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 Beck–Fiala conjecture There exists a universal constant C > 0 such that every set system S_1, …, S_m ⊆ [n] of degree at most t admits a colouring χ : [n] → -1, +1 with |Σ_j ∈ S_i χ(j)| ≤ C √(t) for every i.
Let A be a set of n positive integers. Does A contain a sum-free set of size at least frac n 3 + Ω(n), where Ω(n) → ∞ as n → ∞?
What is the largest subset of [N] with no solution to x + 3y = 2z + 2w in distinct integers x, y, z, w?
Suppose that G is a finite group, and let A ⊂ G × G be a subset of density α. Is it true that there are ≫_α |G|^3 triples x, y, g such that (x, y), (gx, y), (x, gy) all lie in A? Note: A is taken as α-dense, i.e. |A| ≥ α |G|^2 [Au16, Question 2]
Let A ⊂ ℤ be a set of n integers. Is there a set S ⊂ A of size (log n)^100 such that the restricted sumsetS hat+ S is disjoint from A?
Suppose that a_1, …, a_k are integers which do not satisfy Rado's condition: thus if Σ_i ∈ I a_i = 0 then I = ∅. It then follows from Rado's theorem that the equation a_1x_1 + ⋯ + a_kx_k = 0 is not partition regular.
Can we improve the lower bound N^1/2 + O(1), at least for infinitely many N?
Are there infinitely many q for which there is a set A ⊂ ℤ/qℤ, |A| = (√(2) + o(1))q^1/2, with A + A = ℤ/qℤ? [Gr24]
Lower bound for c(p) for 1 < p ≤ ∞, improving the known value √(4/7) at p = 2 or the known value 0.64 at p = ∞.
Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d.
What is the largest product-free set in the alternating group A_n?
Does f(r) → ∞? [Gr24]
How many rotated (about the origin) copies of the 'pyjama set' \(x, y) ∈ ℝ^2 : dist(x, ℤ) ≤ ε\ are needed to cover ℝ^2? That is, determine the minimal number of rotations as a function of ε > 0.
Can we pick residue classes a_p pmodp, one for each prime p ≤ N, such that every integer ≤ N lies in at least 10 of them? Erdős remarks that he does not know how to answer it with 10 replaced by 2; this is Erdos689.erdos_689.
We conjecture that the best-known lower bound can be improved.
Which finite groups have the smallest biggest product-free sets? We formalise this as: determine the supremum of exponents α such that every nontrivial finite group of order n contains a product-free set of size ≥ c n^α for some absolute constant c > 0.
Let A ⊂ 𝔽_2^n be a set of density α > 0. Does 10A contain a coset of some subspace of dimension at least n - O(log(1/α))?
Suppose A, B ⊆ 1, …, N both have size at least N^0.49. Must the sumset A + B contain a composite number?
Is there an absolute constant c > 0 such that, whenever A ⊆ ℕ is a set of squares with |A| ≥ 2, the sumset A + A satisfies |A + A| ≥ |A|^1 + c?
Suppose that A + A contains the first n squares. Is |A| ≥ n^1 - o(1)? It is known that necessarily |A| ≥ n^2/3 - o(1), whilst in the other direction there do exist such A with |A| ≪_C n / log^C n for any C.
Let p be a large prime, and let A be the set of all primes less than p. Is every x ∈ 1, …, p-1 congruent to some product a_1 a_2 where a_1, a_2 ∈ A?
Is there always a sum of two squares between X - 1/10X^1/4 and X? We formalize this as an eventual statement for sufficiently large real X.
The no-k-in-line problem: For which k > 2 does every N × N grid with N ≥ k contain a set of (k - 1) N points with no k on a line, so that AllowedSetSize k N is the pigeonhole bound (k - 1) N?
Given n points in the unit disc, must there be a triangle of area at most n^-2+o(1) determined by them?
Let A ⊂ ℤ be a set of size n. For how many θ ∈ ℝ/ℤ must we have Σ_a ∈ A cos(2π aθ) = 0? The answer is the function minZeros.
Let A ⊂ R be a set of positive measure. Does A contain an affine copy of 1, 1/2, 1/4, . . . ?
Same parity betrothed numbers conjecture. Do there exist betrothed numbers (m, n) where both have the same parity (both even or both odd)? All known betrothed pairs consist of one even and one odd number. The requirement m ≠ n is part of the question: IsBetrothed n n says σ(n) = 2n + 1, i.e.
Starting at any n and iterating the map n ↦ a(n), we will always reach 0. - _Antti Karttunen_, Jun 18,20 2017
Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.
Conjecture 1 (Bondy, 1980). Let k ≥ 1 and let G be a k-connected graph on n vertices. If δ(G) ≥ n + k(k-1)/k+1, then for every longest cycle C of G, every path in G - V(C) has at most k-1 vertices.
Borsuk's conjecture, open range: every bounded subset of ℝ^n with at least two points can be partitioned into n + 1 sets of strictly smaller diameter, for 4 ≤ n ≤ 62. The conjecture is known to be true for n ≤ 3 and false for n ≥ 63.
Brennan's conjecture, part 1: B(-2) = 1.
Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.
Büchi's problem There exists a positive integer M such that, for all integers x and a, if (x+n)^2 + a is a square for M consecutive values of n, then a = 0.
Problem 10.7. Let ε be a positive real number. Are there arbitrarily large real numbers α such that α is not a Pisot number and all the fractional parts α^n, n ≥ 1, are lying in an interval of length ε / α? [Bug12b]
Problem 10.1. Are there a transcendental number α and a positive real number ξ such that lVert ξ α^n rVert tends to~0 as~n tends to infinity? [Har19] (Trivial for |α| < 1)
Problem 10.9. There are no real numbers ξ such that 0 ≤ ξ (3/2)^n < 1/2 for every positive integer n, i.e. no Z-number exists. Posed by Mahler [Mah68].
Problem 10.8 (p-adic Littlewood conjecture). For every real number ξ and every prime number p, inf_q ≥ 1 q · lVert q ξ rVert · |q|_p = 0, where lVert · rVert denotes the distance to the nearest integer and |·|_p denotes the p-adic absolute value. Posed by de Mathan and Teulié [dMT04].
Problem 10.61. Let α > 2 be a Pisot number. For every ξ ∈ C(α) the sequence (ξ α^n)_n ≥ 1 is not uniformly distributed modulo one.
Bunyakovsky conjecture If a polynomial f over integers satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers m such that f(m) is prime.
Determine the value of the Busy Beaver function at n = 6.
Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2 simultaneously? That is, does there exist a prime p that is both a Wieferich prime and a Mirimanoff prime? Wikipedia's list of unsolved problems poses this question, citing J. B. Dobson, On Lerch's formula for the Fermat quotient.
Carmichael's totient function conjecture: For every positive natural number n, there exists a natural number m with m ≠ n, such that φ(n) = φ(m).
The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.
Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation ax^n - by^m = c where (m, n) ≠ (2, 2) and x, y > 1.
If p is a prime with p ≡ 1, 9 pmod20 and p = x^2 + 5y^2 with x, y integers, then Σ_k=0^p-1 a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 01 2010
If p is a prime with (p/7) = 1 and p = x^2 + 7y^2 with x, y integers, then Σ_k=0^p-1 (-1)^k a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 17 2010
An integer n > 3 is prime if and only if a(n) ≡ 1 pmodn^2. We have verified this for n up to 8 · 10^5, and proved that a(p) ≡ 1 pmodp^2 for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016
Does Chua's sequence contain every prime?
