Problems

Let A ⊂ N be infinite with no distinct a, b, c ∈ A such that a | (b + c) with b, c > a. Can |A ∩ [1, N]|/√N have positive lower limit? Must every such A fall below N^1-c infinitely often?

Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)…(n+k))>n^1-ε is at least 1-η, where P(m) is the greatest prime divisor of m? A short argument via the Matomäki-Radziwiłł theorem establishes the…

Let A(n) be the least positive integer not dividing binom2nn. Erdos asked for the behaviour of A(n) for reasonable n. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals…

For n ≥ 4, the natural scalar Poisson-summation certificates cannot prove the Regev-Stephens-Davidowitz Gaussian mass conjecture: any such certificate saturates, so the whole approach is blocked.

Does there exist a good pairwise-coprime sequence u_n with ∑ 1/u_n < ∞ and polynomial growth? What if one only requires u_n ≤ e^o(n)?

Huang, Jiang and Oblomkov conjectured that the Eulerian q-series counting commuting pairs of nilpotent matrices with X^a = Y^b equals an explicit theta-and-eta product, making the point count essentially modular. The conjecture is layered…

Ballantine, Beck, Feigon and Maurischat introduced the subsum polynomial sp(λ,x) := ∏_i (1+x^λ_i) attached to an integer partition λ, studied rational functions built by summing reciprocals of these polynomials over natural classes of…

Let M(n) be the supremum of ∑_a ∈ A 1/(n-a) over pairwise coprime A ⊂ [1,n). Erdos asked whether M(n) ≤ ∑_p<n 1/p + O(1) uniformly. The average order is settled: ∑_n ≤ N M(n) = e^-γ N loglog N + O(N).

Let n_k be the least integer greater than 2k for which ∏_i=1^k (n_k - i) has no prime factor in (k, 2k). How rapidly must n_k grow?

The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg…

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)?

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).

The Lonely Runner Conjecture of Wills and Cusick states that among k+1 runners at distinct constant speeds on a unit circle, each runner is at some time at distance at least 1/(k+1) from all others. Following Rosenfeld's computer-assisted…

For irreducible covering sets of size k, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

Regev and Stephens-Davidowitz conjectured that Z^n maximizes the Gaussian mass Θ_L(t) = ∑_x ∈ L e^-t|x|^2 over stable lattices for every t > 0. The sharp inequality holds for every integral unimodular lattice of rank n ≤ 32, with equality…

Let x_n = tan(∑_k=1^n arctan k). Amdeberhan, Medina and Moll conjectured that x_n ∉ Z for every n ≥ 5. Any integer value x_n = m must satisfy |m| ≥ e^(1/2+o(1)) n log n, which forces #1 ≤ n ≤ N : x_n ∈ Z = O(log N). The conjecture…