Agoh-Giuga conjecture
The Agoh-Giuga Conjecture, Agoh's formulation
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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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Agrawal's conjecture
Agrawal's Primality Conjecture. Does the congruence (X-1)^n ≡ X^n - 1 pmodn, X^r-1 imply n is prime (with a specific exception for n^2 ≡ 1 pmodr)? While the "if" direction is a known theorem, the "only if" direction remains a conjecture.
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Algebraic consequences of the Farrell–Jones conjecture
Vanishing of the reduced projective class group for integral group rings. If G is torsion-free, that is, if its only element of finite order is 1, then every finitely generated projective module over ℤ[G] is stably free.
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Alternating sum of signs of powers of 3/2
For n large enough, does a(n) > √(n) always hold?
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Amicable numbers
Relatively prime amicable numbers conjecture. Do there exist amicable numbers (a, b) with gcd(a, b) = 1? All known amicable pairs share a common factor. It is an open question whether a pair of relatively prime amicable numbers can exist. Reference: Wikipedia
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An Arithmetic Sum Associated with the Classical Theta Function
Conjecture 1.1: For any odd prime k, the sum associated with the classical theta function θ_3, S(k) is positive.
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Andrica's conjecture
Andrica's conjecture The inequality √(p_n+1)-√(p_n) < 1 holds for all n, where p_n is the n-th prime number.
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Apéry numbers
For each n = 1, 2, 3, … the polynomial a_n(x) = Σ_k=0^n C(n, k)^2 C(n+k, k) x^k is irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 21 2013
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Are prime numbers among sums of prime numbers distributed as frac n2ln(n)?
The conjecture claims that π_n∼frac n2ln(n). In other words, primes are distributed among the much sparser sequence (S_n)_n with essentially the same density as in the positive integers, up to a factor of 2. MathOverflow 434111.
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Array read by upward antidiagonals
A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers 2n+1 and 2n+3 for n > 0, at least one will be the product of 2 primes (not necessarily distinct).
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Artin's conjecture on primitive roots
Artin's Conjecture on Primitive Roots, first half. Let a be an integer that is not a square number and not −1. Then the set S(a) of primes p such that a is a primitive root modulo p has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
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Ascending descending base exponent transform of 2^n
The first prime terms in this (always odd) sequence are a(1) = 3, a(3) = 41, and a(4) = 593. What is the next prime? The OEIS comment currently says a(5) = 543, but this conflicts with its defining formula, b-file, and examples: the actual index-five term is the composite number 135457.
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Ascending descending base exponent transform of factorials
Is there a nontrivial power after a(4) = 5^3?
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Ascending descending base exponent transform of squares
The smallest prime in this sequence is a(2) = 5. What is the next prime?
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Asymptotic density of powerful numbers
Can the exponent 1/6 in the error term of the Bateman–Grosswald asymptotic be improved unconditionally? That is, is there δ > 0 such that Q(x) = ζ(3/2)/ζ(3) x^1/2 + ζ(2/3)/ζ(2) x^1/3 + O(x^1/6 - δ)? Improvements are known under the Riemann Hypothesis.
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Babai–Seress Conjectures on the Diameter of Finite Groups
Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant C such that the diameter of the alternating group A_n satisfies diam(A_n) ≤ n^C. Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.580029-0)
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Balanced prime conjecture
Let p_k be the k-th prime number. Are there infinitely many n such that (p_n + p_n+2) / 2 is prime?
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Banach-Mazur Rotation Problem
The Banach–Mazur rotation problem asks whether every separable Banach space whose group of linear isometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert space.
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Barker sequences
Every Barker sequence has length at most 13.
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Bateman-Horn Conjecture
The Bateman-Horn Conjecture Given a finite collection of distinct irreducible polynomials non-constant f_1, f_2, …, f_k ∈ ℤ[x] with positive leading coefficients that satisfy the Schinzel condition, the number of positive integers n ≤ x for which all polynomials f_i are simultaneously prime is…
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Beal conjecture
The Beal Conjecture: if we are given positive integers A, B, C, x, y, z such that x, y, z > 2 and A^x + B^y = C^z then A, B, C have a common divisor.
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Beaver Math Olympiad (BMO)
BMO#1) Let (a_n)_n ≥ 1 and (b_n)_n ≥ 1 be two sequences such that (a_1, b_1) = (1, 2) and (a_n+1, b_n+1) = begincases (a_n-b_n, 4b_n+2) & if a_n ≥ b_n cr (2a_n+1, b_n-a_n) & if a_n < b_n endcases for all positive integers n. Does there exist a positive integer i such that a_i = b_i?
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Beck–Fiala theorem and conjecture
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.
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Ben Green's Open Problem 1
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 → ∞?
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Ben Green's Open Problem 16
What is the largest subset of [N] with no solution to x + 3y = 2z + 2w in distinct integers x, y, z, w?
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Ben Green's Open Problem 18
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]
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Ben Green's Open Problem 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?
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Ben Green's Open Problem 21
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.
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Ben Green's Open Problem 31
Can we improve the lower bound N^1/2 + O(1), at least for infinitely many N?
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Ben Green's Open Problem 33
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]
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Ben Green's Open Problem 35
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 = ∞.
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Ben Green's Open Problem 37
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.
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Ben Green's Open Problem 4
What is the largest product-free set in the alternating group A_n?
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Ben Green's Open Problem 40
Does f(r) → ∞? [Gr24]
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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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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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