Simultaneous insolvability of exponential congruences

TL;DR

This paper characterizes when infinitely many primes exist such that certain exponential congruences are unsolvable, linking to Artin's primitive root conjecture and exploring divisibility and order properties.

Contribution

It provides a necessary and sufficient condition for the infinitude of primes with unsolvable exponential congruences, extending understanding of primitive roots and order divisibility.

Findings

Identifies conditions for infinite primes where specific exponential congruences are insoluble.

Analyzes power residues and order divisibility in multiplicatively independent and dependent cases.

Connects results to Artin's primitive root conjecture and its variants.

Abstract

We determine a necessary and sufficient condition for the infinitude of primes $p$ such that none of the equations $a_i^x \equiv b_i \pmod{p}, 1 \le i \le n,$ are solvable. We control the insolvability of $a^x \equiv b \pmod{p}$ by power residues for multiplicatively independent $a$ and $b$, and by divisibilities and, most importantly, parities of orders in multiplicatively dependent cases. We also consider a more general problem concerning divisibilities of orders. The problems are motivated by Artin's primitive root conjecture and its variants.