Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients

TL;DR

This paper demonstrates that for large cube-free moduli, every residue class can be expressed as a product of 14 small integers within a specific interval, with implications for Fermat quotients and additive number theory.

Contribution

It establishes a new bound on representing residue classes as products of small integers, approaching the theoretical lower limit before Burgess bounds improve.

Findings

Every residue class modulo large cube-free q can be represented as a product of 14 small integers.

The interval size is near the lower limit before Burgess bounds on quadratic nonresidues improve.

Applications to Fermat quotients and additive properties are provided.

Abstract

We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.