Problems

For S(x) = #(a,b) : a + b ≤ x, σ(a) + σ(b) = σ(a+b), is S(x) ~ cx? The preprint claims S(x) grows faster than x (log x)^R for every fixed R, ruling out the linear asymptotic.

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?

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…

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.

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

Let k≥ 3 and A be an additive basis of order k. Does there exist a constant c=c(k)>0 such that if r(n)≥ clog n for all large n (where r(n) counts representations of n as a sum of at most k elements of A) then A must contain a minimal basis…

For every real ξ>0 the sequence of integer parts [ξ 7^n], n=0,1,2,…, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~7.

Let k,r≥ 2. Does there exist a set A⊆ N that contains no non-trivial arithmetic progression of length k+1, yet in any r-colouring of A there must exist a monochromatic non-trivial arithmetic progression of length k? Answered in the…

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…

A monic prime P of F_q[T] is a c-Wieferich prime if ρ_P(1) ≡ 1 bmod P^2 for the Carlitz module ρ. On limited data and proofs in degrees 2 and 3, Thakur suggested in 2015 that in odd characteristic every c-Wieferich prime has degree…

For the least t_k(n) with n | t_k(n)(t_k(n)+1)…(t_k(n)+k-1), do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with c = 1/2048 admissible in the t_2 bound.

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…

The paper gives counterexamples in dimensions eight and nine to two problems: the Cartesian-product problem posed by Cassels for critical determinants and formulated by Zong for lattice packings, and a question raised by Sarnak, formulated…

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

Find a nontrivial lower bound or upper bound of QMX(n).

It is conjectured that if division and reciprocation, as well as subtraction and negation, are also made allowed operations, then they are never necessary to achieve a maximizing combination.