## Abstract

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.

Original language | English |
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Pages (from-to) | 59-80 |

Number of pages | 22 |

Journal | Journal of Number Theory |

Volume | 205 |

DOIs | |

Publication status | Published - Dec 2019 |

## Keywords

- Artin's primitive root conjecture
- Bernoulli number and polynomial
- Euler number and polynomial
- Genocchi number
- Irregular prime
- Primitive root
- Refined class number