Ben Green's Open Problem 41
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.
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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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Ben Green's Open Problem 45
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.
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Ben Green's Open Problem 46
We conjecture that the best-known lower bound can be improved.
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Ben Green's Open Problem 5
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.
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Ben Green's Open Problem 50
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))?
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Ben Green's Open Problem 58
Suppose A, B ⊆ 1, …, N both have size at least N^0.49. Must the sumset A + B contain a composite number?
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Ben Green's Open Problem 60
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?
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Ben Green's Open Problem 61
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.
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Ben Green's Open Problem 62
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?
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Ben Green's Open Problem 66
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.
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Ben Green's Open Problem 72
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?
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Ben Green's Open Problem 77
Given n points in the unit disc, must there be a triangle of area at most n^-2+o(1) determined by them?
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Ben Green's Open Problem 82
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.
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Ben Green's Open Problem 94
Let A ⊂ R be a set of positive measure. Does A contain an affine copy of 1, 1/2, 1/4, . . . ?
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Betrothed numbers
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.
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Binary representation of primes that divide a number, in decimal
Starting at any n and iterating the map n ↦ a(n), we will always reach 0. - _Antti Karttunen_, Jun 18,20 2017
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Bloch and Landau constants
Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.
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Bondy's conjecture on longest cycles in highly connected graphs
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.
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Borsuk's conjecture
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.
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Brennan's Conjecture
Brennan's conjecture, part 1: B(-2) = 1.
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Brocard's Conjecture
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.
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Büchi's problem
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.
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Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers
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]
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Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers
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)
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Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers
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].
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Bugeaud Collection of Conjectures and Open Questions: p-adic Littlewood Conjecture
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].
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Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set
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.
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Bunyakovsky conjecture
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.
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Busy Beaver
Determine the value of the Busy Beaver function at n = 6.
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Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2?
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.
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Carmichael's totient function conjecture
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).
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Casas-Alvero Conjecture
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.
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Catalan-Mersenne numbers
Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
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Catalan's conjecture and related Diophantine equations
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.
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Central binomial sum a(n) = Σ_k=0^n (-4)^k C(n, k)^2 C(n-k, k)^2
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
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Central binomial sum a(n) = Σ_k=0^n C(n, k)^2 C(n-k, k)^2 (-16)^k
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
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Central trinomial coefficients
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
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Chua's Euclidean prime sequence
Does Chua's sequence contain every prime?
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Chvátal's Conjecture
If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.
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Class number problem for real quadratic fields
There are infinitely many real quadratic fields ℚ(√d) with class number one, where d > 1 is a squarefree integer.
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Coefficients of Π_k>0 (1 - x^k/k!)
The coefficients c(n) of A(x)^2 = (Σ_n ≥ 0 a(n) x^n)^2 differ in sign from c(n-1) if and only if n is a triangular number. - _Peter Bala_, Mar 17 2022
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Collatz step differences
Conjecture 1: More than half of the terms are 0. - _Ya-Ping Lu_, May 04 2024
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Concatenation of the next n numbers
"The second term is a prime. When is the next prime, if there is another? - _N. J. A. Sloane_, Dec 16 2016"
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Congruent Number
Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.
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Conjecture 1.35(c)
Do there exist simple pro-orderable groups?
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Conjecture 1.40
Is a group a nilgroup if it is the product of two normal nilsubgroups? Since H and K are normal, the product HK coincides with the join H sqcup K, so "G is the product of H and K" is stated as H sqcup K = G.
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Conjecture 1.74 (Minimal topological groups)
Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete.
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Conjecture 19.25
Let G and H be finite groups of the same order with Σ_g ∈ G φ(|g|) = Σ_h ∈ H φ(|h|), where φ is the Euler totient function. Suppose that G is simple. Is H necessarily simple?
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Conjecture 20.76
Let G be a finite p-group and assume that all abelian normal subgroups of G have order at most p^k. Is it true that every abelian subgroup of G has order at most p^2k?
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Conjecture 8.8
Does there exist a non-cyclic finitely presented group G which contains an element a such that each element of G is conjugate to some power of a? Here a power of a means a^n for some n ∈ ℤ.
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