Problems

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

How long must an interval be to contain distinct representatives x_i, with a_i | x_i, for every n-element set of moduli A = a_1, …, a_n?

Let h(n) count powerful integers in [n^2, (n+1)^2). What is the extremal order of h(n)?

Let a_1 = 2 and a_2 = 3 and continue the sequence by appending to a_1, …, a_n all possible values of a_ia_j - 1 with i ≠ j. Is it true that the set of integers which eventually appear has positive density?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Whether there are infinitely many integers a, b, n with a, b ≥ ε n such that a!· b! divides n!·(a+b-n)! while a+b exceeds n by more than C·log n.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Is the maximum size of a set A⊆ 1,…,N such that ab+1 is never squarefree (for all a,b∈ A) achieved by taking those n≡ 7pmod25? Resolved for all sufficiently large N: any near-maximal A is contained in n≡ 7pmod25 or n≡ 18pmod25, leaving…

Estimate the least excess g_k(N) forcing k integers whose pairwise sums all lie in a dense subset of 1, …, 2N; in particular, determine the positive variant h_4(n).

Let F(n) be the largest A⊆1,…,n with anmid bc for distinct a,b,c∈ A. Is F(n)=π(n)+(C+o(1)) n^2/3(log n)^-2 for some constant C?

Han and Xiong extended the Gaussian binomial coefficient to positive rational index and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at the integer point. Ono's…

If A ⊆ N has unbounded dyadic-shell counts and ∑_n ∈ A |θ n| = ∞ for every 0 < θ < 1, must A be complete - is every sufficiently large integer a sum of distinct elements of A?

Let X_1,…,X_n be independent nonnegative random variables with EX_i ≤ 1, and let S be their sum. Is P(S < ES + 1) ≥ 1/e? Feige proved the constant 1/13 and conjectured the sharp 1/e. Three independent July 2026 proofs settle it, both…

For a closed infinite set F ⊆ C, let μ(F) be the infimum of |z : |f(z)| < 1| over monic polynomials with zeros in F. Is μ(F) determined only by the transfinite diameter of F?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Erdős asked whether every n-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least (1+o(1))n^2. Disproved: an explicit high-dimensional construction beats the conjectured constant.

If W(k) is the least N such that every two-colouring of 1, …, N contains a monochromatic k-term arithmetic progression, must W(k+1) - W(k) → ∞?

Can a nonabelian group admit a Rota-Baxter operator that is surjective but not injective? A construction shows yes.

For a single-source unsplittable flow, find the optimal universal additive constant C s.t. every feasible fractional flow x with arc costs c should admit an unsplittable routing y with c^top y ≤ c^top x and y_a ≤ x_a + C · D on every arc.…