An open, curated catalogue of 100 of the most famous open problems in mathematics — and a place where anyone can propose ideas, partial results, and hypothetical approaches toward solving them.
⭐ If you find this useful or fun, star the repo. Contributions welcome — see CONTRIBUTING.md.
The inspiration for this repo: on 19–20 July 2026, mathematician Levent Alpöge announced an explicit counterexample to the Jacobian Conjecture (#81, open since 1939), constructed with the help of Anthropic's Claude Fable 5. Rather than proving the conjecture, the approach built a counterexample — a polynomial map ℂ³ → ℂ³ with constant Jacobian determinant −2 that is nevertheless not injective.
The map (verified in this repo):
a = (1 + xy)³·z + y²(1 + xy)(4 + 3xy)
b = y + 3x(1 + xy)²·z + 3xy²(4 + 3xy)
c = 2x − 3x²y − x³z
Its Jacobian determinant is the constant −2, yet the three distinct points (0,0,−¼), (1,−3⁄2,13⁄2), (−1,3⁄2,13⁄2) all map to (−¼,0,0). That kills injectivity → the conjecture is false for n > 2 (and, by padding with identity coordinates, in every dimension ≥ 3). The n = 2 case is still open.
Because the counterexample is a concrete algebraic object, anyone can verify it — no need to trust the AI. Full story, references, and a verification script: problems/081-jacobian-conjecture.md.
This is exactly the spirit of the project: not "trust me, I solved it," but "here is a checkable object — verify it yourself."
This is:
- A well-referenced reference for each problem: precise statement, history, current state of the art, and links.
- A community space to share approaches — heuristics, reformulations, computational evidence, partial results, or "here's an angle nobody seems to have tried."
This is not:
- A place to claim you've solved the Riemann Hypothesis in a
.txtfile. 😄
Please read this before contributing a "solution":
The crackpot filter. Famous open problems attract thousands of flawed "proofs" every year (the Collatz conjecture alone is legendary for this). A real proof of any problem here would be a historic event, published in a peer-reviewed journal and verified by experts — not merged into a GitHub repo. That does not mean your ideas are worthless. Reformulations, connections between problems, computational experiments, and honest partial progress are genuinely valuable and welcome. We just ask you to be rigorous and humble: state your assumptions, show your work, and clearly mark what is proven vs. conjectured. See CONTRIBUTING.md for the required format.
⚠️ Status disclaimer. Each problem is marked Open to the best of our knowledge as of the repository's creation (2026). Mathematics moves — some problems on this list may be resolved after that date, and edge cases are debated. If you know a problem here has been solved or disproven, please open a PR with a citation. This catalogue is only as good as its community keeps it.
★M = one of the six still-open Clay Millennium Prize Problems ($1,000,000 each). (The seventh, the Poincaré conjecture, was solved by Grigori Perelman in 2003.)
- Riemann Hypothesis
★M— all non-trivial zeros of ζ(s) have real part ½. - Goldbach's Conjecture — every even integer > 2 is a sum of two primes.
- Twin Prime Conjecture — there are infinitely many primes p with p+2 also prime.
- Collatz Conjecture (3n+1) — the 3n+1 iteration always reaches 1.
- Legendre's Conjecture — there is always a prime between n² and (n+1)².
- Landau's n²+1 Problem — are there infinitely many primes of the form n²+1?
- Brocard's Problem — for which n is n!+1 a perfect square (Brown numbers)?
- Odd Perfect Numbers — does an odd perfect number exist?
- Infinitude of Mersenne Primes — are there infinitely many primes 2^p−1?
- Fermat Primes — are there finitely or infinitely many primes 2^(2^n)+1?
- abc Conjecture — bounds rad(abc) for coprime a+b=c (Mochizuki's claimed proof remains unaccepted by the wider community).
- Beal Conjecture — if aˣ+bʸ=cᶻ with x,y,z>2 then a,b,c share a common factor.
- Fermat–Catalan Conjecture — finitely many coprime solutions to aᵐ+bⁿ=cᵏ with 1/m+1/n+1/k<1.
- Erdős–Straus Conjecture — 4/n = 1/x+1/y+1/z solvable for all n>1.
- Lehmer's Totient Problem — does φ(n) | (n−1) imply n is prime?
- Carmichael's Totient Conjecture — no φ-value is attained exactly once.
- Gilbreath's Conjecture — iterated absolute differences of primes always start with 1.
- Polignac's Conjecture — every even number is the gap between infinitely many consecutive prime pairs.
- Singmaster's Conjecture — a uniform bound on how often a number >1 appears in Pascal's triangle.
- Cramér's Conjecture — prime gaps are O((log p)²).
- Firoozbakht's Conjecture — pₙ^(1/n) is strictly decreasing.
- Bunyakovsky Conjecture — irreducible integer polynomials produce infinitely many primes (under mild conditions).
- Schinzel's Hypothesis H — a joint generalization of many prime-producing conjectures.
- Grimm's Conjecture — distinct prime factors can be assigned to consecutive composite numbers.
- Giuga's Conjecture — a number-theoretic characterization of primes via a power-sum congruence.
- Generalized Riemann Hypothesis — RH for all Dirichlet L-functions.
- Lindelöf Hypothesis — ζ(½+it) grows slower than any positive power of t.
- Chowla Conjecture — the Liouville function has no correlations.
- Elliott–Halberstam Conjecture — strong averaged control of primes in arithmetic progressions.
- Montgomery's Pair Correlation Conjecture — spacing of ζ zeros matches random matrix theory.
- Hardy–Littlewood k-Tuple Conjecture — asymptotic density of prime constellations.
- Second Hardy–Littlewood Conjecture — π(x+y) ≤ π(x)+π(y) (believed to conflict with the k-tuple conjecture).
- Grand Riemann Hypothesis — RH for all automorphic L-functions.
- Birch and Swinnerton-Dyer Conjecture
★M— rank of an elliptic curve equals the order of vanishing of its L-function. - Hall's Conjecture — lower bounds on |x³−y²|.
- Littlewood Conjecture — simultaneous Diophantine approximation of two reals.
- Vojta's Conjecture — a deep height inequality unifying much of Diophantine geometry.
- Bombieri–Lang Conjecture — distribution of rational points on varieties of general type.
- Hilbert's Twelfth Problem — explicit generation of abelian extensions of number fields.
- Zaremba's Conjecture — every denominator admits a continued fraction with bounded partial quotients.
- Union-Closed Sets Conjecture (Frankl) — some element lies in ≥ half the sets of any union-closed family.
- Sunflower Conjecture — bounds on set systems forced to contain sunflowers.
- Hadamard Matrix Conjecture — a Hadamard matrix exists for every order divisible by 4.
- 1/3–2/3 Conjecture — every finite non-total poset has a pair comparable in [1/3, 2/3] of linear extensions.
- Erdős–Turán Conjecture on Additive Bases — additive bases of order 2 have unbounded representation function.
- Erdős Conjecture on Arithmetic Progressions — sets with divergent reciprocal sums contain arbitrarily long APs.
- Ramsey Number R(5,5) — determine its exact value (known to lie between 43 and 48).
- Lonely Runner Conjecture — every runner gets "lonely" on a circular track.
- Caccetta–Häggkvist Conjecture — short cycles forced by minimum out-degree in digraphs.
- Tuza's Conjecture — triangle edge-cover vs. triangle packing ratio.
- Rota's Basis Conjecture — rearranging n bases into n disjoint transversal bases.
- Erdős–Gyárfás Conjecture — every cubic graph has a cycle of length a power of 2.
- Hadwiger Conjecture — a k-chromatic graph has a K_k minor.
- Reconstruction Conjecture (Ulam) — graphs are determined by their vertex-deleted subgraphs.
- Graceful Tree Conjecture (Ringel–Kotzig) — every tree has a graceful labeling.
- Cycle Double Cover Conjecture — every bridgeless graph has a family of cycles covering each edge twice.
- Total Coloring Conjecture — total chromatic number is at most Δ+2.
- Berge–Fulkerson Conjecture — every bridgeless cubic graph has 6 perfect matchings covering each edge twice.
- Oberwolfach Problem — 2-factorizations of complete graphs into prescribed cycles.
- Barnette's Conjecture — every bipartite cubic planar 3-connected graph is Hamiltonian.
- Inscribed Square Problem (Toeplitz) — every Jordan curve contains 4 points forming a square.
- Kissing Number Problem — maximum touching unit spheres in dimensions beyond 1–4, 8, 24.
- Sphere Packing — optimal packing density in dimensions other than 1, 2, 3, 8, 24.
- Hadwiger–Nelson Problem — chromatic number of the plane (known to be 5, 6, or 7).
- Kobon Triangle Problem — maximum triangles formed by n lines.
- Erdős Unit Distance Problem — maximum unit distances among n points in the plane.
- Ulam's Packing Conjecture — the ball is the worst-packing convex body in 3D.
- Reinhardt Conjecture — the smoothed octagon is the worst-packing convex body in 2D.
- Borsuk's Problem — dimensions in which every bounded set splits into d+1 smaller-diameter pieces.
- Happy Ending Problem — points forcing a convex n-gon (Erdős–Szekeres exact bound).
- Moser's Worm Problem — smallest region covering every unit-length curve.
- Danzer's Problem — does a bounded-density set meeting every convex body of volume 1 exist?
- Smooth 4-Dimensional Poincaré Conjecture — is every homotopy 4-sphere diffeomorphic to S⁴?
- Novikov Conjecture — homotopy invariance of higher signatures.
- Baum–Connes Conjecture — K-theory of group C*-algebras.
- Zeeman Conjecture — collapsibility of certain contractible 2-complexes.
- Andrews–Curtis Conjecture — trivializing balanced presentations of the trivial group.
- Hilbert–Smith Conjecture — only Lie groups act faithfully on manifolds.
- Volume Conjecture — colored Jones polynomials recover hyperbolic volume of knots.
- Kaplansky's Zero-Divisor Conjecture — group rings of torsion-free groups have no zero divisors.
- Jacobian Conjecture — polynomial maps with constant nonzero Jacobian are invertible. → ⚡ DISPROVEN for n > 2 (July 2026); the
n = 2case remains open. See the featured case study. - Dixmier Conjecture — endomorphisms of the Weyl algebra are automorphisms.
- Inverse Galois Problem — is every finite group a Galois group over ℚ?
- Zariski Cancellation Problem — does Aⁿ ≅ Bⁿ imply A ≅ B (in characteristic 0)?
- Köthe Conjecture — the sum of two nil left ideals is nil.
- Nakai Conjecture — smoothness detected by differential operators.
- Crouzeix's Conjecture — a spectral-norm bound involving the numerical range (constant 2).
- Navier–Stokes Existence and Smoothness
★M— do smooth solutions always exist globally in 3D? - Invariant Subspace Problem — does every bounded operator on separable Hilbert space have a non-trivial invariant subspace?
- Hilbert's Sixteenth Problem — bounding limit cycles of planar polynomial vector fields.
- Schanuel's Conjecture — a sweeping transcendence statement about exp.
- Four Exponentials Conjecture — a special case of transcendence for 2×2 exponent grids.
- Sendov's Conjecture — critical points near roots of complex polynomials (proven for large degree, open in general).
- Yang–Mills Existence and Mass Gap
★M— rigorous quantum Yang–Mills theory with a positive mass gap. - P versus NP
★M— can every efficiently-checkable problem be efficiently solved? - Unique Games Conjecture — hardness of approximation threshold.
- Exponential Time Hypothesis — 3-SAT has no sub-exponential algorithm.
- Hodge Conjecture
★M— Hodge classes are combinations of algebraic cycles. - Pierce–Birkhoff Conjecture — piecewise-polynomial functions are max/min combinations of polynomials.
- Casas-Alvero Conjecture — a polynomial sharing a root with each derivative is a pure power.
problems/ One Markdown file per problem: statement, history, state of the art, references.
approaches/ Community-submitted approaches, ideas, and partial results.
Detailed problem pages are being filled in — flagship problems first. Writing a missing problems/NNN-*.md page is one of the easiest ways to contribute. Use problems/_TEMPLATE.md.
Read CONTRIBUTING.md. In short:
- Fix / expand a problem page → PR.
- Propose an approach → add a file under
approaches/following the template, or open an issue. - Report a problem as solved → PR with a peer-reviewed citation.
Be rigorous, cite sources, and separate proven from conjectured. Good-faith partial ideas are welcome; grandiose unverifiable "proofs" will be closed with kindness.
- Content (problem descriptions, prose): CC BY 4.0.
- Attribution appreciated. This project stands on the shoulders of Wikipedia's List of unsolved problems in mathematics, the Clay Mathematics Institute, and countless mathematicians.
Curated with care by Simone Ruggiero. Mathematics belongs to everyone — dig in.
unsolved-math-100 is free and open source. If you find it useful, please ⭐️ star the repo — and if you'd like to support my open-source work, you can 💛 sponsor me on GitHub or ☕️ buy me a coffee. Completely optional, always appreciated. 🙏