Deciding hard small Turing machines (Busy Beaver)
Prove halting or non-halting of specific small Turing machines that current deciders cannot resolve.
Sub-problems: Lean formalisation of Busy Beaver deciders and results
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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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Erdős minimum overlap problem
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
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Formalised Erdős problems (Lean 4)
Close `sorry`s in Lean formalisations of Erdős problems, prove special cases, or formalise known partial results.
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Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known, starting with the smallest open orders.
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Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
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Large cap sets in F_3^n
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
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Smaller covering designs
Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
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Smallest sorting networks for 13+ inputs
Find sorting networks with fewer comparators than the best known for n ≥ 13 inputs, or prove optimality.
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Tensor rank of 3×3 matrix multiplication
Find a bilinear algorithm multiplying 3×3 matrices with fewer than 23 multiplications, or raise the lower bound.
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Tensor rank of 4×4 matrix multiplication
Find bilinear algorithms that multiply two 4×4 matrices with fewer multiplications. The records are 48 over Q and C (2025) and 47 over GF(2) (2022).
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The Ramsey number R(4,6)
Narrow the gap 36 ≤ R(4,6) ≤ 40. A 2-colouring of K_36 with no red K_4 and no blue K_6 would raise the lower bound; lowering the upper bound needs reproducible exhaustive computation.
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The Ramsey number R(5,5)
Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
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The kissing number in dimension 5
Determine τ5, the maximum number of non-overlapping unit spheres touching a central unit sphere in R^5. Currently 40 ≤ τ5 ≤ 44.
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(m,k)-perfect numbers
There does not exist a (2,5)-perfect number
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A binomial coefficient sum
Let b(n) = a(2n-1). Then the supercongruence b(n p^k) ≡ b(n p^k-1) pmodp^3k holds for positive integers n and k and all primes p ≥ 5. - Zhi-Wei Sun, Nov 16 2019
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A binomial coefficient summation
Let b(n) = a(2n-1). Then the supercongruence b(n p^k) ≡ b(n p^k-1) pmodp^3k holds for positive integers n and k and all primes p ≥ 5. - Zhi-Wei Sun, Nov 16 2019
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A conjecture by Margulis on matrix groups
Let D be the diagonal group of SL_n(ℝ) where n ≥ 3. Then any relatively compact D-orbit in SL_n(ℝ) / SL_n(ℤ) is closed.
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A GCD-Driven Sequence with Universal Jumps
Does every positive integer occur as a difference in this sequence?
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a(0) = 1, a(n) = a(n-1)a(n-1) + 2
Prime for a(1) = 3, a(2) = 11, a(4) = 15131; semiprime for a(3) = 123 = 3 41, a(5) = 228947163 = 3 76315721. a(6), added by Jonathan Vos Post, has 4 prime factors. a(7) = 41 811^2 106693969 317171188688357726699 8272236925540996054440172449761. When is the next prime in the sequence?
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a(n) = (smallest prime > n^2) - n^2
Conjecture: a(n) ≤ 1 + φ(n) for n > 0. This improves on Oppermann's conjecture, which says a(n) < n. - Thomas Ordowski, Dec 17 2014
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a(n) = 2^(2^n)
I conjecture that a(n) ; n>1 are the numbers such that n^4-1 divides 2^n-1, intersection of A247219 and A247165. - M. F. Hasler, Jul 25 2015 This formalizes the reverse direction.
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a(n) = 3a(n-1) + a(n-2) - 3a(n-3)
The current sequence contains primes, including 3, 5, 41, 21523361. Is there an (a, b, c) weighted tribonacci sequence with a, b, c relatively prime which is prime-free?
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a(n) = lcm1,2,…,n/denom(H(n))
It is conjectured that every odd number occurs in this sequence.
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a(n) = the number of values of k <= 10^n such that √(k(k+1)(k+2)(k+3)+1) is prime
Conjecture: a(n)/A006880(n) → 1.77... where A006880(n) is the number of primes ≤ 10^n.
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a(n) = Σ_j=1^n (3^j + (-2)^j)
First primes are a(11) = 264353 and a(17) = 193622861. Additional primes: a(71), a(91), a(431). What is the next prime?
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a(n) = Σ_k=0^n C(2k, k)^3
Conjecture 1 (Peter Bala, 2024): If prime p is in A003625 then a(p^2) ≡ 8 + p^2 pmodp^3.
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a(n) is the integer whose decimal digits are the first n+1 decimal digits of π
Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than 10^-9.
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abc conjecture
For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)
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Agoh-Giuga conjecture
The Agoh-Giuga Conjecture, Agoh's formulation
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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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