Irregular primes to two billion

TL;DR

This paper computationally verifies all irregular primes below two billion, confirms the Kummer-Vandiver conjecture for these primes, and introduces a more efficient algorithm for calculating irregular indices.

Contribution

It provides the first comprehensive computation of irregular primes up to 2^31 and improves the efficiency of irregular index calculations using an adapted Rader's algorithm.

Findings

All irregular primes less than 2^31 verified.

Kummer-Vandiver conjecture holds for these primes.

New algorithm reduces computation time for irregular indices.

Abstract

We compute all irregular primes less than 2^31 = 2 147 483 648. We verify the Kummer-Vandiver conjecture for each of these primes, and we check that the p-part of the class group of Q(zeta_p) has the simplest possible structure consistent with the index of irregularity of p. Our method for computing the irregular indices saves a constant factor in time relative to previous methods, by adapting Rader's algorithm for evaluating discrete Fourier transforms.