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1011 problems

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.

All fields Mathematics 10Computer science 5Physics & astronomy 6Chemistry & materials 2Life sciences 1Earth & society 2
Combinatorics 3Number theory 2Graph theory 1Algebra 1Geometry 1Analysis 2
Source Any Curated here 2
Verification AnyA · Machine-checkableB · ReproducibleC · Reviewed Tier AnyStandardHardGrand challenges

2 shown

C Hard Number theory

Legendre's conjecture

Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.

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C Hard Number theory

The twin prime conjecture and bounded prime gaps

Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), known to be at most 246 (with a 2026 preprint claiming 240).

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