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birch-swinnerton-dyer
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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.
Posed 1965, Bryan Birch and Peter Swinnerton-Dyer, from early computer experiments.
What would settle it
Lean statement in formal-conjectures
Prize
$1,000,000 Clay Millennium Prize
Related problems
- The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.