A new problem from zero-sum theory
Acta Arithmetica
MSC: Primary 11B30; Secondary 11B75, 20K01
DOI: 10.4064/aa251115-21-1
Published online: 31 May 2026
Abstract
Let $p$ be a prime number. We denote by $\mathsf {s}_{p}^{*}$ the smallest integer $l$ such that, out of any given $l$ integers coprime with $p$, one can select $p$ integers such that the sum of the $p$ integers is a multiple of $p$, but not a multiple of $p^2$. It is conjectured that $\mathsf {s}_{p}^{*}=2p+1$ for any prime number $p\ge 3$. We give a non-trivial upper bound $\mathsf {s}_{p}^{*}\le 3p-2$.
