Simultaneous insolvability of exponential congruences

Abstract

<p>We determine a necessary and sufficient condition for the infinitude of primes <em>p</em> such that none of the equations <em>​​​​​​​a<span>^x</span>_i</em> ≡ b_i (mod <em>p</em>), 1 ≤ <em>i</em> ≤ <em>n</em>, are solvable. We control the insolvability a<span>^x </span>≡ b(mod <em>p</em>)<span></span>of by power residues for multiplicatively independent <em>a</em> and <em>b</em>, 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.<br></p>