On the irregular primes with respect to Euler polynomials

TL;DR

This paper investigates primes that divide the numerator of certain Euler polynomial values, establishing links to class number divisibility and exploring their distribution, contributing to the understanding of irregular primes in number theory.

Contribution

It introduces a new notion of irregular primes related to Euler polynomials and connects their properties to class number divisibility, expanding classical prime irregularity concepts.

Findings

Identifies conditions under which primes are irregular with respect to Euler polynomials.

Establishes a connection between prime irregularity and class number divisibility.

Provides results on the distribution of such irregular primes.

Abstract

An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link the regularity of primes to the divisibility of some class numbers. Besides, we obtain some results on the distribution of such irregular primes. Remark: This preprint has been withdrawn, because all the results in it have been included or improved in a recent preprint. (See arXiv:1809.08431).