Simultaneous insolvability of exponential congruences

Abstract

We determine a necessary and sufficient condition for the infinitude of primes $p$ such that none of the equations $a_i^x\equiv b_i(\mathrm{mod}p),1\le i\le n,$ are solvable. We control the insolvability of $a^x\equiv b(\mathrm{mod}p)$ 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.

13 pages. Rewrite and restructure article. Consider a more general problem on (in)divisibilities of orders