedia.org/wiki/Measurement_problem"]],"statement":"How does a definite measurement outcome arise from a wavefunction that evolves smoothly into a superposition?","y":[1932,null,null,true]},"memory-storage":{"explanation":"Synaptic plasticity is the accepted mechanism, and since 2012 'engram' cells have been tagged and reactivated in mice to trigger specific memories. How memories persist for decades while synaptic proteins turn over in days is not understood.","links":[["Wikipedia: Engram (neuropsychology)","https://en.wikipedia.org/wiki/Engram_(neuropsychology)"]],"statement":"Where and how is a specific memory physically stored in the brain?","y":[1904,null,2012,true]},"mercury-perihelion":{"closed":"1915, Albert Einstein, general relativity","explanation":"Le Verrier, who had found Neptune from anomalies in Uranus's orbit, proposed a hidden planet, Vulcan, which was never found. Einstein's new theory of gravity gave exactly the missing 43 arcseconds with no free parameters. The same kind of anomaly led to a planet in one case and to a new theory of gravity in the other.","links":[["Wikipedia: Tests of general relativity","https://en.wikipedia.org/wiki/Tests_of_general_relativity"]],"posed":"1859, Urbain Le Verrier","statement":"Why does Mercury's orbit precess 43 arcseconds per century more than Newtonian gravity predicts?","y":[1859,1915,null,false]},"metallic-hydrogen":{"explanation":"Predicted to exist at pressures above roughly 400 GPa, and probably present inside Jupiter. A 2017 claim of observation was not reproduced. Experiments at the limits of diamond anvils continue, and the reading of their optical and electrical data is disputed.","links":[["Wikipedia: Metallic hydrogen","https://en.wikipedia.org/wiki/Metallic_hydrogen"]],"posed":"1935, Eugene Wigner and Hillard Bell Huntington","statement":"Does solid hydrogen become a metal at high pressure, and at what pressure?","y":[1935,null,null,false]},"mlc-conjecture":{"explanation":"If true, the Mandelbrot set's whole structure would be described by a simple combinatorial model, pinched disks, and hyperbolicity would be dense among quadratic maps. Yoccoz proved local connectivity at many points, and Jeremy Kahn and Mikhail Lyubich extended it. The full conjecture is open.","legacy":"wiki-mandelbrot","links":[["Wikipedia: Mandelbrot set","https://en.wikipedia.org/wiki/Mandelbrot_set"]],"posed":"1980s, Adrien Douady and John H. Hubbard","statement":"The Mandelbrot set is locally connected.","verifier":"Lean statement in formal-conjectures","y":[1980,null,null,false]},"modularity-theorem":{"closed":"2001, Breuil, Conrad, Diamond and Taylor, completing Wiles","explanation":"An elliptic curve and a modular form are very different objects, and the theorem says they carry the same data. Wiles and Taylor proved the semistable case in 1995, which was enough for Fermat. The full statement followed six years later. It is the template for every later modularity result.","legacy":"wiki-modularity-conjecture","links":[["Wikipedia: Modularity theorem","https://en.wikipedia.org/wiki/Modularity_theorem"]],"posed":"1950s–60s, Taniyama, Shimura and Weil","statement":"Every elliptic curve over the rationals is modular.","y":[1950,2001,null,false]},"molyneux-problem":{"closed":"2011, Held and colleagues test newly sighted patients","explanation":"For three centuries it was a thought experiment about innate ideas. Cataract surgery in India under Project Prakash let it be tested: patients could not immediately match seen shapes to touched ones, but learned within days. A philosophical question became an experiment, though what the result means is still argued.","links":[["Wikipedia: Molyneux's problem","https://en.wikipedia.org/wiki/Molyneux%27s_problem"]],"posed":"1688, William Molyneux, in a letter to John Locke","statement":"Could a person born blind, who learns shapes by touch, recognize them by sight alone on gaining vision?","y":[1688,null,2011,false]},"moving-sofa":{"explanation":"Gerver's 1992 sofa, bounded by 18 curve sections, has area about 2.2195. In November 2024 Jineon Baek posted a proof that Gerver's sofa is optimal. Marked partial here until the proof completes review. That makes it a good candidate for a formal check.","legacy":"wiki-moving-sofa","links":[["Wikipedia: Moving sofa problem","https://en.wikipedia.org/wiki/Moving_sofa_problem"],["Baek, Optimality of Gerver's sofa (arXiv 2411.19826)","https://arxiv.org/abs/2411.19826"]],"posed":"1966, Leo Moser","statement":"What is the largest area of a shape that can be moved around a right-angled corner in a hallway of unit width?","verifier":"Lean statement in formal-conjectures","y":[1966,null,2024,false]},"natural-selection":{"closed":"1859, Charles Darwin and Alfred Russel Wallace; modern synthesis, 1930s–40s","explanation":"Natural selection acting on heritable variation explains adaptation without a designer. Darwin had no mechanism of inheritance, and Mendel's genetics, rediscovered in 1900, supplied it. The modern synthesis combined the two with population genetics.","links":[["Wikipedia: Natural selection","https://en.wikipedia.org/wiki/Natural_selection"]],"statement":"How do the varieties of living things come to exist and fit their surroundings?","y":[1809,1859,null,true]},"navier-stokes":{"explanation":"The equations predict every weather forecast and airfoil, yet nobody knows whether their solutions can blow up in finite time. In two dimensions they are known to stay smooth. Tao showed in 2016 that a modified, averaged version can blow up, so any proof must use structure that the averaged equation lacks. Recent computer-assisted work on the related Euler equations has produced rigorous blow-up scenarios.","legacy":"millennium-navier-stokes","links":[["Wikipedia: Navier–Stokes existence and smoothness","https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness"]],"posed":"1822–1845, Navier and Stokes wrote the equations; the Clay form dates to 2000","prize":"$1,000,000 Clay Millennium Prize","statement":"Do smooth initial conditions for incompressible 3D fluid flow always have smooth solutions for all time?","verifier":"Lean statement in formal-conjectures","y":[1822,null,null,false]},"neural-correlates":{"explanation":"Integrated information theory and global workspace theory make conflicting predictions. A pre-registered adversarial collaboration published in 2023 and 2025 found results that challenged both. Adversarial collaboration of this kind is a useful template for problems without a clean verifier.","links":[["Wikipedia: Neural correlates of consciousness","https://en.wikipedia.org/wiki/Neural_correlates_of_consciousness"]],"statement":"Which brain activity is minimally sufficient for a conscious experience?","y":[1990,null,null,true]},"neuron-doctrine":{"closed":"1888, Santiago Ramón y Cajal; confirmed by electron microscopy in the 1950s","explanation":"Golgi, whose stain made the question answerable, believed in a continuous net. Cajal used the same stain to argue for discrete neurons connected at gaps. They shared the 1906 Nobel Prize while still disagreeing, and electron microscopes later showed the synaptic gap.","links":[["Wikipedia: Neuron doctrine","https://en.wikipedia.org/wiki/Neuron_doctrine"]],"statement":"Is the nervous system a continuous network, or made of separate cells?","y":[1873,1888,null,true]},"neutrino-mass":{"closed":"1998, Super-Kamiokande: neutrinos oscillate, so they have mass","explanation":"The Standard Model had neutrinos massless. Oscillations showed they are not, which earned the 2015 Nobel Prize, and the solar neutrino problem was resolved along the way. The absolute masses, their ordering, and whether neutrinos are Majorana particles remain open. KATRIN bounds the mass below 0.45 eV.","links":[["Wikipedia: Neutrino oscillation","https://en.wikipedia.org/wiki/Neutrino_oscillation"]],"statement":"What are the neutrino masses, how are they ordered, and are neutrinos their own antiparticles?","y":[1957,null,1998,true]},"newcombs-problem":{"explanation":"Two principles of rational choice, maximizing expected utility and dominance, give opposite answers. Philosophers split roughly evenly in surveys. The problem matters for how artificial agents should reason when other agents can predict them.","links":[["Wikipedia: Newcomb's paradox","https://en.wikipedia.org/wiki/Newcomb%27s_paradox"]],"posed":"1960, William Newcomb; published by Robert Nozick, 1969","statement":"A reliable predictor has filled a box based on its forecast of your choice. Should you take one box or both?","y":[1960,null,null,false]},"nitrogen-fixation":{"closed":"1909–1913, Fritz Haber and Carl Bosch","explanation":"The Haber–Bosch process now feeds roughly half the world's population. It also made explosives cheap for the First World War, and Haber led Germany's chemical weapons program. The open problem it left is doing the same thing without high temperature and pressure, the way bacteria do.","links":[["Wikipedia: Haber process","https://en.wikipedia.org/wiki/Haber_process"]],"posed":"1898, William Crookes warns of a coming fertilizer famine","statement":"Can atmospheric nitrogen be turned into ammonia at industrial scale?","y":[1898,1909,null,false]},"one-way-functions":{"explanation":"Almost all of modern cryptography assumes the answer is yes, and a proof would imply P ≠ NP. Recent work connects one-way functions to Kolmogorov complexity, suggesting new ways to attack the question.","links":[["Wikipedia: One-way function","https://en.wikipedia.org/wiki/One-way_function"]],"statement":"Is there a function that is easy to compute but hard to invert on average?","y":[1976,null,null,true]},"origin-of-language":{"explanation":"The Paris Linguistic Society banned the topic in 1866 as unanswerable. Genetics, archaeology and comparative cognition now offer partial evidence. Estimates of when language arose still range from 50,000 to over 1,000,000 years ago.","links":[["Wikipedia: Origin of language","https://en.wikipedia.org/wiki/Origin_of_language"]],"statement":"When and how did human language evolve?","y":[1866,null,null,true]},"origin-of-life":{"explanation":"The RNA world hypothesis, in which RNA both stored information and catalyzed reactions, is the leading framework. Prebiotic chemistry has made nucleotides under plausible early-Earth conditions. No one has yet made a self-sustaining, evolving system from simple starting materials, which is the clear verifier for this problem.","links":[["Wikipedia: Abiogenesis","https://en.wikipedia.org/wiki/Abiogenesis"]],"statement":"How did the first self-replicating, evolving system arise from non-living chemistry?","verifier":"A self-replicating chemical system built from prebiotic starting materials","y":[1871,null,null,true]},"origin-of-water":{"explanation":"Hydrogen isotope ratios point toward water-rich asteroids more than comets, but no source matches perfectly. Some recent work argues much of the water was present in the planet's original building blocks.","links":[["Wikipedia: Origin of water on Earth","https://en.wikipedia.org/wiki/Origin_of_water_on_Earth"]],"statement":"Did Earth's oceans come from comets, asteroids, or the material that formed the planet?","y":[1955,null,null,true]},"ozone-hole":{"closed":"1974 mechanism, Molina and Rowland; 1985 detection, Farman, Gardiner and Shanklin; 1987 Montreal Protocol","explanation":"Chlorofluorocarbons release chlorine that destroys ozone catalytically, especially on polar stratospheric clouds. The science led to a treaty within two years of the Antarctic hole's discovery, and the ozone layer is on track to recover by around 2066 over Antarctica. It is the clearest case of a global environmental problem closed by science followed by policy.","links":[["Wikipedia: Ozone depletion","https://en.wikipedia.org/wiki/Ozone_depletion"]],"statement":"What is destroying stratospheric ozone, and can it be stopped?","y":[1974,1987,null,false]},"p-vs-np":{"explanation":"Most researchers believe P ≠ NP: that finding is genuinely harder than checking. Three known barriers, relativization, natural proofs and algebrization, rule out whole families of proof techniques, which explains why fifty years of effort have not produced a proof. If P = NP with a practical algorithm, most public-key cryptography would fail and much of mathematics could be automated. The formal-conjectures entry also contains NP ≠ coNP.","legacy":"millennium-pvs-np","links":[["Wikipedia: P versus NP problem","https://en.wikipedia.org/wiki/P_versus_NP_problem"],["Clay Mathematics Institute","https://www.claymath.org/millennium/p-vs-np/"]],"posed":"1971, Stephen Cook and Leonid Levin; Gödel asked a version in a 1956 letter to von Neumann","prize":"$1,000,000 Clay Millennium Prize","statement":"Can every problem whose solutions can be checked quickly also be solved quickly?","verifier":"Lean statement `P_ne_NP` in formal-conjectures","y":[1971,null,null,false]},"peptic-ulcers":{"closed":"1982, Barry Marshall and Robin Warren identify Helicobacter pylori","explanation":"The consensus that no bacteria could survive stomach acid was wrong. Marshall drank a culture of H. pylori to prove it caused gastritis. Ulcers became curable with antibiotics, and the pair won the 2005 Nobel Prize.","links":[["Wikipedia: Helicobacter pylori","https://en.wikipedia.org/wiki/Helicobacter_pylori"]],"statement":"Are peptic ulcers caused by stress and acid, or by an infection?","y":[1910,1982,null,true]},"periodic-table":{"closed":"1869, Dmitri Mendeleev; 1913, Henry Moseley ties order to atomic number","explanation":"Mendeleev left gaps for elements not yet found and predicted their properties; gallium, scandium and germanium filled them within 17 years. Moseley's X-ray measurements showed the order was nuclear charge, not weight. Quantum mechanics later explained why the pattern exists.","links":[["Wikipedia: Periodic table","https://en.wikipedia.org/wiki/Periodic_table"]],"statement":"Is there an organizing principle behind the chemical elements?","y":[1817,1869,1913,true]},"plate-tectonics":{"closed":"1960s: seafloor spreading, magnetic striping and plate tectonics","explanation":"Wegener had strong evidence, matching coastlines, fossils and rock layers, but no mechanism, and most geologists rejected him for fifty years. Magnetic stripes on the seafloor, mapped after the Second World War, showed new crust forming at mid-ocean ridges. A correct idea waited decades for the measurements that could confirm it.","links":[["Wikipedia: Plate tectonics","https://en.wikipedia.org/wiki/Plate_tectonics"]],"posed":"1912, Alfred Wegener","statement":"Do the continents move, and if so, how?","y":[1912,1960,null,false]},"poincare-conjecture":{"closed":"2003, Grigori Perelman","explanation":"Higher dimensions fell first: Smale in dimension 5 and up in 1961, Freedman in dimension 4 in 1982. Perelman finished dimension 3 by running Hamilton's Ricci flow, cutting out singularities as they formed, and proving Thurston's geometrization along the way. He declined both the Fields Medal and the Millennium Prize. The smooth four-dimensional version is still open.","legacy":"millennium-poincare","links":[["Wikipedia: Poincaré conjecture","https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture"]],"posed":"1904, Henri Poincaré","prize":"Clay Millennium Prize awarded 2010, declined","statement":"Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.","y":[1904,2003,null,false]},"polio-eradication":{"explanation":"Cases fell by over 99% from about 350,000 a year in 1988. Wild types 2 and 3 are certified eradicated. Wild type 1 persists in Pakistan and Afghanistan, and outbreaks derived from the oral vaccine complicate the endgame.","links":[["Wikipedia: Polio eradication","https://en.wikipedia.org/wiki/Polio_eradication"]],"posed":"1988, Global Polio Eradication Initiative","statement":"Can poliovirus be eradicated worldwide?","y":[1988,null,null,false]},"post-quantum":{"closed":"2024, NIST publishes FIPS 203, 204 and 205","explanation":"Shor's algorithm would break RSA and elliptic-curve cryptography on a large enough quantum computer. NIST standardized lattice-based ML-KEM and ML-DSA and the hash-based SLH-DSA in 2024. SIKE, a finalist candidate, was broken in 2022 by a classical attack in about an hour, a reminder that hardness assumptions are conjectures.","links":[["Wikipedia: Post-quantum cryptography","https://en.wikipedia.org/wiki/Post-quantum_cryptography"]],"statement":"Which public-key schemes resist attack by large quantum computers?","y":[1994,null,2024,false]},"powered-flight":{"closed":"1903, Wilbur and Orville Wright","explanation":"The Wrights built their own wind tunnel, found that the published lift data were wrong, and solved control, by wing warping, before power. In 1903 the New York Times had predicted flight would take a million to ten million years.","links":[["Wikipedia: Wright Flyer","https://en.wikipedia.org/wiki/Wright_Flyer"]],"statement":"Can a heavier-than-air machine fly under its own power with a pilot in control?","y":[1799,1903,null,true]},"primes-in-p":{"closed":"2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena","explanation":"Randomized tests had been fast for decades, but no deterministic polynomial-time test was known. AKS found one using elementary algebra, and two of the three authors were undergraduates. In practice randomized tests are still used; the result settled the question of principle.","links":[["Wikipedia: AKS primality test","https://en.wikipedia.org/wiki/AKS_primality_test"]],"statement":"Can primality be tested deterministically in polynomial time?","y":[1976,2002,null,true]},"problem-of-induction":{"explanation":"Every argument for induction seems to rely on induction. Popper answered that science does not need induction, only falsification. Bayesian and algorithmic approaches, such as Solomonoff induction, give formal accounts of learning from data, yet each rests on a prior that is itself not justified from experience. The problem is newly practical: machine learning is induction at industrial scale.","links":[["Wikipedia: Problem of induction","https://en.wikipedia.org/wiki/Problem_of_induction"]],"posed":"1739, David Hume","statement":"What justifies believing that the future will resemble the past?","y":[1739,null,null,false]},"productivity-slowdown":{"explanation":"Solow noted in 1987 that computers appeared everywhere except in the productivity statistics. Explanations include measurement error, ideas getting harder to find, and slow diffusion of technology. Whether AI changes the trend is the live question.","links":[["Wikipedia: Productivity paradox","https://en.wikipedia.org/wiki/Productivity_paradox"]],"statement":"Why has productivity growth in rich countries slowed since the 1970s, despite computing?","y":[1987,null,null,false]},"protein-design":{"explanation":"De novo design, pioneered by David Baker's lab, now routinely produces binders and new folds. Designing enzymes that match natural catalytic efficiency remains much harder.","links":[["Wikipedia: Protein design","https://en.wikipedia.org/wiki/Protein_design"]],"statement":"Can we design a protein from scratch to carry out any chosen chemical function?","y":[1988,null,2019,true]},"protein-structure-prediction":{"closed":"2020, AlphaFold 2 at CASP14","explanation":"For fifty years this was a grand challenge. CASP, a blind biennial benchmark, served as the verifier: predictors submit structures before experiments are released. AlphaFold 2 reached accuracy comparable to experiment on most targets, and Hassabis, Jumper and Baker shared the 2024 Nobel Prize in Chemistry. Dynamics, disordered proteins and complexes are still open.","links":[["Wikipedia: AlphaFold","https://en.wikipedia.org/wiki/AlphaFold"],["Wikipedia: CASP","https://en.wikipedia.org/wiki/CASP"]],"posed":"1972, Christian Anfinsen: sequence determines structure","statement":"Can a protein's three-dimensional structure be predicted from its amino-acid sequence?","verifier":"CASP blind assessment","y":[1972,2020,null,false]},"quantum-gravity":{"explanation":"Naively quantizing gravity produces infinities that cannot be absorbed. String theory, loop quantum gravity and asymptotic safety each offer a candidate, and none has made a confirmed prediction. The effects appear near the Planck length, far below anything experiments can reach, so the problem is hard for lack of data as much as lack of ideas. Tabletop tests of whether gravity can entangle two masses are the most promising near-term probe.","links":[["Wikipedia: Quantum gravity","https://en.wikipedia.org/wiki/Quantum_gravity"]],"statement":"What theory reduces to general relativity at large scales and quantum field theory at small ones, and stays consistent where both matter?","y":[1930,null,null,true]},"quintic-insolvability":{"closed":"1824, Niels Henrik Abel (after Ruffini, 1799); explained by Galois, 1832","explanation":"There is not. Galois explained why: solvability corresponds to a property of the equation's symmetry group, and the symmetric group on five letters lacks it. Galois theory, the tool built to close this question, became one of the foundations of modern algebra.","links":[["Wikipedia: Abel–Ruffini theorem","https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem"]],"posed":"16th century, after Cardano's cubic and Ferrari's quartic formulas","statement":"Is there a formula in radicals for the roots of every polynomial of degree five?","y":[1550,1824,null,false]},"ramsey-r55":{"explanation":"It is known that $43 \\le R(5,5) \\le 46$, with the upper bound brought down in 2024 by Angeltveit and McKay. Erdős joked that if aliens demanded R(5,5) we should marshal all our computers, and if they demanded R(6,6) we should try to destroy them. The asymptotic growth rate improved in 2023, when Campos, Griffiths, Morris and Sahasrabudhe beat the $4^n$ barrier.","legacy":"wiki-ramsey-numbers","links":[["Wikipedia: Ramsey's theorem","https://en.wikipedia.org/wiki/Ramsey%27s_theorem"]],"statement":"What is the smallest $n$ such that every red/blue coloring of the edges of $K_n$ has a monochromatic $K_5$?","verifier":"Lean statement in formal-conjectures; any exact value is a finite check","y":[1955,null,2024,true]},"regeneration":{"explanation":"Axolotls regrow limbs, spinal cord and parts of the heart. The difference lies in how injury responses are controlled rather than in missing genes, which suggests the capacity might be restored.","links":[["Wikipedia: Regeneration (biology)","https://en.wikipedia.org/wiki/Regeneration_(biology)"]],"statement":"Why can salamanders regrow limbs while mammals cannot, and can mammals be made to?","y":[1768,null,null,true]},"resolution-of-singularities-char-p":{"explanation":"Hironaka proved in 1964 that in characteristic zero every singular variety can be replaced by a smooth one, and won a Fields Medal for it. In positive characteristic the method breaks, and the question is settled only up to dimension 3. Many results in arithmetic geometry work around it with alterations, de Jong's weaker substitute.","links":[["Wikipedia: Resolution of singularities","https://en.wikipedia.org/wiki/Resolution_of_singularities"]],"posed":"After Hironaka's 1964 proof in characteristic 0","statement":"Every variety over a field of positive characteristic admits a resolution of singularities.","y":[1964,null,null,false]},"reusable-rockets":{"closed":"2015 first orbital-class booster landing; 2017 first reflight, SpaceX Falcon 9","explanation":"The Space Shuttle was reusable in principle and expensive in practice. Propulsive landing and rapid reuse made boosters fly dozens of times each and cut launch costs sharply. Full reuse including the upper stage is the next open step.","links":[["Wikipedia: Reusable launch vehicle","https://en.wikipedia.org/wiki/Reusable_launch_vehicle"]],"statement":"Can an orbital rocket's first stage land and fly again economically?","y":[1972,2017,2015,true]},"riemann-hypothesis":{"explanation":"The zeta function encodes the primes, and the location of its zeros controls how far the count of primes up to x can stray from its smooth estimate. If the hypothesis holds, the error is about the square root of x, the best that could be hoped for. Over ten trillion zeros have been computed and all lie on the line. Hundreds of theorems are proved conditionally on it, so a proof would settle them all at once. The analogue for curves over finite fields was proved by Deligne in 1974, which is the strongest evidence that the statement is true and the main source of ideas for attacking it.","legacy":"millennium-riemann-hypothesis","links":[["Wikipedia: Riemann hypothesis","https://en.wikipedia.org/wiki/Riemann_hypothesis"],["Clay Mathematics Institute","https://www.claymath.org/millennium/riemann-hypothesis/"]],"posed":"1859, Bernhard Riemann","prize":"$1,000,000 Clay Millennium Prize","statement":"Every nontrivial zero of the Riemann zeta function has real part 1/2.","verifier":"Lean statement in formal-conjectures","y":[1859,null,null,false]},"room-temperature-superconductor":{"explanation":"Hydrides under megabar pressure superconduct at over 200 K, but only inside diamond anvils. Claims of ambient-pressure room-temperature superconductivity have repeatedly failed replication: the 2023 LK-99 claim was explained by copper sulfide impurities within weeks, and several high-profile hydride papers were retracted. The problem has a clean verifier: zero resistance and a Meissner effect, reproduced in independent labs.","links":[["Wikipedia: Room-temperature superconductor","https://en.wikipedia.org/wiki/Room-temperature_superconductor"],["Wikipedia: LK-99","https://en.wikipedia.org/wiki/LK-99"]],"statement":"Is there a material that superconducts at room temperature and normal pressure?","verifier":"Four-probe zero resistance plus Meissner effect, independently replicated","y":[1911,null,null,true]},"sato-tate":{"closed":"2008–2011, Clozel, Harris, Shepherd-Barron and Taylor, then Barnet-Lamb, Geraghty, Harris and Taylor","explanation":"Point counts of an elliptic curve modulo primes look random, and the conjecture names their distribution exactly. The proof needed potential modularity of all symmetric powers of the curve. It is an example of the Langlands machinery giving statistical laws for arithmetic.","legacy":"wiki-sato-tate-conjecture","links":[["Wikipedia: Sato–Tate conjecture","https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture"]],"posed":"1960s, Mikio Sato and John Tate","statement":"For a non-CM elliptic curve, the normalized errors in point counts mod p follow the semicircle distribution.","y":[1960,2008,null,false]},"scurvy":{"closed":"1747, James Lind's trial; 1932, vitamin C identified","explanation":"Lind ran one of the first controlled clinical trials, and the citrus arm recovered. The knowledge was lost and found again several times, partly because lime juice, substituted for lemon, had less vitamin C. It is an example of an answer that was found and then did not stay found.","links":[["Wikipedia: Scurvy","https://en.wikipedia.org/wiki/Scurvy"]],"statement":"What causes scurvy on long voyages, and how is it prevented?","y":[1497,1932,1747,true]},"sensitivity-conjecture":{"closed":"2019, Hao Huang","explanation":"Thirty years of partial results ended with a two-page proof using an ingenious signed adjacency matrix of the hypercube. Scott Aaronson called it one of the most beautiful proofs he had seen. Huang's argument is short enough to formalize quickly.","links":[["Wikipedia: Sensitivity theorem","https://en.wikipedia.org/wiki/Sensitivity_theorem"]],"posed":"1992, Noam Nisan and Mario Szegedy","statement":"The sensitivity of a Boolean function is polynomially related to its other complexity measures, such as block sensitivity.","y":[1992,2019,null,false]},"smallpox-eradication":{"closed":"1980, declared eradicated by the World Health Assembly","explanation":"Smallpox killed an estimated 300 million people in the twentieth century alone. It was eradicated by surveillance and ring vaccination, vaccinating the contacts of each case, rather than by vaccinating everyone. It remains the only human disease ever eradicated.","links":[["Wikipedia: Eradication of smallpox","https://en.wikipedia.org/wiki/Eradication_of_smallpox"]],"posed":"1796, Edward Jenner's vaccine; 1959 and 1967 WHO eradication programs","statement":"Can a human disease be removed from the planet?","y":[1796,1980,null,false]},"smooth-poincare-4":{"explanation":"Dimension four is the one place where topological and smooth categories come apart most wildly: $\\mathbb{R}^4$ alone carries uncountably many smooth structures. The topological version is Freedman's theorem. Whether an exotic smooth 4-sphere exists is open, and it is one of the central questions of low-dimensional topology.","links":[["Wikipedia: Generalized Poincaré conjecture","https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture"]],"statement":"Every smooth homotopy 4-sphere is diffeomorphic to the standard 4-sphere.","verifier":"Lean statement in formal-conjectures (Poincaré variants)","y":[1982,null,null,true]},"snowball-earth":{"explanation":"Glacial deposits at the equator 700 million years ago support near-global glaciation. Whether the oceans froze completely or kept open water near the equator is debated, and the answer bears on how complex life survived.","links":[["Wikipedia: Snowball Earth","https://en.wikipedia.org/wiki/Snowball_Earth"]],"posed":"1992, Joseph Kirschvink","statement":"Did ice cover the whole planet during the Cryogenian, and how did life survive?","y":[1992,null,null,false]},"solar-neutrino-problem":{"closed":"2001, Sudbury Neutrino Observatory","explanation":"For thirty years either the solar model or the physics of neutrinos had to be wrong. SNO measured all three neutrino flavors and found the full flux: the neutrinos were changing flavor on the way. The solar model was right, and neutrinos have mass.","links":[["Wikipedia: Solar neutrino problem","https://en.wikipedia.org/wiki/Solar_neutrino_problem"]],"posed":"1968, Ray Davis's Homestake experiment","statement":"Why do detectors see only about a third of the neutrinos the Sun should produce?","y":[1968,2001,null,false]},"spectrum-auctions":{"closed":"1994, simultaneous multiple-round auction, designed by Paul Milgrom and Robert Wilson","explanation":"Auction theory turned a political giveaway into a designed market that has raised hundreds of billions of dollars worldwide. Milgrom and Wilson received the 2020 economics Nobel. It is mechanism design, the engineering side of economics, working at scale.","links":[["Wikipedia: Spectrum auction","https://en.wikipedia.org/wiki/Spectrum_auction"]],"statement":"How should a government allocate interdependent licenses efficiently?","y":[1959,1994,null,false]},"sphere-packing-8-24":{"closed":"2016, Maryna Viazovska; dimension 24 with Cohn, Kumar, Miller and Radchenko","explanation":"Cohn and Elkies had reduced the problem to finding one special function. Viazovska built it from modular forms, and a week later the same idea settled dimension 24. She received the Fields Medal in 2022. The optimal packing in every other dimension above 3 is open.","links":[["Wikipedia: Sphere packing","https://en.wikipedia.org/wiki/Sphere_packing"]],"statement":"The E8 lattice and the Leech lattice are the densest sphere packings in dimensions 8 and 24.","y":[1967,2016,null,true]},"spontaneous-generation":{"closed":"1859–1862, Louis Pasteur's swan-neck flask experiments","explanation":"Pasteur's broth stayed sterile in flasks open to air through an S-shaped neck that trapped dust. The refutation of everyday spontaneous generation left the deeper origin-of-life question in place.","links":[["Wikipedia: Spontaneous generation","https://en.wikipedia.org/wiki/Spontaneous_generation"]],"statement":"Does life arise routinely from non-living matter, as in maggots from meat?","y":[-350,1862,null,true]},"squaring-the-circle":{"closed":"1882, Ferdinand von Lindemann","explanation":"It stood for two thousand years because it was impossible. Constructible lengths are algebraic of a special kind, and Lindemann proved that π is not algebraic at all. The answer came from algebra and analysis, far from the geometry in which it was asked. Impossibility proofs close questions just as firmly as constructions do.","links":[["Wikipedia: Squaring the circle","https://en.wikipedia.org/wiki/Squaring_the_circle"]],"posed":"Antiquity, Greek geometers","statement":"Construct, with compass and straightedge, a square with the same area as a given circle.","y":[-400,1882,null,false]},"strong-cp":{"explanation":"Quantum chromodynamics has a parameter θ that could take any value, and the neutron's electric dipole moment bounds it below about 10^-10. The Peccei–Quinn mechanism explains the small value with a new particle, the axion, which is also a dark matter candidate. Axion searches are running now.","links":[["Wikipedia: Strong CP problem","https://en.wikipedia.org/wiki/Strong_CP_problem"]],"statement":"Why does the strong force appear to respect CP symmetry when the theory allows it to be broken?","y":[1977,null,null,true]},"structure-of-dna":{"closed":"1953, James Watson and Francis Crick, from Rosalind Franklin's and Maurice Wilkins's diffraction data","explanation":"A double helix with paired bases, whose pairing rule immediately suggested how it copies itself. Franklin's Photo 51 was shown to Watson without her knowledge; her role is now recognized as central.","links":[["Wikipedia: DNA","https://en.wikipedia.org/wiki/DNA"],["Wikipedia: Photo 51","https://en.wikipedia.org/wiki/Photo_51"]],"statement":"What is the physical structure of the molecule that carries heredity?","y":[1944,1953,null,false]},"thermophotovoltaic-efficiency":{"explanation":"Thermophotovoltaic cells convert light from a hot emitter directly to electricity with no moving parts, which makes them a candidate for storing energy as heat. Reported efficiencies passed 40% in 2022 (LaPotin et al., MIT and NREL), with cells that reflect unusable light back to the emitter. Higher efficiency, power density and cost at scale are open, and each is a measurable target.","links":[["Wikipedia: Thermophotovoltaic energy conversion","https://en.wikipedia.org/wiki/Thermophotovoltaic_energy_conversion"]],"statement":"Can heat be turned into electricity by photovoltaic cells at efficiencies that rival turbines?","verifier":"Certified conversion efficiency on a calibrated test stand","y":[1956,null,2022,false]},"three-body-problem":{"closed":"1890, Henri Poincaré showed no general closed-form solution exists; Sundman's convergent series, 1912","explanation":"Poincaré won the King Oscar prize by showing the problem is not integrable, and while correcting an error in his prize essay he discovered what is now called chaos. Sundman later found a convergent power series, but it converges too slowly to use. The question that replaced it is stability: whether the solar system stays bounded forever, which KAM theory answers only in part.","links":[["Wikipedia: Three-body problem","https://en.wikipedia.org/wiki/Three-body_problem"]],"posed":"1687, Isaac Newton; prize question of King Oscar II, 1885","statement":"Find a general closed-form solution for three bodies moving under gravity.","y":[1687,1890,null,false]},"transistor":{"closed":"1947, John Bardeen, Walter Brattain and William Shockley at Bell Labs","explanation":"The point-contact transistor made modern computing possible. Chips now hold tens of billions of transistors each, and the problem shifted from building one to making them smaller.","links":[["Wikipedia: Transistor","https://en.wikipedia.org/wiki/Transistor"]],"statement":"Can a solid-state device replace the vacuum tube as an amplifier and switch?","y":[1926,1947,null,true]},"turbulence":{"explanation":"Kolmogorov's 1941 theory predicts how energy cascades from large eddies to small ones, and measurements confirm it roughly but with systematic deviations called intermittency. Engineers rely on empirical models. Feynman called turbulence the most important unsolved problem of classical physics.","links":[["Wikipedia: Turbulence","https://en.wikipedia.org/wiki/Turbulence"]],"statement":"Can the statistics of turbulent flow be predicted from the equations of fluid motion?","y":[1883,null,null,true]},"turing-test":{"explanation":"Turing proposed conversation as a test to replace an ill-defined question. By the mid-2020s language models were judged human in controlled imitation-game studies at or above chance. Few think this settles what Turing's original question was about, which is why it is marked partial.","links":[["Wikipedia: Turing test","https://en.wikipedia.org/wiki/Turing_test"]],"posed":"1950, Alan Turing, Computing Machinery and Intelligence","statement":"Can a machine converse indistinguishably from a human?","y":[1950,null,2025,false]},"twin-prime-conjecture":{"explanation":"Primes thin out, but they seem to keep arriving in close pairs forever. Sieve methods can count almost-primes but run into the parity barrier, which stops them from telling a number with an even count of prime factors from one with an odd count. The 2013 breakthrough on bounded gaps got within a constant of the conjecture: some gap of at most 246 occurs infinitely often. Closing 246 down to 2 appears to need an idea that breaks parity.","legacy":"wiki-twin-primes","links":[["Wikipedia: Twin prime","https://en.wikipedia.org/wiki/Twin_prime"]],"posed":"1849, Alphonse de Polignac (general form)","statement":"There are infinitely many primes p such that p + 2 is also prime.","verifier":"Lean statement in formal-conjectures","y":[1849,null,null,false]},"ultraviolet-catastrophe":{"closed":"1900, Max Planck's quantum hypothesis","explanation":"Planck fitted the measured blackbody spectrum by assuming energy is exchanged in discrete amounts, an assumption he regarded as a mathematical trick. Einstein took it literally in 1905 to explain the photoelectric effect. A mismatch between theory and a curve on a graph started quantum mechanics.","links":[["Wikipedia: Ultraviolet catastrophe","https://en.wikipedia.org/wiki/Ultraviolet_catastrophe"]],"posed":"1900, the Rayleigh–Jeans law","statement":"Why does a hot body not radiate infinite energy at short wavelengths, as classical physics predicts?","y":[1900,1900,null,false]},"union-closed-sets":{"explanation":"For decades nobody could prove any constant fraction. In 2022 Justin Gilmer, then at Google, proved 1%, using an information-theoretic argument. Within weeks others pushed it to about 38%. The last stretch to 50% appears to need a new idea.","legacy":"wiki-union-closed","links":[["Wikipedia: Union-closed sets conjecture","https://en.wikipedia.org/wiki/Union-closed_sets_conjecture"]],"posed":"1979, Péter Frankl","statement":"In every finite union-closed family of sets, other than the one containing only the empty set, some element belongs to at least half the sets.","verifier":"Lean statement in formal-conjectures","y":[1979,null,2022,false]},"vagueness":{"explanation":"Most words are vague, and classical logic assumes sharp boundaries. Supervaluationism, epistemicism, which holds that sharp boundaries exist but are unknowable, and fuzzy logic each give up something. The paradox matters wherever law or policy must draw a line on a continuum.","links":[["Wikipedia: Sorites paradox","https://en.wikipedia.org/wiki/Sorites_paradox"]],"posed":"4th century BCE, Eubulides of Miletus","statement":"If removing one grain never turns a heap into a non-heap, how are there any heaps?","y":[-350,null,null,false]},"voynich-manuscript":{"explanation":"Radiocarbon dating puts the vellum in the early 1400s. Its text has statistical regularities like natural language, yet every proposed decipherment has failed to predict anything. It may be a cipher, a lost language, or an elaborate hoax.","links":[["Wikipedia: Voynich manuscript","https://en.wikipedia.org/wiki/Voynich_manuscript"]],"statement":"What, if anything, does the 15th-century Voynich manuscript say?","y":[1912,null,null,false]},"weak-goldbach":{"closed":"2013, Harald Helfgott","explanation":"Vinogradov proved in 1937 that it holds for all sufficiently large odd numbers, but the threshold was astronomically large. Helfgott sharpened the circle method until the threshold fell to a range that a computer had already checked. It shows a common endgame: an asymptotic proof, then decades of bookkeeping to make its constants small.","links":[["Wikipedia: Goldbach's weak conjecture","https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture"]],"posed":"1742, from the Goldbach–Euler correspondence","statement":"Every odd integer greater than 5 is the sum of three primes.","y":[1742,2013,null,false]},"weil-conjectures":{"closed":"1974, Pierre Deligne (the Riemann hypothesis part)","explanation":"Weil saw that counting points on varieties over finite fields should behave like topology. Grothendieck built étale cohomology to make that literal, and Deligne used it to prove the hardest part. This is the finite-field shadow of the Riemann hypothesis, and the reason many believe the original is true.","links":[["Wikipedia: Weil conjectures","https://en.wikipedia.org/wiki/Weil_conjectures"]],"posed":"1949, André Weil","statement":"Zeta functions of varieties over finite fields are rational, satisfy a functional equation and obey a Riemann hypothesis.","y":[1949,1974,null,false]},"whitehead-problem":{"closed":"1974, Saharon Shelah: independent of ZFC","explanation":"An ordinary-looking question in algebra turned out to have no answer in standard set theory. It is true under V = L and false under Martin's axiom with the continuum hypothesis false. It was the first major problem from outside logic shown independent.","links":[["Wikipedia: Whitehead problem","https://en.wikipedia.org/wiki/Whitehead_problem"]],"posed":"1950s, J. H. C. Whitehead","statement":"Is every abelian group A with Ext(A, Z) = 0 a free abelian group?","y":[1950,1974,null,false]},"why-something":{"explanation":"Physics can describe how a universe evolves from an initial state, and even how a quantum state might arise from 'nothing'. Each such answer assumes laws and a framework that exist already. It may be the only question on this map that could have no answer.","links":[["Wikipedia: Why is there anything at all?","https://en.wikipedia.org/wiki/Why_is_there_anything_at_all%3F"]],"posed":"1714, Gottfried Wilhelm Leibniz","statement":"Why does anything exist at all?","y":[1714,null,null,false]},"yang-mills-mass-gap":{"explanation":"Physicists compute with Yang–Mills theory every day, and lattice simulations show the gap clearly. Nobody can yet construct the theory with mathematical rigor. It is a physics problem whose resolution would be a theorem.","links":[["Wikipedia: Yang–Mills existence and mass gap","https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap"]],"posed":"2000, Clay Mathematics Institute, after Yang and Mills (1954)","prize":"$1,000,000 Clay Millennium Prize","statement":"Prove that quantum Yang–Mills theory exists rigorously in four dimensions and has a positive mass gap.","y":[2000,null,null,false]},"zenos-paradoxes":{"explanation":"The mathematics of limits and convergent series, made rigorous in the 19th century, shows that infinitely many shrinking steps can take finite time. 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and matter","Life","Mind","Medicine","Society","Engineering","Philosophy","Number theory","The Langlands program","Algebraic geometry","Geometry and topology","Combinatorics","Analysis and PDE","Logic and foundations","Algebra and groups","Dynamical systems","Convex and discrete geometry","Probability","Several complex variables","Special functions","Complexity classes","Algorithms","Cryptography","Learning and AI","Gravity and the quantum","The Standard Model's loose ends","Quantum foundations","Fluids and many bodies","The dark sector","Origins of the universe","Stars and black holes","Life beyond Earth","The deep Earth","Climate","Early Earth","Materials","Energy and catalysis","Molecules of life","Heredity and the code","Origins","Development and aging","The brain","Consciousness","Language","Machine minds","Infectious disease","Chronic disease","Wealth and growth","Collective choice","Scripts and the past","Energy","Machines that move","Computing hardware","Knowledge","Being and self","Value and reason","Riemann hypothesis","Twin prime conjecture","Bounded gaps between primes","Goldbach's conjecture","Weak Goldbach conjecture","abc conjecture","Collatz conjecture","Fermat's Last Theorem","Catalan's conjecture","Birch and Swinnerton-Dyer conjecture","The Curling Number Conjecture","Cunningham chains — Jones's conjecture","Unique Crystal Components","An Arithmetic Sum Associated with the Classical Theta Function","A conjecture by Margulis on matrix groups","Integer values of $\\tan(\\arctan 1 + \\arctan 2 + \\cdots + \\arctan n)$","Minimum modulus for the unique multiset-sum problem","Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers","Bugeaud Collection of Conjectures and Open Questions: Spectrum of Sequence","Bugeaud Collection of Conjectures and Open Questions: Lacunary Sequences in Real Number Fields","Bugeaud Collection of Conjectures and Open Questions: Simultaneously Small Entropies","Bugeaud Collection of Conjectures and Open Questions: Rapidly Increasing Sequences Dense Modulo One","Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set","Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers","Bugeaud Collection of Conjectures and Open Questions: $p$-adic Littlewood Conjecture","Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers","Equidistributed Sequences","Erdős Problem 1000","Erdős Problem 1002","Erdős Problem 1003","Erdős Problem 1004","Erdős Problem 1005","Erdős Problem 1049","Erdős Problem 1050","Erdős Problem 1051","Erdős Problem 1052","Erdős Problem 1054","Erdős Problem 1055","Erdős Problem 1056","Erdős Problem 1057","Erdős Problem 1058","Erdős Problem 1059","Erdős Problem 1060","Erdős Problem 1061","Erdős Problem 1062","Erdős Problem 1063","Erdős Problem 1064","Erdős Problem 1065","Erdős Problem 1072","Erdős Problem 1073","Erdős Problem 1074","Erdős Problem 1094","Erdős Problem 1095","Erdős Problem 1096","Erdős Problem 1097","Erdős Problem 1099","Erdős Problem 11","Erdős Problem 1101","Erdős Problem 1102","Erdős Problem 1106","Erdős Problem 1107","Erdős Problem 1108","Erdős Problem 1113","Erdős Problem 1134","Erdős Problem 1135","Erdős Problem 1136","Erdős Problem 1137","Erdős Problem 1138","Erdős Problem 1139","Erdős Problem 1141","Erdős Problem 1142","Erdős Problem 1146","Erdős Problem 1147","Erdős Problem 1148","Erdős Problem 1180","Erdős Problem 1188","Erdős Problem 1196","Erdős Problem 1197","Erdős Problem 12","Erdős Problem 1201","Erdős Problem 1203","Erdős Problem 1209","Erdős Problem 121","Erdős Problem 1210","Erdős Problem 1212","Erdős Problem 1213","Erdős Problem 1214","Erdős Problem 123","Erdős Problem 124","Erdős Problem 125","Erdős Problem 126","Erdős Problem 137","Erdős Problem 138","Erdős Problem 14","Erdős Problem 142","Erdős Problem 143","Erdős Problem 145","Erdős Problem 15: Convergence of Series with Primes","Erdős Problem 16","Erdős Problem 164","Erdős Problem 17","Erdős Problem 175","Erdős Problem 178","Erdős Problem 18","Erdős Problem 2","Erdős Problem 205","Erdős Problem 206","Erdős Problem 208","Erdős Problem 218","Erdős Problem 220","Erdős Problem 221","Erdős Problem 233","Erdős Problem 234","Erdős Problem 238","Erdős Problem 239","Erdős Problem 242","Erdős Problem 244","Erdős Problem 246","Erdős Problem 247","Erdős Problem 248","Erdős Problem 249","Erdős Problem 25: Logarithmic density of size-dependent congruences","Erdős Problem 250","Erdős Problem 251","Erdős Problem 252","Erdős Problem 253","Erdős Problem 254","Erdős Problem 257","Erdős Problem 258","Erdős Problem 259","Erdős Problem 26","Erdős Problem 260","Erdős Problem 261","Erdős Problem 263","Erdős Problem 264","Erdős Problem 266","Erdős Problem 267","Erdős Problem 269","Erdős Problem 270","Erdős Problem 275","Erdős Problem 276","Erdős Problem 277","Erdős Problem 279","Erdős Problem 28","Erdős Problem 280","Erdős Problem 281","Erdős Problem 283","Erdős Problem 286","Erdős Problem 287","Erdős Problem 288","Erdős Problem 289","Erdős Problem 29","Erdős Problem 290","Erdős Problem 291","Erdős Problem 292","Erdős Problem 296","Erdős Problem 298","Erdős Problem 299","Erdős Problem 3","Erdős Problem 30","Erdős Problem 300","Erdős Problem 302","Erdős Problem 304","Erdős Problem 305","Erdős Problem 306","Erdős Problem 307","Erdős Problem 308","Erdős Problem 309","Erdős Problem 31","Erdős Problem 310","Erdős Problem 313","Erdős Problem 314","Erdős Problem 315","Erdős Problem 317","Erdős Problem 318","Erdős Problem 32","Erdős Problem 321","Erdős Problem 322","Erdős Problem 323","Erdős Problem 324","Erdős Problem 325","Erdős Problem 328","Erdős Problem 33","Erdős Problem 331","Erdős Problem 332","Erdős Problem 333","Erdős Problem 341","Erdős Problem 346","Erdős Problem 347","Erdős Problem 348","Erdős Problem 349","Erdős Problem 35","Erdős Problem 351","Erdős Problem 354","Erdős Problem 355","Erdős Problem 357","Erdős Problem 359","Erdős Problem 361","Erdős Problem 362","Erdős Problem 363","Erdős Problem 364","Erdős Problem 366","Erdős Problem 367","Erdős Problem 369","Erdős Problem 370","Erdős Problem 371","Erdős Problem 372","Erdős Problem 373","Erdős Problem 375","Erdős Problem 376","Erdős Problem 377","Erdős Problem 379","Erdős Problem 38","Erdős Problem 383","Erdős Problem 384","Erdős Problem 385","Erdős Problem 386","Erdős Problem 387","Erdős Problem 389","Erdős Problem 39","Erdős Problem 390","Erdős Problem 391","Erdős Problem 392","Erdős Problem 394","Erdős Problem 396","Erdős Problem 397","Erdős Problem 398","Erdős Problem 399","Erdős Problem 4","Erdős Problem 40","Erdős Problem 400","Erdős Problem 401","Erdős Problem 402","Erdős Problem 403","Erdős Problem 405","Erdős Problem 406","Erdős Problem 407","Erdős Problem 409","Erdős Problem 41","Erdős Problem 410","Erdős Problem 412","Erdős Problem 413","Erdős Problem 414","Erdős Problem 416","Erdős Problem 417","Erdős Problem 418","Erdős Problem 419","Erdős Problem 421","Erdős Problem 422","Erdős Problem 424: Sequence generated by $a_i a_j - 1$","Erdős Problem 427","Erdős Problem 428","Erdős Problem 429","Erdős Problem 431","Erdős Problem 433","Erdős Problem 434","Erdős Problem 437","Erdős Problem 438","Erdős Problem 440","Erdős Problem 441","Erdős Problem 442","Erdős Problem 443","Erdős Problem 445","Erdős Problem 448","Erdős Problem 450","Erdős Problem 452","Erdős Problem 453","Erdős Problem 454","Erdős Problem 455","Erdős Problem 456","Erdős Problem 457","Erdős Problem 458","Erdős Problem 459","Erdős Problem 462","Erdős Problem 463","Erdős Problem 464","Erdős Problem 466","Erdős Problem 469","Erdős Problem 47","Erdős Problem 470","Erdős Problem 473","Erdős Problem 478","Erdős Problem 479","Erdős Problem 48","Erdős Problem 480","Erdős Problem 485","Erdős Problem 486: Logarithmic density for sets avoiding modular subsets","Erdős Problem 487","Erdős Problem 489","Erdős Problem 490","Erdős Problem 491","Erdős Problem 493","Erdős Problem 495","Erdős Problem 5","Erdős Problem 50","Erdős Problem 51","Erdős Problem 510","Erdős Problem 512","Erdős Problem 52","Erdős Problem 520","Erdős Problem 521","Erdős Problem 53","Erdős Problem 536","Erdős Problem 537","Erdős Problem 538","Erdős Problem 541","Erdős Problem 542","Erdős Problem 56","Erdős Problem 586","Erdős Problem 587","Erdős Problem 6","Erdős Problem 646","Erdős Problem 647","Erdős Problem 648","Erdős Problem 649","Erdős Problem 66","Erdős Problem 67","Erdős Problem 672","Erdős Problem 673","Erdős Problem 674","Erdős Problem 677","Erdős Problem 678","Erdős Problem 68","Erdős Problem 680","Erdős Problem 681","Erdős Problem 682","Erdős Problem 683","Erdős Problem 686","Erdős Problem 688","Erdős Problem 689","Erdős Problem 69","Erdős Problem 690","Erdős Problem 692","Erdős Problem 694","Erdős Problem 695","Erdős Problem 697","Erdős Problem 699","Erdős Problem 7","Erdős Problem 700","Erdős Problem 726","Erdős Problem 727","Erdős Problem 728","Erdős Problem 729","Erdős Problem 730","Erdős Problem 749","Erdős Problem 763","Erdős Problem 764","Erdős Problem 770","Erdős Problem 773","Erdős Problem 779","Erdős Problem 785","Erdős Problem 786","Erdős Problem 793","Erdős Problem 796","Erdős Problem 806","Erdős Problem 821","Erdős Problem 822","Erdős Problem 825","Erdős Problem 826","Erdős Problem 828","Erdős Problem 829","Erdős Problem 830","Erdős Problem 839","Erdős Problem 841","Erdős Problem 845","Erdős Problem 846","Erdős Problem 847","Erdős Problem 848","Erdős Problem 849","Erdős Problem 850","Erdős Problem 851","Erdős Problem 853","Erdős Problem 855","Erdős Problem 858","Erdős Problem 859","Erdős Problem 869","Erdős Problem 871","Erdős Problem 873","Erdős Problem 884","Erdős Problem 885","Erdős Problem 886","Erdős Problem 887","Erdős Problem 888","Erdős Problem 889","Erdős Problem 890","Erdős Problem 891","Erdős Problem 897","Erdős Problem 912","Erdős Problem 913","Erdős Problem 930","Erdős Problem 931","Erdős Problem 932","Erdős Problem 933","Erdős Problem 936","Erdős Problem 937","Erdős Problem 938","Erdős Problem 939","Erdős Problem 940","Erdős Problem 942","Erdős Problem 943","Erdős Problem 944","Erdős Problem 945","Erdős Problem 946","Erdős Problem 947","Erdős Problem 950","Erdős Problem 951","Erdős Problem 952","Erdős Problem 955","Erdős Problem 961","Erdős Problem 962","Erdős Problem 967","Erdős Problem 968","Erdős Problem 970","Erdős Problem 971","Erdős Problem 972","Erdős Problem 973","Erdős Problem 974","Erdős Problem 975","Erdős Problem 977","Erdős Problem 978","Erdős Problem 979","Erdős Problem 985","Erdős Problem 987","Erdős Problem 994","Erdős Problem 997","Erdős Problem 999","Ben Green's Open Problem 2","Ben Green's Open Problem 3","Green's Open Problem 44","Ben Green's Open Problem 45","Ben Green's Open Problem 46","Green's Open Problem 47","Ben Green's Open Problem 60","Ben Green's Open Problem 61","Ben Green's Open Problem 62","Green's Open Problem 64","Ben Green's Open Problem 66","Ben Green's Open Problem 7","Ben Green's Open Problem 82","Lam--Litt conjecture","Mathoverflow 17560","Are prime numbers among sums of prime numbers distributed as $\\frac n{2\\ln(n)}$?","Mathoverflow 75792","Expansion of g.f. $(1+x)(3+x)/(1+6x^2+x^4)$","Conjectures associated with A100474","Prime-th recurrence with reversal at each step","Pentanacci $\\pi$ sequence","Conjectures associated with A100800","Conjectures associated with A101779","Conjectures associated with A102371","Floor of sum of $\\{n/k\\}$","$a(0) = 1$, $a(n) = a(n-1)a(n-1) + 2$","Number of decompositions of $2n+1$ into $2p+q$, where $p$ and $q$ are both odd primes","Fibonacci transform satisfying $|a(n)| = F(n+1)$","$a(n) = 3a(n-1) + a(n-2) - 3a(n-3)$","Smallest power with base>1 and exponent $n$ without digit 0","$a(n) = [x^{2n}] \\left(\\frac{1 + x}{1 - x}\\right)^n$","Number of zeros in ternary representation of $2^n$","Array read by upward antidiagonals","Conjectures associated with A105210","Indicator sequence for 5 Fibonacci numbers with n digits","Triangular matchstick numbers in the class of prime numbers","Imaginary part of $\\prod_{k=0}^n (1 + k \\cdot i)$, $i = \\sqrt{-1}$","Fibonacci-Collatz sequence","Sum of squares of nonacci numbers","Fractional parts of sums of reciprocals of Catalan numbers $C(n)$","Riesel Problem","$a(n) = 16n^2 + 1$","Digital sum of the Fermat number $2^{2^n} + 1$","A108306: Expansion of $(3x+1)/(1-3x-3x^2)$","Numbers $n$ such that $\\phi(n) = \\phi(n + \\phi(n))$","Numbers $n$ such that the perfect deficiency of $n$ is $\\le 10$.","Numerator of $\\sum_{k=1}^n 2^k/k$.","Numerator of $\\binom{6n-2}{2n} / \\left(2 \\binom{4n-1}{2n}\\right)$","Conjectures associated with A109227","Conjectures associated with A109671","Conjectures associated with A109845","Conjectures associated with A109905","Conjectures associated with A109908","Conjectures associated with A109909","Number of symbols '*' and '^' to write the canonical prime factorization of n","$a(n) = \\operatorname{lcm}\\{1,2,\\dots,n\\}/\\operatorname{denom}(H(n))$","Smallest $m > 0$ such that there are no primes between $nm$ and $n(m+1)$ inclusive.","Conjectures associated with A110854","Integer part of $\\mathrm{prime}(n)/\\pi(n)$","Number of refactorable numbers (A033950) $\\le 10^n$","Sequence related to NOR bracketings","A generalized Stern sequence","Number of digits of n raised to the power of the sum of the digits of n","Number of digits of n raised to the power of the digital root of n","Smallest number $m$ such that $2^n - m$ and $2^n + m$ are primes","Expansion of g.f. $-(1 - 48x^2 - 256x^3) / ((1 - 4x)(1 + 4x)(1 + 4x + 16x^2))$","Conjectures associated with A113252","Square odd-indexed terms in the recurrence $a(n) = 8a(n-1) - 8a(n-2) + 8a(n-3) - a(n-4)$","Conjectures associated with A113255","Ascending descending base exponent transform of squares","Ascending descending base exponent transform of factorials","Ascending descending base exponent transform of $2^n$","Number of prime powers $q<=n$ such that also $q+2$ is a prime power","Difference between first odd semiprime $> 2^n$ and $2^n$","Largest odd divisor of $a(n-1) + \\textrm{prime}(n)$","Numerator of $\\zeta(4n)/\\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$)","$a(n) = 2^(2^n)$","Each term is previous term plus floor of harmonic mean of two previous terms.","Partial sums of $\\binom{2n}{n}^2$","$a(n)$ = the number of values of $k <= 10^n$ such that $\\sqrt{k(k+1)(k+2)(k+3)+1}$ is prime","$a(n)$ is the integer whose decimal digits are the first $n+1$ decimal digits of $\\pi$","Sum of squares of divisors of $n$","$a(n) = \\sum_{j=1}^{n} (3^j + (-2)^j)$","Determinants of 2 X 2 matrices of non-overlapping blocks of 4 consecutive primes","Number of primes in $n$-th row of triangle $k^2 - k + p_n$","Least $k$ such that cyclotomic polynomial $\\Phi_k(n)$ is prime","Sum of Fermat number and Mersenne number minus 1: $2^{2^n} + 2^n - 1$","Least $k \\ge 1$ such that $2 \\cdot n^k - 1$ is prime","Half-Fibonacci sequence","Prime gaps","Ratio of product of GCDs to product of factorials of floor divisions","Odious primes minus evil primes among first $n$ primes","Recurrence involving LCM: $a(n) = x(n+1)/x(n) - 2$","Lesser of twin primes","Number of Abelian cubes of length $3n$ over an alphabet of size 3","Factorial distance to nearest square","Collatz step differences","Representations as $p + 2^x + 7 \\cdot 2^y$ with $p \\equiv 5 \\pmod 6$","Representations as $p + 2^x + 11 \\cdot 2^y$ with $p \\equiv 1 \\pmod 6$","Smallest $m$ such that $n^3 + m^3 + 1$ is prime","Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers","Rowland-style prime-generating recurrence","Chua's Euclidean prime sequence","Smallest index $k > n$ such that $(p_k+p_{k+1})/(p_n+p_{n+1})$ is an integer $\\ge 2$","Denominators of $\\sum_{k=1}^n \\frac{1}{k 2^k}$","Recurrence with fourth powers of binomial coefficients","Denominator of sum of reciprocals of divisors","Central binomial sum $a(n) = \\sum_{k=0}^n (-4)^k \\binom{n}{k}^2 \\binom{n-k}{k}^2$","Central binomial sum $a(n) = \\sum_{k=0}^n \\binom{n}{k}^2 \\binom{n-k}{k}^2 (-16)^k$","Difference of digit sums in base 3 and base 2","Sum of fourth powers of Fibonacci-like binomial coefficients","Squares of double factorials","Product of two consecutive primes modulo the next prime","Recurrence with bitwise XOR","Number of odd primes between $n^2$ and $(n+1)^2$ with $(n/p) = 1$","Coefficients of $\\prod_{k>0} (1 - x^k/k!)$","Size of smallest subset of $\\{1, 2, \\dots, n\\}$ with distinct subset sums","Integrality and supercongruences of the factorial ratio $\\frac{(6n)! n!}{(3n)! (2n)!^2}$","A nonlinear recurrence sequence","Number of squares $\\bmod n$","Existence of integers $k$ with $k^2 \\operatorname{XOR} (k+1)^2 = (2n+1)^2$","Representation of sequence terms by harmonic numbers $\\lfloor \\frac{1}{2H(n) - H(n^2+n-1) - \\gamma} \\rfloor$","Eighth root of the generating function of an Apéry-like sequence","Numbers $n$ such that $n^2 + \\pi(n)$ is prime","Sum of two numbers with prime conditions","Representations with prime conditions","Multiplicative order of 2 mod $2n+1$","Number of parts in the symmetric representation of $\\sigma(n)$","Primitive roots of the form k² + 1","Cuban Primes","Central trinomial coefficients","Signed digit sums in bases $b \\ge 5$","Determinant of Hankel matrix of the first $2n-1$ prime numbers","Central factorial numbers: $((2n)!!)^2$","Smallest prime factors of $2^{2^n+2} + 3$","Partitions of $n$ into distinct non-squarefree parts for $n > 23$","Asymptotics of an inner sum formula for $a(n)$","A GCD-Driven Sequence with Universal Jumps","Recurrence for Rule 167 cellular automaton sequence","Maximal run lengths in the second MSB of Tribonacci numbers $T(n)$","Congruences $a(n+k) \\equiv a(n) \\pmod k$ for hypergeometric values ${}_2F_1(n, -n; ; -1)$","The 1680-Conjecture","Sum of four squares with square conditions","Period of $p$-th powers modulo $n$","Sum of a triangular number, a generalized pentagonal number, and a generalized heptagonal number","Combinatorial representation of the recurrence $a(n+2) = 2a(n+1) + 2a(n)$","Symmetry of digit sum differences $\\operatorname{sign}(S_5(k) - S_1(k))$","Coefficient of $(xyz)^n$ in $((x+y)(y+z)(z+x))^n$ equaling $\\binom{2n}{n}^3$","Step difference bound $a(n+1) - a(n) \\in \\{1, 2\\}$ in mass-redistribution cellular automata","Sum of two squares and two central-binomial-type terms","Sum of two squares, a power of 3, and a power of 5","Maximality of $k = 43$ with restricted digit counts in bases $3 \\le b < k$","The 2-4-6-8 Conjecture","Vanishing of bases with $b^n \\equiv -1 \\pmod{2n+1}$ for absolute Euler pseudoprimes","Four-square conjecture with powers of 2, 3, and 5","Characterization of Carmichael numbers via squarefree denominators $a(n)$","A binomial coefficient sum","A binomial coefficient summation","Nonnegativity of the Dirichlet square root of A046644","Pronic indices for odd coefficients of $\\sum_{n \\ge 0} x^n \\frac{(1+x^n)^n}{(1+x^{n+1})^{n+1}}$","Pronic indices for odd coefficients of $\\sum_{n \\ge 0} x^n \\frac{(1+x^n)^n}{(1-x^{n+1})^{n+1}}$","Convergence of fraction numerators $a(n)/b(n)$ to $e$","Bounds on fractional parts $n r^2 - a(n)$ for $r = (2+\\sqrt{5})/2$","Smallest number $k$ such that $kn + 1$ is prime","Smallest prime $\\equiv 1 \\pmod n$","Numerator of a sum involving binomial coefficients","Factor bounds for Fermat numbers","Primes congruent to $\\{3, 5, 6\\} \\pmod 7$","Primality of continued fraction denominators $a(n)$ for $n \\ge 3$","Realization of primes $p \\equiv \\pm 1 \\pmod{10}$ by continued fraction denominators","Integrality of the factorial ratio $\\frac{(12n+6)! (6n+9)!}{108 (4n+2)! (2n+3)! ((6n+5)!)^2}$","Home primes (OEIS A037274)","Unique realization of odd primes $p \\notin \\{3, 5\\}$ by continued fraction values","Number of primes $< n^3$","Number of primes $< n^2$","Periodicity of $k$-th prime factors in coupled nonlinear recurrence $a(n)$","Conjectures associated with A038552","Smallest composite $c$ such that $\\textrm{primorial}(n) + c$ is prime","The prime numbers","No powers as partition numbers","Least positive multiple of $n$ in base 10 with digits 0 and 1","Denominators of coefficients in Stirling's expansion for $\\log(\\Gamma(z))$","$a(n) = \\sum_{k=1}^n (k^2 \\bmod n)$","Nearest integer to $n/\\sqrt{2}$","Asymptotics of subsets of $\\{1, 2, \\dots, n\\}$ with integer average","Practical numbers","Maximum exponent in the prime factorization of $n$","Apéry numbers","$a(n) = (\\text{smallest prime} > n^2) - n^2$","Concatenation of the next $n$ numbers","Catalan-Larcombe-French sequence","Least $m$ such that $\\phi(m) = n!$","Divisibility of $2^n + 1$ by $n$","Numerator of $1/\\det(M)$ for $M[i,j] = 1/\\operatorname{lcm}(i,j)$","Reversible multiples $a(3^n) = 10^{3^{n-2}} - 1$ for powers of $3$","Conjectures associated with A063880","Numerator - denominator in $n$-th harmonic number","Decimal encoding of the prime factorization of $n$","Conjectures associated with A067720","Sum of $a(k)/k!$ over divisors equals harmonic number","Number of times $n^2 + s^2$ is prime for positive integers $s < n$","Number of primes $p$ such that $n^n \\le p \\le n^n + n^2$","Number of primes $p$ such that $2^n \\le p \\le 2^n + \\mathrm{prime}(n)$","Catalan-Mersenne numbers","Value of $n$-th cyclotomic polynomial at $n$","Recurrence $a(n+2) = |a(n+1)a(n) - a(n)a(n+1)|$ via concatenation","Determinant of matrix with entries indicating primality of $i^2 + j^2$","Alternating sum of signs of powers of $3/2$","Smallest factorial containing exactly $n$ 6's","Wolstenholme numbers","Sum of next $n$ primes","Number of times $n$ occurs as a binary sub-pattern of $n^2$","Smallest $x$ such that $\\sigma(x) \\bmod x = n$","Trajectory of 103 under the Reverse and Add! operation in base 3","$a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$","Smallest $m > 0$ such that $n \\cdot 2^m + 1$ is prime","Least $k > 0$ such that $(k+1)(k+2)\\cdots(k+n) + 1$ is prime","Least prime $\\ge n$","$a(n) = \\sum_{k=0}^n \\binom{2k}{k}^3$","Number of prime powers strictly between $n$-th prime and $(n+1)$-th prime","Conjecture relating two characterizations of a set of integers.","Denominator of $\\sum_{k=1}^n k^{\\mu(k)}$","Primes of the form 2^n + 2^i + 1","Smallest palindrome with exactly $n$ divisors","Smallest prime $p$ such that $p + n$ is an $n$-th power","Smallest $r$ such that (concatenation of $n$, $r$ times) $\\cdot 10 + 1$ is prime","Binary representation of primes that divide a number, in decimal","Expansion of $(1 - x)/(1 - 2 x + 3 x^2)$","Smallest prime formed by concatenation $n, n-1, \\dots, n-k$","Number of pairs of twin primes between $n^2$ and $(n+1)^2$","Primality and primitive root property from divisibility $n \\mid (a(n-1) + 2^{n-2})$","Tug of war score between prime gap increases and decreases","Product of composite numbers in triangular intervals","$\\gcd(\\mathrm{numerator}(H_n), n!)$","Euclid-Mullin sequence","Recurrence $a(n) = (a(n-1) + a(n-2)) \\pmod n$","The Catch-Up game and conjecture","Dubner's conjecture","Gourevitch's series identity","Kurepa's conjecture","Prime Tuples Conjecture","Serre's uniformity conjecture over the rationals","Zagier's Conjecture on Multiple Zeta Values","Agoh-Giuga conjecture","Agrawal's conjecture","Normality of Irrational Algebraic Numbers","Non-Power-of-2 Almost Perfect Numbers Conjecture","Amicable numbers","Andrica's conjecture","Artin's conjecture on primitive roots","Balanced prime conjecture","Bateman-Horn Conjecture","Beal conjecture","Betrothed numbers","Brocard's Conjecture","Büchi's problem","Bunyakovsky conjecture","Carmichael's totient function conjecture","Class number problem for real quadratic fields","Congruent Number","Dickson's conjecture","Diophantine $m$-tuples","Elliott–Halberstam conjecture","Some conjectures about ranks of elliptic curves over ℚ","The Erdős–Moser equation","Euclid Numbers conjecture","Euler's sum of powers conjecture","Factorial primes","Feit-Thompson conjecture on primes","Open questions about Fermat numbers","Fermat-Catalan conjecture","Fibonacci Primes","Firoozbakht's conjecture","Fortune's Conjecture","Gauss circle problem","Gilbreath's conjecture","The Goormaghtigh conjecture","Grimm's conjecture","Hall's conjecture","First Hardy–Littlewood conjecture","The Hasse--Weil conjecture for elliptic curves","Idoneal numbers completeness conjecture","Juggler conjecture","Kummer–Vandiver conjecture","Lander, Parkin, and Selfridge Conjecture","Legendre's conjecture","Lehmer's Mahler measure problem","Lehmer's totient problem","Lemoine's conjectures","Littlewood conjectures","Lonely runner conjecture","Lychrel numbers in base 10","Magic Squares","Mahler's 3/2 Problem","Conjectures about Mersenne primes","Mills' Theorem","Normality of mathematical constants","Oppermann's Conjecture","Infinitude of Pell number primes","Perfect numbers","Pierpont primes","Pollock's (tetrahedral numbers) conjecture","Asymptotic density of powerful numbers","Prime Triplet Conjecture","Primes and perfect squares","Quasiperfect Numbers","Ramanujan τ-function","Rational distance problem","Infinite Regular Primes","Particular values of the Riemann zeta function","Rudin's conjecture on squares in arithmetic progressions","Schanuel's Conjecture","Hypothesis H","Scholz conjecture on addition chains","Selfridge's conjectures","Sierpiński number","Singmaster's conjecture","Solitary Numbers","Sum of three cubes","(m,k)-perfect numbers","Taxicab numbers","Open questions on transcendence of numbers","Infinitude of Wall–Sun–Sun primes","Can a prime $p$ satisfy $2^{p-1} \\equiv 1 \\pmod{p^2}$ and $3^{p-1} \\equiv 1 \\pmod{p^2}$?","Wieferich primes","Wilson primes","Wolstenholme Prime","Woodall Primes","Erdős Problem 8","Erdős Problem 27","Erdős Problem 37","Erdős Problem 49","Erdős Problem 54","Erdős Problem 55","Erdős Problem 122","Erdős Problem 131","Erdős Problem 144","Erdős Problem 148","Erdős Problem 177","Erdős Problem 222","Erdős Problem 235","Erdős Problem 237","Erdős Problem 240","Erdős Problem 255","Erdős Problem 262","Erdős Problem 265","Erdős Problem 278","Erdős Problem 284","Erdős Problem 293","Erdős Problem 294","Erdős Problem 297","Erdős Problem 301","Erdős Problem 311","Erdős Problem 320","Erdős Problem 327","Erdős Problem 334","Erdős Problem 335","Erdős Problem 336","Erdős Problem 338","Erdős Problem 339","Erdős Problem 343","Erdős Problem 344","Erdős Problem 345","Erdős Problem 356","Erdős Problem 360","Erdős Problem 365","Erdős Problem 368","Erdős Problem 374","Erdős Problem 378","Erdős Problem 380","Erdős Problem 381","Erdős Problem 382","Erdős Problem 388","Erdős Problem 393","Erdős Problem 404","Erdős Problem 408","Erdős Problem 411","Erdős Problem 415","Erdős Problem 420","Erdős Problem 425","Erdős Problem 430","Erdős Problem 432","Erdős Problem 436","Erdős Problem 439","Erdős Problem 444","Erdős Problem 446","Erdős Problem 449","Erdős Problem 451","Erdős Problem 460","Erdős Problem 461","Erdős Problem 465","Erdős Problem 467","Erdős Problem 468","Erdős Problem 471","Erdős Problem 472","Erdős Problem 475","Erdős Problem 482","Erdős Problem 492","Erdős Problem 496","Erdős Problem 526","Erdős Problem 530","Erdős Problem 531","Erdős Problem 543","Erdős Problem 635","Erdős Problem 656","Erdős Problem 663","Erdős Problem 675","Erdős Problem 676","Erdős Problem 679","Erdős Problem 684","Erdős Problem 685","Erdős Problem 687","Erdős Problem 691","Erdős Problem 693","Erdős Problem 696","Erdős Problem 708","Erdős Problem 709","Erdős Problem 710","Erdős Problem 711","Erdős Problem 731","Erdős Problem 768","Erdős Problem 771","Erdős Problem 782","Erdős Problem 783","Erdős Problem 784","Erdős Problem 795","Erdős Problem 820","Erdős Problem 823","Erdős Problem 824","Erdős Problem 843","Erdős Problem 852","Erdős Problem 854","Erdős Problem 856","Erdős Problem 860","Erdős Problem 861","Erdős Problem 866","Erdős Problem 870","Erdős Problem 874","Erdős Problem 878","Erdős Problem 879","Erdős Problem 880","Erdős Problem 882","Erdős Problem 892","Erdős Problem 896","Erdős Problem 928","Erdős Problem 929","Erdős Problem 935","Erdős Problem 941","Erdős Problem 954","Erdős Problem 963","Erdős Problem 964","Erdős Problem 969","Erdős Problem 976","Erdős Problem 980","Erdős Problem 981","Erdős Problem 983","Erdős Problem 988","Erdős Problem 989","Erdős Problem 991","Erdős Problem 992","Erdős Problem 1001","Erdős Problem 1053","Erdős Problem 1081","Erdős Problem 1100","Erdős Problem 1103","Erdős Problem 1122","Erdős Problem 1140","Erdős Problem 1143","Erdős Problem 1144","Erdős Problem 1149","Erdős Problem 1164","Erdős Problem 1165","Erdős Problem 1166","Erdős Problem 1181","Erdős Problem 1184","Erdős Problem 1189","Erdős Problem 1195","Erdős Problem 1200","Erdős Problem 1202","Erdős Problem 1204","Erdős Problem 1205","Erdős Problem 1211","Erdős Problem 1217","Modularity theorem","Geometric Langlands conjecture","Sato–Tate conjecture","Langlands reciprocity (general)","Jacobian conjecture, n ≥ 3","Jacobian conjecture in the plane","Hodge conjecture","Weil conjectures","Resolution of singularities in characteristic p","Generic and maximal rank of 3-tensors","The Poisson Conjecture","Hartshorne's conjecture on Vector Bundles","Poincaré conjecture","Smooth 4-dimensional Poincaré conjecture","Kepler conjecture","Sphere packing in dimensions 8 and 24","Kakeya conjecture in three dimensions","Kakeya set conjecture, all dimensions","Einstein problem (aperiodic monotile)","Moving sofa problem","Squaring the circle","The first Atiyah--Sutcliffe conjecture","Erdős Problem 1082","Erdős Problem 1088","Erdős Problem 1089","Erdős Problem 352","Erdős Problem 502","Erdős Problem 503","Erdős Problem 506","Erdős Problem 507","Erdős Problem 660","Erdős Problem 898","Ben Green's Open Problem 41","Green's Open Problem 42","Mathoverflow 34145","Integer part of area of a regular polygon with $n$ sides each of length 1","Conjecture about cardinality of Lindelöf spaces","Conjectures around homogeneous topological spaces","Conjectures about Weakly First Countable spaces","The Bing-Borsuk Conjecture","Inscribed square problem","Packing","Toronto spaces","Four color theorem","Ramsey number R(5,5)","Hadwiger–Nelson problem","Union-closed sets conjecture","Boolean Pythagorean triples problem","Kahn–Kalai conjecture","The length of an $s$-increasing sequence of $r$-tuples","Ringel's Conjecture and Kotzig's Conjecture for large $n$","Barker sequences","Independent Domination of Regular Graphs, Conjecture 1.6","Digit $2$ in base $3$ representation of $2^n$","TxGraffiti Conjecture 1: independence, annihilation, and residue","TxGraffiti Conjecture 2: zero forcing versus independence (cubic graphs)","TxGraffiti Conjecture 3: independent domination versus saturation (regular)","TxGraffiti Conjecture 4: the saturation number versus the harmonic index","First Proof, Theorem 6","Dean's conjecture on cycles of length divisible by `k`","Bondy's conjecture on longest cycles in highly connected graphs","The Alon-Tarsi short cycle cover conjecture","Erdős Problem 1","Erdős Problem 10","Erdős Problem 1007","Erdős Problem 1008","Erdős Problem 1009","Erdős Problem 1010","Erdős Problem 1014","Erdős Problem 1020","Erdős Problem 1021","Erdős Problem 1022","Erdős Problem 1023","Erdős Problem 1025","Erdős Problem 1026","Erdős Problem 1027","Erdős Problem 1028","Erdős Problem 1029","Erdős Problem 1030","Erdős Problem 1034","Erdős Problem 1035","Erdős Problem 1036","Erdős Problem 1037","Erdős Problem 105","Erdős Problem 1067","Erdős Problem 1068","Erdős Problem 1077","Erdős Problem 1079","Erdős Problem 108","Erdős Problem 1080","Erdős Problem 109","Erdős Problem 1090","Erdős Problem 1092","Erdős Problem 1093","Erdős Problem 1098","Erdős Problem 1104","Erdős Problem 1105","Erdős Problem 1109","Erdős Problem 1110","Erdős Problem 113","Erdős Problem 1145","Erdős Problem 1159","Erdős Problem 1167","Erdős Problem 1175","Erdős Problem 1179","Erdős Problem 1187","Erdős Problem 1190","Erdős Problem 1192","Erdős Problem 1193","Erdős Problem 1199","Erdős Problem 120","Erdős Problem 1206","Erdős Problem 127","Erdős Problem 128","Erdős Problem 13","Erdős Problem 130","Erdős Problem 133","Erdős Problem 134","Erdős Problem 136","Erdős Problem 139","Erdős Problem 141","Erdős Problem 146","Erdős Problem 147","Erdős Problem 150","Erdős Problem 152","Erdős Problem 153","Erdős Problem 154","Erdős Problem 155","Erdős Problem 156","Erdős Problem 158","Erdős Problem 159","Erdős Problem 160","Erdős Problem 163","Erdős Problem 166","Erdős Problem 168","Erdős Problem 170","Erdős Problem 171","Erdős Problem 172","Erdős Problem 180","Erdős Problem 181","Erdős Problem 183","Erdős Problem 184","Erdős Problem 185","Erdős Problem 188","Erdős Problem 189","Erdős Problem 191","Erdős Problem 192","Erdős Problem 193","Erdős Problem 194","Erdős Problem 195","Erdős Problem 196","Erdős Problem 197","Erdős Problem 198","Erdős Problem 199","Erdős Problem 20","Erdős Problem 200","Erdős Problem 202","Erdős Problem 203","Erdős Problem 204","Erdős Problem 21","Erdős Problem 214","Erdős Problem 219","Erdős Problem 22","Erdős Problem 228","Erdős Problem 23","Erdős Problem 231","Erdős Problem 236","Erdős Problem 24","Erdős Problem 241","Erdős Problem 245","Erdős Problem 272","Erdős Problem 273","Erdős Problem 282","Erdős Problem 285","Erdős Problem 295","Erdős Problem 303","Erdős Problem 312","Erdős Problem 316","Erdős Problem 319","Erdős Problem 326","Erdős Problem 329: Maximum Density of Sidon Sets","Erdős Problem 330","Erdős Problem 337","Erdős Problem 34","Erdős Problem 340","Erdős Problem 342","Erdős Problem 350","Erdős Problem 358","Erdős Problem 36","Erdős Problem 395","Erdős Problem 42: Maximal Sidon Sets and Disjoint Difference Sets","Erdős Problem 423","Erdős Problem 426","Erdős Problem 43","Erdős Problem 435","Erdős Problem 44: Extending Sidon Sets","Erdős Problem 447","Erdős Problem 45","Erdős Problem 46","Erdős Problem 476","Erdős Problem 481","Erdős Problem 484","Erdős Problem 488","Erdős Problem 494","Erdős Problem 497","Erdős Problem 498","Erdős Problem 501","Erdős Problem 532","Erdős Problem 533","Erdős Problem 534","Erdős Problem 535","Erdős Problem 539","Erdős Problem 540","Erdős Problem 544","Erdős Problem 545","Erdős Problem 546","Erdős Problem 547","Erdős Problem 548","Erdős Problem 549","Erdős Problem 550","Erdős Problem 551","Erdős Problem 552","Erdős Problem 559","Erdős Problem 562","Erdős Problem 563","Erdős Problem 564","Erdős Problem 566","Erdős Problem 567","Erdős Problem 568","Erdős Problem 569","Erdős Problem 57","Erdős Problem 570","Erdős Problem 571","Erdős Problem 572","Erdős Problem 579","Erdős Problem 58","Erdős Problem 581","Erdős Problem 582","Erdős Problem 583","Erdős Problem 593","Erdős Problem 595","Erdős Problem 596","Erdős Problem 599","Erdős Problem 60","Erdős Problem 600","Erdős Problem 609","Erdős Problem 61 -- Erdős–Hajnal Conjecture","Erdős Problem 613","Erdős Problem 615","Erdős Problem 617","Erdős Problem 618","Erdős Problem 619","Erdős Problem 621","Erdős Problem 624","Erdős Problem 628","Erdős Problem 63","Erdős Problem 633","Erdős Problem 639","Erdős Problem 64","Erdős Problem 645","Erdős Problem 65","Erdős Problem 650","Erdős Problem 653","Erdős Problem 655","Erdős Problem 658","Erdős Problem 664","Erdős Problem 666","Erdős Problem 698","Erdős Problem 701","Erdős Problem 703","Erdős Problem 705","Erdős Problem 707: Embedding Sidon Sets in Perfect Difference Sets","Erdős Problem 71","Erdős Problem 713","Erdős Problem 714","Erdős Problem 72","Erdős Problem 720","Erdős Problem 723: The prime power conjecture.","Erdős Problem 73","Erdős Problem 74","Erdős Problem 740: Infinitary version of chromatic number and odd cycles","Erdős Problem 741","Erdős Problem 742","Erdős Problem 748","Erdős Problem 75","Erdős Problem 750","Erdős Problem 751","Erdős Problem 753","Erdős Problem 757","Erdős Problem 760","Erdős Problem 762","Erdős Problem 765","Erdős Problem 767","Erdős Problem 77","Erdős Problem 772","Erdős Problem 774","Erdős Problem 775","Erdős Problem 780","Erdős Problem 781","Erdős Problem 789","Erdős Problem 79","Erdős Problem 794","Erdős Problem 798","Erdős Problem 80","Erdős Problem 800","Erdős Problem 801","Erdős Problem 803","Erdős Problem 804","Erdős Problem 812","Erdős Problem 814","Erdős Problem 816","Erdős Problem 817","Erdős Problem 818","Erdős Problem 82","Erdős Problem 83","Erdős Problem 835","Erdős Problem 842","Erdős Problem 844","Erdős Problem 85","Erdős Problem 857","Erdős Problem 86","Erdős Problem 862","Erdős Problem 865","Erdős Problem 867","Erdős Problem 868","Erdős Problem 87","Erdős Problem 872","Erdős Problem 877","Erdős Problem 881","Erdős Problem 883","Erdős Problem 893","Erdős Problem 894","Erdős Problem 895","Erdős Problem 899","Erdős Problem 9","Erdős Problem 904","Erdős Problem 905","Erdős Problem 914","Erdős Problem 915","Erdős Problem 918","Erdős Problem 920","Erdős Problem 921","Erdős Problem 922","Erdős Problem 923","Erdős Problem 926","Erdős Problem 927","Erdős Problem 94","Erdős Problem 948","Erdős Problem 949","Erdős Problem 95","Erdős Problem 958","Erdős Problem 966","Erdős Problem 986","Ben Green's Open Problem 1","Green's Open Problem 12","Ben Green's Open Problem 14","Green's Open Problem 15","Ben Green's Open Problem 16","Ben Green's Open Problem 18","Ben Green's Open Problem 19","Ben Green's Open Problem 21","Green's Open Problem 22","Green's Open Problem 23","Green's Open Problem 24","Green's Open Problem 25","Green's Open Problem 26","Green's Open Problem 27","Ben Green's Open Problem 31","Green's Open Problem 32","Ben Green's Open Problem 33","Green's Open Problem 36","Ben Green's Open Problem 37","Green's Open Problem 38","Green's Open Problem 39","Ben Green's Open Problem 40","Green's Open Problem 49","Ben Green's Open Problem 5","Ben Green's Open Problem 50","Green's Open Problem 51","Green's Open Problem 52","Green's Open Problem 53","Ben Green's Open Problem 57","Ben Green's Open Problem 58","Ben Green's Open Problem 72","Ben Green's Open Problem 77","Green's Open Problem 9","Optimal monotone families for the discrete isoperimetric inequality","$a(n) = \\sum_{i=0}^n \\binom{2n-i}{n+i}$","Expansion of g.f. $(1-\\sqrt{1-4x-4x^2})/(2(1+x))$","Number of different products of subsets of $\\{1, 2, \\dots, n\\}$","Open Quantum Problem 13: Mutually unbiased bases","Open Quantum Problem 35: existence of absolutely maximally entangled pure states","Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets","Beaver Math Olympiad (BMO)","Suffix-prefix avoidance bound","VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²","Chvátal's Conjecture","Claude's Cycles","DegreeSequencesTriangleFree","Main conjecture on fusible numbers","Kotzig's Conjecture","Conjectures about Latin Squares","Latin Tableau Conjecture","Monochromatic quantum graphs (inherited vertex colorings)","Reed's omega, delta, and chi conjecture","Ringel's Conjecture","Size, Order, and Connected Domination","Tu-Deng Conjecture","Beck–Fiala theorem and conjecture","Conway's 99-graph problem","Dedekind Numbers","Babai–Seress Conjectures on the Diameter of Finite Groups","Graceful Tree Conjecture (Ringel–Kotzig conjecture)","Komlós conjecture","The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture","Pebbling number conjecture","Sidorenko's conjecture (1993)","Six standard deviations suffice (Spencer's theorem)","Snake in the box","Sparse Ruler","Steiner Systems","Vizing's conjecture (1968)","The $\\frac 1 3$–$\\frac 2 3$ conjecture","Written on the Wall II - Conjecture 160","Written on the Wall II - Conjecture 1","Written on the Wall II - Conjecture 100","Written on the Wall II - Conjecture 101","Written on the Wall II - Conjecture 103","Written on the Wall II - Conjecture 109","Written on the Wall II - Conjecture 13","Written on the Wall II - Conjecture 133","Written on the Wall II - Conjecture 141","Written on the Wall II - Conjecture 142","Written on the Wall II - Conjecture 143","Written on the Wall II - Conjecture 144","Written on the Wall II - Conjecture 145","Written on the Wall II - Conjecture 146","Written on the Wall II - Conjecture 16","Written on the Wall II - Conjecture 17","Written on the Wall II - Conjecture 172","Written on the Wall II - Conjecture 176","Written on the Wall II - Conjecture 18","Written on the Wall II - Conjecture 181","Written on the Wall II - Conjecture 19","Written on the Wall II - Conjecture 194","Written on the Wall II - Conjecture 198a","Written on the Wall II - Conjecture 2","Written on the Wall II - Conjecture 20","Written on the Wall II - Conjecture 200","Written on the Wall II - Conjecture 217","Written on the Wall II - Conjecture 23","Written on the Wall II - Conjecture 291","Written on the Wall II - Conjecture 3","Written on the Wall II - Conjecture 309","Written on the Wall II - Conjecture 31","Written on the Wall II - Conjecture 314","Written on the Wall II - Conjecture 315","Written on the Wall II - Conjecture 316","Written on the Wall II - Conjecture 32","Written on the Wall II - Conjecture 322","Written on the Wall II - Conjecture 327","Written on the Wall II - Conjecture 33","Written on the Wall II - Conjecture 34","Written on the Wall II - Conjecture 36","Written on the Wall II - Conjecture 4","Written on the Wall II - Conjecture 40","Written on the Wall II - Conjecture 430a","Written on the Wall II - Conjecture 438b","Written on the Wall II - Conjecture 5","Written on the Wall II - Conjecture 58","Written on the Wall II - Conjecture 59","Written on the Wall II - Conjecture 6","Written on the Wall II - Conjecture 61","Written on the Wall II - Conjecture 63","Written on the Wall II - Conjecture 65","Written on the Wall II - Conjecture 7","Written on the Wall II - Conjecture 85","Erdős Problem 19","Erdős Problem 59","Erdős Problem 62","Erdős Problem 76","Erdős Problem 78","Erdős Problem 81","Erdős Problem 84","Erdős Problem 88","Erdős Problem 110","Erdős Problem 111","Erdős Problem 112","Erdős Problem 117","Erdős Problem 129","Erdős Problem 140","Erdős Problem 149","Erdős Problem 151","Erdős Problem 157","Erdős Problem 161","Erdős Problem 162","Erdős Problem 165","Erdős Problem 167","Erdős Problem 169","Erdős Problem 176","Erdős Problem 179","Erdős Problem 182","Erdős Problem 186","Erdős Problem 187","Erdős Problem 190","Erdős Problem 201","Erdős Problem 207","Erdős Problem 271","Erdős Problem 483","Erdős Problem 500","Erdős Problem 518","Erdős Problem 553","Erdős Problem 554","Erdős Problem 555","Erdős Problem 556","Erdős Problem 557","Erdős Problem 558","Erdős Problem 560","Erdős Problem 561","Erdős Problem 565","Erdős Problem 573","Erdős Problem 574","Erdős Problem 575","Erdős Problem 576","Erdős Problem 577","Erdős Problem 578","Erdős Problem 580","Erdős Problem 584","Erdős Problem 585","Erdős Problem 597","Erdős Problem 601","Erdős Problem 603","Erdős Problem 608","Erdős Problem 610","Erdős Problem 611","Erdős Problem 612","Erdős Problem 614","Erdős Problem 616","Erdős Problem 620","Erdős Problem 622","Erdős Problem 625","Erdős Problem 626","Erdős Problem 627","Erdős Problem 629","Erdős Problem 630","Erdős Problem 631","Erdős Problem 632","Erdős Problem 636","Erdős Problem 637","Erdős Problem 638","Erdős Problem 640","Erdős Problem 641","Erdős Problem 642","Erdős Problem 643","Erdős Problem 644","Erdős Problem 665","Erdős Problem 667","Erdős Problem 702","Erdős Problem 704","Erdős Problem 706","Erdős Problem 712","Erdős Problem 715","Erdős Problem 716","Erdős Problem 717","Erdős Problem 718","Erdős Problem 719","Erdős Problem 721","Erdős Problem 722","Erdős Problem 724","Erdős Problem 725","Erdős Problem 732","Erdős Problem 733","Erdős Problem 734","Erdős Problem 736","Erdős Problem 737","Erdős Problem 738","Erdős Problem 739","Erdős Problem 743","Erdős Problem 744","Erdős Problem 745","Erdős Problem 746","Erdős Problem 747","Erdős Problem 752","Erdős Problem 758","Erdős Problem 759","Erdős Problem 761","Erdős Problem 766","Erdős Problem 776","Erdős Problem 777","Erdős Problem 778","Erdős Problem 787","Erdős Problem 788","Erdős Problem 790","Erdős Problem 791","Erdős Problem 792","Erdős Problem 797","Erdős Problem 799","Erdős Problem 802","Erdős Problem 805","Erdős Problem 807","Erdős Problem 808","Erdős Problem 809","Erdős Problem 810","Erdős Problem 811","Erdős Problem 813","Erdős Problem 815","Erdős Problem 819","Erdős Problem 832","Erdős Problem 833","Erdős Problem 834","Erdős Problem 836","Erdős Problem 837","Erdős Problem 840","Erdős Problem 863","Erdős Problem 864","Erdős Problem 875","Erdős Problem 876","Erdős Problem 900","Erdős Problem 901","Erdős Problem 902","Erdős Problem 903","Erdős Problem 910","Erdős Problem 911","Erdős Problem 916","Erdős Problem 917","Erdős Problem 919","Erdős Problem 924","Erdős Problem 925","Erdős Problem 934","Erdős Problem 984","Erdős Problem 993","Erdős Problem 1006","Erdős Problem 1011","Erdős Problem 1012","Erdős Problem 1013","Erdős Problem 1015","Erdős Problem 1016","Erdős Problem 1017","Erdős Problem 1018","Erdős Problem 1019","Erdős Problem 1024","Erdős Problem 1031","Erdős Problem 1032","Erdős Problem 1033","Erdős Problem 1066","Erdős Problem 1075","Erdős Problem 1076","Erdős Problem 1078","Erdős Problem 1091","Erdős Problem 1111","Erdős Problem 1112","Erdős Problem 1123","Erdős Problem 1155","Erdős Problem 1156","Erdős Problem 1157","Erdős Problem 1158","Erdős Problem 1160","Erdős Problem 1161","Erdős Problem 1162","Erdős Problem 1163","Erdős Problem 1177","Erdős Problem 1178","Erdős Problem 1182","Erdős Problem 1183","Erdős Problem 1185","Erdős Problem 1186","Erdős Problem 1191","Erdős Problem 1194","Erdős Problem 1198","Erdős Problem 1216","Erdős Problem 1218","Erdős Problem 1219","Erdős Problem 1220","Navier–Stokes existence and smoothness","Kadison–Singer problem","Bieberbach conjecture","Invariant subspace problem","Spectral sets and weak tiling","First Proof, Theorem 4","Banach-Mazur Rotation Problem","Convergence of the Borwein Series with Sinusoidal Coefficient","Erdős Problem 1038","Erdős Problem 1043","Erdős Problem 1044","Erdős Problem 1046","Erdős Problem 1047","Erdős Problem 1048","Erdős Problem 1125","Erdős Problem 1126","Erdős Problem 1133","Erdős Problem 115","Erdős Problem 116","Erdős Problem 119","Erdős Problem 225","Erdős Problem 226","Erdős Problem 229","Erdős Problem 230","Erdős Problem 243","Erdős Problem 268","Erdős Problem 353","Erdős Problem 509","Erdős Problem 513","Erdős Problem 516","Erdős Problem 517","Erdős Problem 519","Erdős Problem 525","Erdős Problem 906","Erdős Problem 907","Erdős Problem 996","Ben Green's Open Problem 35","Ben Green's Open Problem 54","Green's Open Problem 85","Ben Green's Open Problem 94","Mathoverflow 235893","Mathoverflow 347178","De Giorgi's conjecture","Energy-critical NLS in three dimensions.","Voronovskaja-type Formula for the Bezier Variant of the Bernstein Operators","Weak tiling problems","Bloch and Landau constants","Brennan's Conjecture","Falconer's distance set conjecture","Convergence of the Flint Hills and Cookson Hills series","Fuglede's conjecture in dimensions 1 and 2","Goodman's conjecture on coefficients of $p$-valent functions","The Pompeiu problem","Erdős Problem 114","Erdős Problem 227","Erdős Problem 256","Erdős Problem 511","Erdős Problem 514","Erdős Problem 515","Erdős Problem 523","Erdős Problem 524","Erdős Problem 527","Erdős Problem 671","Erdős Problem 908","Erdős Problem 909","Erdős Problem 995","Erdős Problem 998","Erdős Problem 1039","Erdős Problem 1040","Erdős Problem 1042","Erdős Problem 1045","Erdős Problem 1114","Erdős Problem 1115","Erdős Problem 1116","Erdős Problem 1117","Erdős Problem 1118","Erdős Problem 1120","Erdős Problem 1129","Erdős Problem 1130","Erdős Problem 1131","Erdős Problem 1132","Erdős Problem 1151","Erdős Problem 1152","Erdős Problem 1153","Erdős Problem 1154","Erdős Problem 1215","Erdős Problem 1221","Continuum hypothesis","Hilbert's second problem: consistency of arithmetic","Whitehead problem","Boxdot Conjecture","Erdős Problem 1119","Erdős Problem 1128","Erdős Problem 1176","Erdős Problem 590","Erdős Problem 591","Erdős Problem 592","Erdős Problem 594","Erdős Problem 598","Erdős Problem 602","Erdős Problem 623","Erdős Problem 70","Erdős Problem 965","Decidability of reachability for branching vector addition systems","Vaught conjecture","Erdős Problem 118","Erdős Problem 474","Erdős Problem 1168","Erdős Problem 1169","Erdős Problem 1170","Erdős Problem 1171","Erdős Problem 1172","Erdős Problem 1173","Erdős Problem 1174","Solving the quintic by radicals","Classification of finite simple groups","Inverse Galois problem","Burnside problem","Zariski Cancellation","The circulant Hadamard conjecture","Group structure via subgroup counts","Fernandes' conjecture on the 2-generation of even direct product permutation groups","The microscopic weighting on a metric space","Finite generation of the cohomology of finite-dimensional Hopf algebras","Algebraic consequences of the Farrell–Jones conjecture","Erdős Problem 1150","Erdős Problem 274","Erdős Problem 477","Erdős Problem 499","Erdős Problem 522","Erdős Problem 990","Ben Green's Open Problem 29","Ben Green's Open Problem 4","Hilbert's 17th problem","Hilbert's Fifth Problem and the Hilbert–Smith Conjecture","Conjecture 19.25","Conjecture 1.35(c)","Conjecture 1.40","Conjecture 1.74 (Minimal topological groups)","Conjecture 20.76","Conjecture 8.8","Mathoverflow 21003","Mathoverflow 31809","Mathoverflow 339137","Mathoverflow 486451","Mathoverflow 507128","Open Quantum Problem 23: SIC-POVMs","Equational Theories","The Gerstenhaber problem for three commuting matrices","Schur's theorem on Galois groups of truncated exponential polynomials","The Auslander-Reiten conjecture","Casas-Alvero Conjecture","The $S_3$-conjecture (conjugacy classes of distinct sizes)","The Andrews-Curtis conjecture","Determinantal conjecture","Open questions regarding the existence of Euler bricks","Gap conjecture","Gromov's theorem on groups of polynomial growth","Hadamard's conjecture","Jacobson Conjecture","Kaplansky's Conjectures","Köthe conjecture","Leinster Groups","Local uniformization","Mean value problem","Rational_variety","Pierce–Birkhoff conjecture","Sendov's conjecture","Gottschalk's surjunctivity conjecture","MLC: the Mandelbrot set is locally connected","The three-body problem","Furstenberg's `times p, times q` conjectures","The Rule 30 Prize Problems","Planar $L_p$-Rogers-Shephard, the equality case","Erdős Problem 100","Erdős Problem 101","Erdős Problem 104","Erdős Problem 107","Erdős Problem 1071","Erdős Problem 1083","Erdős Problem 1084","Erdős Problem 1085","Erdős Problem 1121","Erdős Problem 1207","Erdős Problem 209","Erdős Problem 212","Erdős Problem 213","Erdős Problem 215","Erdős Problem 224","Erdős Problem 505","Erdős Problem 508","Erdős Problem 659","Erdős Problem 755","Erdős Problem 756","Erdős Problem 769","Erdős Problem 89","Erdős Problem 90: The unit distance problem","Erdős Problem 91","Erdős Problem 92","Erdős Problem 93","Erdős Problem 959","Erdős Problem 96","Erdős Problem 97","Erdős Problem 98","Erdős Problem 982","Erdős Problem 99","Is Every Convex Polyhedron Rupert?","Borsuk's conjecture","Moser's Worm","Erdős Problem 102","Erdős Problem 103","Erdős Problem 106","Erdős Problem 132","Erdős Problem 135","Erdős Problem 173","Erdős Problem 174","Erdős Problem 210","Erdős Problem 211","Erdős Problem 216","Erdős Problem 217","Erdős Problem 223","Erdős Problem 232","Erdős Problem 504","Erdős Problem 528","Erdős Problem 529","Erdős Problem 588","Erdős Problem 589","Erdős Problem 604","Erdős Problem 605","Erdős Problem 606","Erdős Problem 607","Erdős Problem 634","Erdős Problem 651","Erdős Problem 652","Erdős Problem 654","Erdős Problem 657","Erdős Problem 661","Erdős Problem 662","Erdős Problem 668","Erdős Problem 669","Erdős Problem 670","Erdős Problem 735","Erdős Problem 754","Erdős Problem 827","Erdős Problem 831","Erdős Problem 838","Erdős Problem 953","Erdős Problem 956","Erdős Problem 957","Erdős Problem 960","Erdős Problem 1069","Erdős Problem 1070","Erdős Problem 1086","Erdős Problem 1087","Erdős Problem 1124","Erdős Problem 1127","Erdős Problem 1208","Central Binomial Tail Bounds, Conjecture 6.3","Green's Open Problem 28","Erdős Problem 1041","Mathoverflow 1973","Exponentials conjectures and theorems","Open questions on irrationality of numbers","P versus NP","Sensitivity conjecture","PRIMES is in P","Graph isomorphism complexity","Subword complexity of the morphism a → aab, b → b","Strong Sensitivity Conjecture (`bs(f) ≤ s(f)^2`)","Černý Conjecture","Polynomial-time computability of factoring","The exponent of matrix multiplication","Busy Beaver BB(5)","Busy Beaver BB(6)","Hilbert's tenth problem","Drawing the map of knowledge","Existence of one-way functions","Post-quantum cryptography","Why deep networks generalize","ARC-AGI","A quantum theory of gravity","Black hole information paradox","The precession of Mercury","Direct detection of gravitational waves","The Higgs boson","The hierarchy problem","The strong CP problem","Neutrino masses","Yang–Mills existence and mass gap","The measurement problem","Local realism (Bell's theorem)","The ultraviolet catastrophe","Turbulence","Mechanism of high-temperature superconductivity","The nature of the glass transition","The arrow of time","What dark matter is","What dark energy is","The Hubble tension","Is the universe expanding?","Big Bang or steady state","Cosmic inflation","Matter–antimatter asymmetry","The solar neutrino problem","Imaging a black hole","Early supermassive black holes","Planets around other stars","The Fermi paradox","Continental drift","The age of the Earth","What killed the dinosaurs","The geodynamo and magnetic reversals","Earthquake prediction","Equilibrium climate sensitivity","The ozone hole","Atlantic overturning circulation tipping point","Where Earth's water came from","Snowball Earth","Room-temperature, ambient-pressure superconductor","Metallic hydrogen","The order of the elements","The island of stability","Fixing nitrogen from the air","Ammonia at ambient conditions","Artificial photosynthesis","Protein structure prediction","Designing proteins with arbitrary function","The structure of benzene","How species arise","The structure of DNA","Cracking the genetic code","The complete human genome","The genotype–phenotype map","The origin of life","Spontaneous generation","The origin of complex cells","The Cambrian explosion","Reversing cell differentiation","The causes of aging","Limb regeneration","The neuron doctrine","How memories are stored","What sleep is for","Complete brain wiring diagrams","The hard problem of consciousness","Neural correlates of consciousness","Can artificial systems be conscious?","Is language localized in the brain?","How children learn language","The origin of language","Mechanistic interpretability","AI alignment","Can machines think? (the imitation game)","What causes infectious disease","Smallpox eradication","Polio eradication","Antimicrobial resistance","A cure for HIV","A malaria vaccine","Alzheimer's disease","Cancer","What causes stomach ulcers","Scurvy","Chronic pain","Why some nations grow rich","The productivity slowdown","A fair voting rule","Cooperation on global commons","Selling the airwaves","Egyptian hieroglyphs","Linear B","Linear A","The Indus script","The Voynich manuscript","Reading the Herculaneum scrolls","Fusion as a power plant","Cheap long-duration energy storage","High-efficiency thermophotovoltaics","Carbon removal at gigaton scale","Longitude at sea","Heavier-than-air powered flight","Reusable orbital rockets","General-purpose robots","Autonomous driving","A solid-state amplifier","Fault-tolerant quantum computing","The problem of induction","The Gettier problem","Molyneux's problem","Why there is something rather than nothing","Free will","Zeno's paradoxes of motion","The sorites paradox","The is–ought problem","Newcomb's problem","The liar paradox"],"parent":[0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,8,9,9,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,17,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18,18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and structure. The largest collaborative proof in history lives here, along with problems that have been open since Galois.","algebraic-geometry":"The geometry of solution sets of polynomial equations. Its biggest open questions ask which shapes can be cut out by polynomials and when a polynomial map can be undone.","algorithms":"The search for faster procedures, where a single new idea can move a bound that stood for decades.","analysis":"Limits, functions and the equations of motion. The hardest questions ask whether the equations physics runs on actually have solutions.","chronic-disease":"The diseases of longer lives.","climate":"How the atmosphere and oceans respond to what we add to them.","collective-choice":"How groups decide and cooperate. Some of these questions have proofs.","combinatorics":"Counting, coloring and arrangement. The Erdős problems live mostly here, and so do the results most often settled by computer search and SAT solvers.","complexity":"The map of how hard problems are. Almost every line on it is conjectured rather than proved.","computing-hardware":"The machines the other continents now depend on.","consciousness":"The one problem where even agreeing on the question is hard.","cryptography":"Security that rests on problems believed to be hard. None of those beliefs is proved.","dark-sector":"Two components that dominate the universe's contents and have never been seen directly.","deep-earth":"The interior we can reach only with seismic waves, and the history we can read only in rocks.","development-aging":"How one cell becomes a body, and why bodies wear out.","dynamics":"What happens when a rule is applied over and over. Chaos, fractals, and the stability of the solar system.","early-earth":"The first billions of years, when the planet and its life shaped each other.","economics":"Why some places are rich, and what could make everywhere rich.","energy":"Making, storing and moving power without burning carbon.","energy-chemistry":"Chemistry that decides how the world is fed and powered.","geometry-topology":"Shape, packing and space. Some of the oldest problems in mathematics live here, and some of the newest were settled by computer-checked proof.","gravity-and-quantum":"General relativity and quantum mechanics are each confirmed to extraordinary precision, and they contradict each other where both should apply.","heredity":"How living things pass on instructions, now read at the level of single letters.","infectious-disease":"The oldest killers, several now beaten and a few coming back.","knowledge":"What it is to know something, and how anyone learns from experience.","langlands-program":"A web of conjectures linking number theory, representation theory and geometry, begun in Robert Langlands's 1967 letter to André Weil. It is the closest thing mathematics has to a grand unified theory.","language":"The capacity that makes an atlas like this possible.","learning-theory":"Why learning machines work, what they can do, and whether they can be understood. The newest region of the continent.","life-beyond-earth":"Whether we are alone, which has become a question telescopes can address.","logic-foundations":"What can be proved at all. The answers here include the most famous impossibility results in mathematics, and they set the limits every other continent works within.","machine-minds":"Understanding and steering the models that now share the continent.","machines":"Getting things, and people, from one place to another, and making machines act in the physical world.","materials":"Substances with properties nobody has yet made at room temperature and pressure.","matter-in-motion":"Systems with too many interacting parts to solve exactly, where the collective behavior is the question.","metaphysics":"What exists, and what we are.","molecules-of-life":"Where chemistry turns into biology: the shapes of proteins and how to design new ones.","msc-32":"2 problems from formal-conjectures and erdosproblems filed under Several complex variables.","msc-33":"2 problems from formal-conjectures and erdosproblems filed under Special functions.","msc-52":"84 problems from formal-conjectures and erdosproblems filed under Convex and discrete geometry.","msc-60":"2 problems from formal-conjectures and erdosproblems filed under Probability.","neuroscience":"About 86 billion neurons and a hundred trillion connections, read one small region at a time.","number-theory":"Questions about the whole numbers, easy to state and often centuries old. The primes sit at the center: how they are spaced, how they add, and what the zeros of the zeta function say about them.","origins":"How the universe began, and why it contains anything at all.","origins-of-life":"How chemistry became biology, and how simple life became complex.","quantum-foundations":"The theory works perfectly and nobody agrees what it says about reality.","scripts-and-the-past":"Writing systems nobody can read, and the ones that yielded. Decipherment is a problem with a clear verifier: a reading that predicts new texts.","standard-model":"The best-tested theory in science, with a list of numbers it cannot explain and particles it does not include.","stars-and-black-holes":"The engines of the universe, and the objects that test gravity at its extremes.","value":"What we ought to do, and how to decide 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