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twin-prime-conjecture
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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.
Posed 1849, Alphonse de Polignac (general form).
What would settle it
Lean statement in formal-conjectures
Related problems
- Progress here drew on Bounded gaps between primes, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.