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 aix≡bi(modp),1≤i≤n, are solvable. We control the insolvability of ax≡b(modp) 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.