Problems

For A ⊆ F_p let A^* = (A+A) ∪ (AA). Sárközy conjectured that for all large primes, every set of size at least c√p has A^* = F_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together…

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.

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

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…

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

If a/b ∈ Q_>0 and b is squarefree, can a/b always be written as a finite sum of reciprocals of distinct products of two distinct primes?

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.

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?

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?

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?

Nathanson asked which subsets of N can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.