Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

[摘要] We introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-(ir)regularity and is based on Genocchi rather than Bernoulli numbers. We say that an odd prime p is G-irregular if it divides at least one of the Genocchi numbers G(2), G(4), . . . , G(p-3), and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound x as x tends to infinity. As a byproduct, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-01

[关键词] Irregular prime;Bernoulli number and polynomial;Euler number and polynomial;Genocchi number;Refined class number;Primitive root;Artin's primitive root conjecture