Do odd perfect numbers exist?
Decide whether an odd perfect number exists. Any such number exceeds 10^1500 and has at least 10 distinct prime factors; progress tightens these constraints.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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The (binary) Goldbach conjecture
Prove that every even integer greater than 2 is the sum of two primes. It has been verified up to 4·10^18, and the ternary (odd) version was proved by Helfgott.
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The Collatz (3n + 1) conjecture
Prove that iterating n ↦ n/2 (n even), 3n + 1 (n odd) reaches 1 from every positive integer. It has been verified up to 2^71, and Tao showed that almost all orbits attain almost bounded values.
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The Erdős–Straus conjecture
Prove that 4/n = 1/x + 1/y + 1/z has a solution in positive integers for every n ≥ 2. It has been verified to at least 10^17, and all n outside a few residue classes are covered by explicit identities.
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The perfect cuboid problem
Decide whether a box exists whose three edges, three face diagonals and space diagonal are all integers. Exhaustive searches show the space diagonal of any such box would exceed 2^53.
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