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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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.
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The circulant Hadamard conjecture
Every circulant Hadamard matrix has order at most four.
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The Curling Number Conjecture
The sequence will eventually reach 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 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 Hasse–Weil conjecture for elliptic curves
The L-series of an elliptic curve over a number field has a meromorphic continuation to ℂ.
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The length of an s-increasing sequence of r-tuples
F(n) ≤ n^3/2.
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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 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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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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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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Tug of war score between prime gap increases and decreases
Is the score a(n) > 0 for some n > 250000?
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Twin prime conjecture
Are there infinitely many primes p such that p + 2 is prime?
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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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TxGraffiti Conjecture 3: independent domination versus saturation (regular)
TxGraffiti Conjecture 3: for every r-regular graph G (r ≥ 1), i(G) ≤ μ^(G). This conjecture is open**.
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Unique Crystal Components
If n = ab is a crystal, then there are no other pairs of positive integers c, d > 1, different from the couple a, b, such that n = cd and B(c, d) ∈ ℕ, i.e., the components of the crystals are unique.
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Value of n-th cyclotomic polynomial at n
a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree?
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Vaught conjecture
The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, aleph_0 or 2^aleph_0.
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VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²
Every convex set in ℝ^3 has VC_2 dimension at most 2.
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Vizing's conjecture (1968)
Vizing's conjecture (1968). For all finite simple graphs G and H, the domination number of the Cartesian (box) product satisfies γ(G square H) ≥ γ(G) γ(H).
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Weak tiling problems
Problem 4.1. Let Ω ⊂ ℝ be a finite union of intervals and ν a weak tiling measure for Ω. Must supp(ν) have bounded density?
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Wieferich primes
There are infinitely many Wieferich primes.
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Wilson primes
There are infinitely many Wilson primes.
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Wolstenholme numbers
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). - Amiram Eldar and Thomas Ordowski, Jul 28 2019
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Wolstenholme Prime
It is conjectured that there are infinitely many Wolstenholme primes. Reference: Wikipedia
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Woodall Primes
There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1.
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Written on the Wall II - Conjecture 100
WOWII Conjecture 100 (status O): For a simple connected graph G, α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where α(G) = G.indepNum is the independence number, max_v l(v) is the maximum over all vertices of the independence number of the neighbourhood (in G), and degreeL2Norm(Gᶜ) is the…
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Written on the Wall II - Conjecture 133
WOWII Conjecture 133: For a simple connected graph G, path(G) ≥ rad(G) + (avg_v l(v))^cC_4(G), where path(G) is the path number of the graph (number of vertices of a largest induced path), rad(G) is the radius (minimum eccentricity, as a natural number), avg_v l(v) = l(G) is the average…
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Written on the Wall II - Conjecture 19
WOWII Conjecture 19 If G is connected then the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G))), where ecc(v) denotes eccentricity and l(G) is the independence number of neighbourhoods.
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Written on the Wall II - Conjecture 198a
WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G.
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Written on the Wall II - Conjecture 40
WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.
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Written on the Wall II - Conjecture 61
WOWII Conjecture 61 For a simple connected graph G, the size f(G) of a largest induced forest satisfies f(G) ≥ residue(G) + ⌈ diam(G) / 3 ⌉, where residue(G) is the Havel-Hakimi residue and diam(G) is the diameter of G.
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Zagier's Conjecture on Multiple Zeta Values
Zagier's conjecture The ℚ-dimension of the vector space spanned by all multiple zeta values of weight n equals d_n, where d_n is the Zagier dimension sequence satisfying d_0 = 1, d_1 = 0, d_2 = 1, and d_n = d_n-2 + d_n-3 for n ≥ 3.
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Zariski Cancellation
The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative.
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