Open Problems
The following conjectures remain open, organized by mathematical field.
By Field
Counts are intentionally omitted here to avoid duplicated bookkeeping. Each field page is the source of truth for current totals.
| # | Field | Page |
|---|---|---|
| 1 | Number Theory | Primes, Diophantine equations, arithmetic functions |
| 2 | Graph Theory | Colorings, matchings, graph structure |
| 3 | Algebra | Algebraic geometry, topology, K-theory, ring theory, group theory |
| 4 | Geometry & Topology | Differential, discrete, geometric topology, Diophantine geometry |
| 5 | Analysis | Complex, harmonic, functional analysis, operator theory |
| 6 | Combinatorics & Order Theory | Enumerative, extremal combinatorics, partial orders |
| 7 | Other Fields | Dynamical systems, computational complexity, probability, and more |
Alphabetical Index
| Conjecture | Field | Page |
|---|---|---|
| 1/3–2/3 conjecture | Order theory | Combinatorics |
| abc conjecture | Number theory | Number Theory |
| Agoh–Giuga conjecture | Number theory | Number Theory |
| Agrawal’s conjecture | Number theory | Number Theory |
| Andrica’s conjecture | Number theory | Number Theory |
| Andrews–Curtis conjecture | Combinatorial group theory | Algebra |
| Artin conjecture (L-functions) | Number theory | Number Theory |
| Artin’s conjecture on primitive roots | Number theory | Number Theory |
| Bateman–Horn conjecture | Number theory | Number Theory |
| Baum–Connes conjecture | Operator K-theory | Analysis |
| Beal’s conjecture | Number theory | Number Theory |
| Beilinson conjecture | Number theory | Number Theory |
| Berry–Tabor conjecture | Geodesic flow | Other |
| Big-line-big-clique conjecture | Discrete geometry | Geometry & Topology |
| Birch and Swinnerton-Dyer conjecture | Number theory | Number Theory |
| Birch–Tate conjecture | Number theory | Number Theory |
| Birkhoff conjecture | Integrable systems | Other |
| Bloch–Beilinson conjectures | Number theory | Number Theory |
| Bloch–Kato conjecture | Algebraic K-theory | Algebra |
| Bochner–Riesz conjecture | Harmonic analysis | Analysis |
| Bombieri–Lang conjecture | Diophantine geometry | Geometry & Topology |
| Borel conjecture | Geometric topology | Geometry & Topology |
| Bost conjecture | Geometric topology | Geometry & Topology |
| Brennan conjecture | Complex analysis | Analysis |
| Brocard’s conjecture | Number theory | Number Theory |
| Brumer–Stark conjecture | Number theory | Number Theory |
| Bunyakovsky conjecture | Number theory | Number Theory |
| Carathéodory conjecture | Differential geometry | Geometry & Topology |
| Carmichael totient conjecture | Number theory | Number Theory |
| Casas-Alvero conjecture | Polynomials | Other |
| Catalan–Dickson conjecture | Number theory | Number Theory |
| Catalan’s Mersenne conjecture | Number theory | Number Theory |
| Cherlin–Zilber conjecture | Group theory | Algebra |
| Chowla conjecture | Möbius function | Other |
| Collatz conjecture | Number theory | Number Theory |
| Conway’s thrackle conjecture | Graph theory | Graph Theory |
| Cramér’s conjecture | Number theory | Number Theory |
| De Polignac’s conjecture | Number theory | Number Theory |
| Deligne conjecture | Monodromy | Other |
| Dittert conjecture | Combinatorics | Combinatorics |
| Eilenberg–Ganea conjecture | Algebraic topology | Algebra |
| Elliott–Halberstam conjecture | Number theory | Number Theory |
| Erdős–Faber–Lovász conjecture | Graph theory | Graph Theory |
| Erdős–Gyárfás conjecture | Graph theory | Graph Theory |
| Erdős–Straus conjecture | Number theory | Number Theory |
| Farrell–Jones conjecture | Geometric topology | Geometry & Topology |
| Filling area conjecture | Differential geometry | Geometry & Topology |
| Firoozbakht’s conjecture | Number theory | Number Theory |
| Fortune’s conjecture | Number theory | Number Theory |
| Four exponentials conjecture | Number theory | Number Theory |
| Frankl conjecture | Combinatorics | Combinatorics |
| Gauss circle problem | Number theory | Number Theory |
| Gilbert–Pollack conjecture | Metric geometry | Geometry & Topology |
| Gilbreath conjecture | Number theory | Number Theory |
| Goldbach’s conjecture | Number theory | Number Theory |
| Gold partition conjecture | Order theory | Combinatorics |
| Goldberg–Seymour conjecture | Graph theory | Graph Theory |
| Goormaghtigh conjecture | Number theory | Number Theory |
| Green’s conjecture | Algebraic curves | Algebra |
| Grimm’s conjecture | Number theory | Number Theory |
| Grothendieck–Katz p-curvature conjecture | Differential equations | Other |
| Hadamard conjecture | Combinatorics | Combinatorics |
| Herzog–Schönheim conjecture | Group theory | Algebra |
| Hilbert–Smith conjecture | Geometric topology | Geometry & Topology |
| Hodge conjecture | Algebraic geometry | Algebra |
| Homological conjectures | Commutative algebra | Algebra |
| Hopf conjectures | Geometry | Geometry & Topology |
| Ibragimov–Iosifescu conjecture | Probability theory | Other |
| Invariant subspace problem | Functional analysis | Analysis |
| Jacobian conjecture | Polynomials | Other |
| Jacobson’s conjecture | Ring theory | Algebra |
| Kaplansky conjectures | Ring theory | Algebra |
| Keating–Snaith conjecture | Number theory | Number Theory |
| Köthe conjecture | Ring theory | Algebra |
| Kung–Traub conjecture | Iterative methods | Other |
| Legendre’s conjecture | Number theory | Number Theory |
| Lemoine’s conjecture | Number theory | Number Theory |
| Lenstra–Pomerance–Wagstaff conjecture | Number theory | Number Theory |
| Leopoldt’s conjecture | Number theory | Number Theory |
| List coloring conjecture | Graph theory | Graph Theory |
| Littlewood conjecture | Diophantine approximation | Other |
| Lovász conjecture | Graph theory | Graph Theory |
| Manin conjecture | Diophantine geometry | Geometry & Topology |
| Marshall Hall’s conjecture | Number theory | Number Theory |
| Mazur’s conjectures | Diophantine geometry | Geometry & Topology |
| MNOP conjecture | Algebraic geometry | Algebra |
| Montgomery’s pair correlation conjecture | Number theory | Number Theory |
| n conjecture | Number theory | Number Theory |
| New Mersenne conjecture | Number theory | Number Theory |
| Novikov conjecture | Algebraic topology | Algebra |
| Oppermann’s conjecture | Number theory | Number Theory |
| Petersen coloring conjecture | Graph theory | Graph Theory |
| Pierce–Birkhoff conjecture | Real algebraic geometry | Other |
| Pillai’s conjecture | Number theory | Number Theory |
| Quantum PCP conjecture | Computational complexity | Other |
| Quantum unique ergodicity conjecture | Dynamical systems | Other |
| Reconstruction conjecture | Graph theory | Graph Theory |
| Riemann hypothesis | Number theory | Number Theory |
| Ringel–Kotzig conjecture | Graph theory | Graph Theory |
| Rudin’s conjecture | Additive combinatorics | Combinatorics |
| Sarnak conjecture | Topological entropy | Other |
| Sato–Tate conjecture | Number theory | Number Theory |
| Schanuel’s conjecture | Number theory | Number Theory |
| Schinzel’s hypothesis H | Number theory | Number Theory |
| Scholz conjecture | Addition chains | Combinatorics |
| Second Hardy–Littlewood conjecture | Number theory | Number Theory |
| Selfridge’s conjecture | Number theory | Number Theory |
| Sendov’s conjecture | Complex polynomials | Analysis |
| Serre’s multiplicity conjectures | Commutative algebra | Algebra |
| Singmaster’s conjecture | Binomial coefficients | Combinatorics |
| Standard conjectures on algebraic cycles | Algebraic geometry | Algebra |
| Tate conjecture | Algebraic geometry | Algebra |
| Toeplitz’ conjecture | Jordan curves | Geometry & Topology |
| Tuza’s conjecture | Graph theory | Graph Theory |
| Twin prime conjecture | Number theory | Number Theory |
| Ulam’s packing conjecture | Packing | Geometry & Topology |
| Unicity conjecture for Markov numbers | Number theory | Number Theory |
| Uniformity conjecture | Diophantine geometry | Geometry & Topology |
| Unique games conjecture | Computational complexity | Other |
| Vandiver’s conjecture | Number theory | Number Theory |
| Virasoro conjecture | Algebraic geometry | Algebra |
| Vizing’s conjecture | Graph theory | Graph Theory |
| Vojta’s conjecture | Number theory | Number Theory |
| Waring’s conjecture | Number theory | Number Theory |
| Weight monodromy conjecture | Algebraic geometry | Algebra |
| Weinstein conjecture | Periodic orbits | Other |
| Whitehead conjecture | Algebraic topology | Algebra |
| Zauner’s conjecture | Operator theory | Analysis |