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

TL;DR

This paper introduces G-irregularity, a new concept based on Genocchi numbers, and explores its properties, distribution, and relation to class numbers and primitive roots, extending classical prime irregularity theory.

Contribution

It defines G-irregularity for primes using Genocchi numbers and studies its distribution and connection to class numbers and primitive roots, providing new insights beyond classical irregularity.

Findings

G-irregular primes are infinitely many in each primitive residue class.

Established lower bounds for the count of G-irregular primes up to x.

Connected G-irregularity to divisibility of class numbers and primitive roots.

Abstract

In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, 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 by-product, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root.