On the modular Erd\H{o}s-Burgess constant

Abstract

Let $n$ be a positive integer. For any integer $a$, we say that $a$ is idempotent modulo $n$ if $a^2\equiv a\pmod n$. The $n$-modular Erd\H{o}s-Burgess constant is the smallest positive integer $\ell$ such that any $\ell$ integers contain one or more integers whose product is idempotent modulo $n$. We gave a sharp lower bound of the $n$-modular Erd\H{o}s-Burgess constant, in particular, we determined the $n$-modular Erd\H{o}s-Burgess constant in the case when $n$ is a prime power or a product of pairwise distinct primes.