On the Distribution of Pseudopowers

TL;DR

This paper investigates the distribution of pseudopowers, providing improved bounds on the smallest such numbers by combining exponential sum bounds with new insights into the multiplicative order of integers modulo primes.

Contribution

It introduces a novel approach that combines exponential sum bounds with new average results on multiplicative orders to improve bounds on pseudopowers.

Findings

Improved upper bounds for the least pseudopower to base g.

New results on the average behavior of multiplicative orders.

Enhanced understanding of pseudopower distribution.

Abstract

An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of $g$ modulo prime numbers.