Sumner's universal tournament conjecture (1971)
Sumner's universal tournament conjecture (1971). Every tournament on 2n - 2 vertices contains every oriented tree on n ≥ 2 vertices as a subgraph.
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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.
All fields Mathematics 1261Computer science 31Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
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Sums of a cube, a square, and a positive triangular number
Conjecture: a(n) > 0 for all n > 0.
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Sums of a square and three weighted cubes
Conjecture (i): For any m ∈ 3, 4, 5, 6 and n ≥ 0, there are nonnegative integers w, x, y, z such that n = w^2 + x^3 + 2y^3 + mz^3.
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Sums of a triangular number and three weighted cubes
Conjecture (i): Any positive integer can be written as w(w+1)/2 + x^3 + by^3 + cz^3 with w > 0 and x, y, z ≥ 0, provided that (b, c) ∈ (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,20), (2,21), (2,34), (3,3), (3,4), (3,5), (3,6), (4,10).
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Tarski's exponential function problem
Tarski's exponential function problem. Is the first-order theory of the real exponential field ℝ_exp = (ℝ, +, ·, -, 0, 1, ≤, exp) decidable?
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Taxicab numbers
Taxicab number for k=5, m=2, and n=2 is not known. Whether such a number exists is also not known.
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The 1680-Conjecture
Zhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^4 + 1680 y^3 z is a square.
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The Alon-Tarsi short cycle cover conjecture
Conjecture 4 (Alon-Tarsi, 1985), simple-graph specialization. Every bridgeless simple graph has a list of cycles covering every edge, with Σ_C |E(C)| ≤ 7/5|E(G)|.
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The Andrews-Curtis conjecture
The Andrews-Curtis conjecture. Every normally generating n-tuple in the free group of rank n is Andrews-Curtis equivalent to the standard tuple of free generators.
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The Auslander-Reiten conjecture
The Auslander-Reiten conjecture [AR75]. Let Λ be an Artin algebra and M a finitely generated Λ-module with Ext^i_Λ(M, Λ) = 0 and Ext^i_Λ(M, M) = 0 for all i > 0. Then M is projective.
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The Balister–Győri–Schelp conjecture
The Balister–Győri–Schelp conjecture (Conjecture 1): for s ≥ 2, every zero-sum list of 2^s-1 nonzero vectors in 𝔽_2^s is the prescribed-difference profile of a perfect matching of 𝔽_2^s.
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The Berge–Fulkerson conjecture
The Berge–Fulkerson conjecture (Fulkerson 1971). Every bridgeless cubic graph has six perfect matchings such that every edge lies in exactly two of them.
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The Bing-Borsuk Conjecture
The Bing-Borsuk Conjecture: every n-dimensional homogeneous absolute neighborhood retract is a topological n-manifold. A topological space X is an n-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X).
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The Bollobás–Eldridge–Catlin conjecture on graph packing (1978)
The Bollobás–Eldridge–Catlin conjecture (1978). If G_1 and G_2 are graphs on the same n vertices with maximum degrees Δ_1, Δ_2 satisfying (Δ_1 + 1)(Δ_2 + 1) ≤ n + 1, then G_1 and G_2 pack.
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The burning number conjecture
The burning number conjecture (Bonato–Janssen–Roshanbin [BJR16]). Every connected graph on n vertices can be burned in at most ⌈√n⌉ rounds: fires started at one new vertex per round, each spreading to all neighbouring vertices every round, can burn the whole graph in ⌈√n⌉ rounds.
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The Catch-Up game and conjecture
Let T_N = Σ_k=1^N k = N(N+1)/2. If T_N is even (equivalently N ≡ 0 pmod 4 or N ≡ 3 pmod 4), then under optimal play the game Catch-Up(1, …, N) ends in a draw.
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The circulant Hadamard conjecture
Every circulant Hadamard matrix has order at most four.
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The cup-of-coffee sequence
Conjectured to be a permutation of the positive integers.
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The Curling Number Conjecture
Starting from any nonempty finite integer sequence, repeatedly appending its curling number eventually gives a sequence with curling number 1 [Chaffin and Sloane, Conjecture 1].
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The Eisenbud-Green-Harris conjecture
The Eisenbud-Green-Harris conjecture. Let I ⊆ k[x_1, …, x_n] be a homogeneous ideal containing a regular sequence of forms of degrees d_1 ≤ … ≤ d_c. Then there is a lex ideal L such that I has the same Hilbert function as L + (x_1^d_1, …, x_c^d_c).
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The Erdős–Moser equation
The only positive solution of S_k(m)=m^k is (k,m)=(1,3).
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The Erdős–Ulam problem
The Erdős–Ulam problem (Ulam 1945). Is there a dense subset of the plane whose points are at pairwise rational distances? Conjecturally, no such set exists.
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The first Atiyah–Sutcliffe conjecture
Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent.
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The Gerstenhaber problem for three commuting matrices
The Gerstenhaber problem: if A, B, and C are pairwise commuting n × n matrices over a field K, is the dimension of the unital K-algebra K[A, B, C] they generate always at most n?
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The Goormaghtigh conjecture
The only Goormaghtigh numbers are 31 and 8191.
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The Gyárfás–Sumner conjecture
The Gyárfás–Sumner conjecture (Gyárfás 1975, Sumner 1981). For every finite tree T the class of T-free graphs (graphs with no induced subgraph isomorphic to T) is χ-bounded: there is a function f such that every T-free graph G satisfies χ(G) ≤ f(ω(G)).
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The Hasse–Weil conjecture for elliptic curves
The L-series of an elliptic curve over a number field has a meromorphic continuation to ℂ. This is the Hasse–Weil conjecture in the form stated in [Wikipedia], and a weak form of [Gross2011], Conjecture 2.7.
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The Irving–Leather conjecture on stable matchings
Thurber's §2 attributes to Irving and Leather the conjecture that their construction attains the maximum number of stable matchings at every power-of-two size.
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The length of an s-increasing sequence of r-tuples
F(n) ≤ n^3/2.
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The linear arboricity conjecture
The linear arboricity conjecture (Akiyama–Exoo–Harary [AEH81]). The edges of every graph of maximum degree Δ can be partitioned into at most ⌈(Δ + 1) / 2⌉ linear forests.
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The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture
Sheehan's conjecture (1977). Every 4-regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a different edge set).
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The Manickam–Miklós–Singhi conjecture (1988)
The Manickam–Miklós–Singhi conjecture (1988). If n ≥ 4k and x_1, …, x_n are real numbers with x_1 + … + x_n = 0, then at least C(n-1, k-1) of the k-element subsets S ⊆ 1, …, n have Σ_i ∈ S x_i ≥ 0.
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The Petersen colouring conjecture (Jaeger 1988)
The Petersen colouring conjecture (Jaeger 1988). Every bridgeless cubic graph has a Petersen colouring: an edge colouring by the edges of the Petersen graph such that the three edges at each vertex are mapped onto the three edges at some vertex of the Petersen graph.
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The prime numbers
Conjecture from Thomas Ordowski (2023): log log a(n+1) - log log a(n) < 1/n for n > 0.
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The Rule 30 Prize Problems
Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period p and threshold N past which the column repeats with period p.
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The S_3-conjecture (conjugacy classes of distinct sizes)
Markel's S_3-conjecture (1973): any nontrivial finite ah-group is isomorphic to S_3. The conjecture is open in general; it is known to be true for solvable groups.
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The small Cohen-Macaulay modules conjecture
The small Cohen-Macaulay modules conjecture. If R is a complete Noetherian local ring, then there is a finitely generated R-module M ≠ 0 such that some system of parameters of R is a regular sequence on M. Hochster stated the conjecture for complete local domains [Ho17, Conjecture 2.1].
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The symbol length of K^M_n(ℂ(x_1, …, x_m))/p
The symbol length problem for complex rational function fields [Krashen2024, Problem 2.1.3.12 and §2.1.3.4]: determine, as a function of m, n and the prime p, the symbol length of K^M_n(ℂ(x_1, …, x_m))/p, that is the least k such that every class is a sum of at most k symbols, or ∞ if there is no…
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The total colouring conjecture (Behzad 1965, Vizing 1968)
The total colouring conjecture (Behzad 1965, Vizing 1968). Every finite simple graph G has a total colouring with at most Δ(G) + 2 colours, i.e. χ''(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree.
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Toronto spaces
Any T2, Toronto space is discrete.
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Trajectory of 103 under the Reverse and Add! operation in base 3
103 is conjectured to be the smallest number such that the Reverse and Add! algorithm in base 3 does not lead to a palindrome. Its trajectory is conjectured to never reach a palindrome.
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Trajectory of 64 under the map n → A006368(n)
It is conjectured that this trajectory does not close on itself.
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Triangular matchstick numbers in the class of prime numbers
Terms are squares at only(?) three values of n = 3, 6, 4072: corresponding terms are 6^2, 13^2, and 15735^2.
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Triples summing to n with 2m+1 and 2m^3+1 prime
Conjecture (i): a(n) > 0 for all n > 2.
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Tripling until the seed's digits reappear
OEIS A331631 states: The sequence is conjectured to be a permutation of the positive integers (see the Crossrefs section).
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Tug of war score between prime gap increases and decreases
Is the score a(n) > 0 for some n > 250000?
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Tuza's conjecture
Tuza's conjecture (1981): every finite graph G satisfies τ(G) ≤ 2ν(G), where τ(G) is its triangle covering number and ν(G) is its triangle packing number. The bound would be sharp, for example for K_4 and K_5.
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Twin prime conjecture
Are there infinitely many primes p such that p + 2 is prime?
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Twin primes of the form (3^n - k) 2^n ± 1
Conjecture: There is always one such k for each n > 0 such that (3^n - k) 2^n - 1 and (3^n - k) 2^n + 1 are a twin prime pair.
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TxGraffiti Conjecture 2: zero forcing versus independence (cubic graphs)
TxGraffiti Conjecture 2: for every connected graph G with Δ(G) ≤ 3 and G ≠ K_4, Z(G) ≤ α(G) + 1. This conjecture is open.
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