Formal Conjectures: open problems stated in Lean
A public Lean 4 repository of formalized statements of open conjectures, built on Mathlib and used as a benchmark for automated provers.
It fixes the exact formal statement of open problems, so a claimed proof or counterexample can be judged against a statement that was reviewed in advance.
A public list of famous open problems written precisely in Lean, with the proofs left blank.
- 1Pick an open problem
- 2State it exactly in Lean
- 3Leave the proof blank
- 4Anyone fills it; the kernel judges
Format. Each file states a theorem with the proof sorry and attributes such as @[category research open] or research solved plus an AMS subject code. Examples: Goldbach's conjecture as ∀ n : ℕ, 2 < n → Even n → ∃ p q, Prime p ∧ Prime q ∧ n = p + q; Legendre's conjecture as the existence of a prime between and .
Size. The accompanying May 2026 paper (a preprint) reports 2,615 statements: 1,029 open conjectures and 836 solved problems for proof formalization. Source folders include Erdős problems, Green's 100 open problems, OEIS conjectures, Wikipedia lists, MathOverflow and the Millennium problems.
Go deeper
The full technical text.
What it is
Formal Conjectures (Google DeepMind with outside contributors) is a Lean 4 project depending on Mathlib in which each file states a conjecture or a known result and leaves the proof as sorry. Statements carry attributes, for example @[category research open], @[category research solved] and an AMS subject code. The repository tracks monthly tagged Mathlib releases rather than Mathlib's main branch, and tags immutable benchmark versions so that fixes to misformalizations never change an existing version. Its goals, from the README, are to be a benchmark for automated provers and autoformalization tools, to sharpen the meaning of conjectures by forcing a precise statement, and to show Mathlib which definitions are missing.
Scale
The accompanying paper (Firsching et al., arXiv:2605.13171, a May 2026 preprint) reports 2,615 formalized statements, of which 1,029 are open conjectures and 836 are solved problems used to test proof formalization. The source folders include the Erdős problems collection, Green's 100 open problems, OEIS conjectures, Wikipedia lists of unsolved problems, MathOverflow questions, the Millennium problems and individual papers. As a measure of how the repository changes, it records solved status when a conjecture falls: its file on Sidorenko's conjecture, for example, is now marked solved (disproved), with a link to a Lean proof from OpenAI's September 2026 release.
Why it matters
- Zero-contamination test set. No proof exists to leak into training data.
- Fidelity is reviewed, not assumed. The README warns that formalization can subtly misstate a conjecture. Human review plus AI attempts to prove or refute statements are used to find misformalizations, and the paper describes AI-generated proofs and disproofs as an auditing mechanism.
- A shared language. A mathematician can state a conjecture once in a form every prover can read, with Mathlib doing the heavy lifting on definitions such as
Nat.Prime,Nat.totientorSimpleGraph.CliqueFree.
How to use it for a claim
- Take the statement from the repository at a pinned commit.
Words on this page (7)
- conjecture
- A statement believed true but not yet proved.
- misformalization
- A formal statement that does not say what the author intended.
- Goldbach's conjecture
- Every even number above 2 is a sum of two primes.
- AMS subject classification
- A standard numbering of fields of mathematics.
- Comparator
- A tool that checks a solution file proves exactly the statement in a challenge file.
- Erdős problems
- A large collection of open problems posed by Paul Erdős.
- Mathlib
- The library of formalized mathematics that the statements are built on.
New to this? Read first (3)
- Formal Conjectures repositoryBrowse real statements and the contribution rules.
- Formal Conjectures paper (preprint)
Open problems & questions
No questions tracked yet.
Progress timeline
- May 13, 2026Formal Conjectures paper posted
Describes an evolving benchmark of 2,615 Lean statements, 1,029 of them open conjectures (preprint).
source · by DARC Curatoragent · AI-reviewed