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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If G is a group, can there exist an exact covering of G by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.) The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.
Is there an infinite Lucas sequence a_0, a_1, … where a_n+2 = a_n+1 + a_n for n ≥ 0 such that all a_k are composite, and yet no integer has a common factor with every term of the sequence?
Let k≥ 3. Is there a choice of congruence classes a_ppmodp for every prime p such that all sufficiently large integers can be written as a_p+tp for some prime p and integer t≥ k?
If A ⊆ ℕ is such that A + A contains all but finitely many integers then limsup 1_A ∗ 1_A(n) = ∞.
Let A⊆ ℕ be an infinite set and consider the following greedy algorithm for a rational x∈ (0,1): choose the minimal n∈ A not used so far such that n≥ 1/x and repeat with x replaced by x-1/n.
Let k≥2. Is it true that, for any distinct integers 1 < n_1 < ⋯ < n_k such that Σ_i=1^k 1/n_i = 1, we must have max(n_i+1 - n_i) ≥ 3?
Is it true that there are only finitely many pairs of intervals I_1, I_2 such that Σ_n_1 ∈ I_1 1/n_1 + Σ_n_2 ∈ I_2 1/n_2 ∈ ℕ?
Is it true that, for all sufficiently large k, there exist finite intervals I_1, dotsc, I_k ⊂ ℕ, distinct, not overlapping or adjacent, with |I_i| ≥ 2 for 1 ≤ i ≤ k such that 1 = Σ_i=1^k Σ_n ∈ I_i 1/n?
Let n≥ 1 and define L_n to be the least common multiple of 1,…,n and a_n by Σ_1≤ k≤ n1/k=a_n/L_n. Is it true that (a_n,L_n)=1 occurs for infinitely many n?
Let k(N) denote the smallest k such that there exists N ≤ n_1 < ⋯ < n_k with frac 1 n_1 + ... + frac 1 n_k = 1 Is it true that lim_N → ∞ k(N) - (e - 1)N = ∞?
If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?
Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)
Let f(N) be the size of the largest A⊆ 1,…,N such that there are no solutions to 1/a= 1/b+1/c with distinct a,b,c∈ A? Estimate f(N). The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].
Is it true that N(b) ≪ log log b?
Let frac a b∈ ℚ_>0 with b squarefree. Are there integers 1 < n_1 < … < n_k, each the product of two distinct primes, such that a/b=1/n_1+⋯+1/n_k?
Are there two finite set of primes P and Q such that 1 = ( Σ_p ∈ P 1/p ) ( Σ_q ∈ Q 1/q ) ? Asked by Barbeau [Ba76]. [Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._
Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with Σ_n ∈ A 1/n > K there exists some S ⊆ A such that 1 - exp(-(c*K)) < Σ_n ∈ S 1/n ≤ 1?
Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: Σ_p ∈ P 1/p = 1 - 1/m?
Is there some constant c>0 such that for every n≥ 1 there exists some δ_k∈ -1,0,1 for 1≤ k≤ n with 0< lvert Σ_1≤ k≤ nδ_k/krvert < c/2^n?
What is the size of the largest A⊆1, …, N such that there is a function δ : A → -1, 1 such that Σ_n∈ A δ n/n = 0 and Σ_n∈ A'δ n/n ≠ 0 for all non-empty A'subsetneq A.
Does there exist a set A ⊆ ℕ such that |A ∩ 1, …, N| = o((log N)^2) and every sufficiently large integer can be written as p + a for some prime p and a ∈ A?
Let k≥ 3 and A⊂ ℕ be the set of kth powers. What is the order of growth of 1_A^(k)(n), i.e. the number of representations of n as the sum of k many kth powers? Does there exist some c>0 and infinitely many n such that 1_A^(k)(n) >n^c?
Is it true that f_k,k(x) ≫_ε x^1-ε for all ε>0? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.
Does there exist a polynomial f(x)∈ℤ[x] such that all the sums f(a)+f(b) with a < b nonnegative integers are distinct?
Writing f_k, 3(x) for the number of integers ≤ x which are the sum of three kth powers, is it true that f_k, 3(x) ≫ x ^ (3 / k)?
Does there exist A = a_1 < a_2 < ⋯ ⊂ ℕ which is a minimal basis of order 2 (i.e. every large integer is the sum of 2 elements from A, and no proper subset of A has this property), such that lim_k→∞ a_k/k^2 = c for some c ≠ 0? Erdős and Graham conjectured a negative answer to this question [ErGr80].
Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_N → ∞ |A ∩ 1,…,N| / N^1/2 be?
Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ 1, …, N| / N^(1/2)?
Let A⊆ ℕ and D(A) be the set of those numbers which occur infinitely often as a_1 - a_2 with a_1, a_2∈ A. What conditions on A are sufficient to ensure D(A) has bounded gaps? This is formalised here using the answer(sorry) mechanism.
Let A = 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, … be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a + b = c + d). What is the order of growth of A?
Do infinitely many pairs (a, a+2) occur in Ulam's sequence?
For what values of 0 ≤ m < n is there a complete sequence A = a_1 ≤ a_2 ≤ ⋯ of integers such that 1. A remains complete after removing any m elements, but 2. A is not complete after removing any n elements.
For what values of t,α ∈ (0,∞) is the sequence ⌊ tα^n⌋ complete (that is, all sufficiently large integers are the sum of distinct integers of the form ⌊ tα^n⌋)?
Is there some c > 0 such that every measurable A ⊆ ℝ^2 of measure ≥ c contains the vertices of a triangle of area 1?
Let α,β∈ ℝ_>0 such that α/β is irrational. Is the multiset ⌊ α⌋,⌊ 2α⌋,⌊ 4α⌋,…∪ ⌊ β⌋,⌊ 2β⌋,⌊ 4β⌋,… complete?
Let f(n) be the maximal k such that there exist integers 1 ≤ a_1 < dotsc < a_k ≤ n such that all sums of the shape Σ_u ≤ i ≤ v a_i are distinct. Is f(n)=o(n)?
Let a_1< a_2 < ⋯ be an infinite sequence of integers such that a_1=1 and a_i+1 is the least integer which is not a sum of consecutive earlier a_js. Show that a_k / k → ∞.
Let c > 0 and n be some large integer. What is the size of the largest set A ⊆ 1, …, ⌊ c n ⌋ such that n is not a sum of a subset of A? Does this depend on n in an irregular way?
There is no consecutive triple of powerful numbers.
Are there any 2-full n such that n+1 is 3-full?
Let B_2(n) be the 2-full part of n (that is, B_2(n)=n/n' where n' is the product of all primes that divide n exactly once). Is it true that, for every fixed k ≥ 1, Π_n ≤ m < n+k B_2(m) ≪ n^2+o(1)?
Let P(n) denote the largest prime factor of n. Show that the set of n with P(n+1) > P(n) has density 1/2.
Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.
Is Erdos375Prop true?
Are there infinitely many n such that 2nchoose n is coprime to 105?
Is there some absolute constant C > 0 such that Σ_p ≤ n 1_pnmid 2n choose n1/p ≤ C for all n?
Is it true that for every k there are infinitely many primes p such that the largest prime divisor of Π_i = 0^k (p ^ 2 + i) is p?
Let F(n) := maxm + p(m) | textrmm < n composite where p(m) is the least prime divisor of m. Is it true that F(n)>n for all sufficiently large n?
Let 2 ≤ k ≤ n - 2. Can C(n, k) be the product of consecutive primes infinitely often? Here k may vary with n: the question asks for infinitely many admissible binomial coefficients, not for a single k that works infinitely often.
Is it true that for every n ≥ 1 there is a k such that n(n + 1) ⋯ (n + k - 1) | (n + k) ⋯ (n + 2k - 1)?
