Simultaneous insolvability of exponential congruences

2019/12/05 by Järviniemi, Olli
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.02526

Abstract

We determine a necessary and sufficient condition for the infinitude of primes p such that none of the equations a_i^x ≡ b_i \pmodp, 1 ≤ i ≤ n, are solvable. We control the insolvability of a^x ≡ b \pmodp 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.

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