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

TL;DR

This paper investigates the $n$-modular Erd ext{"o}s-Burgess constant, providing sharp bounds and exact values for prime power and distinct prime product cases, advancing understanding of idempotent products modulo $n$.

Contribution

It establishes a sharp lower bound for the $n$-modular Erd ext{"o}s-Burgess constant and determines its exact value for prime power and pairwise distinct prime cases.

Findings

Sharp lower bound for the constant established

Exact values determined for prime power cases

Exact values determined for products of distinct primes

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.