Irregular primes to 163 million

TL;DR

This paper reports the computation of all irregular primes below 163 million, confirming the Kummer-Vandiver conjecture and lambda-invariant properties for these primes, thus providing extensive empirical evidence in number theory.

Contribution

It extends the known range of irregular primes significantly and verifies key conjectures for all primes within this range, offering valuable computational data.

Findings

All irregular primes less than 163 million satisfy the Kummer-Vandiver conjecture.

For these primes, the lambda-invariant equals the index of irregularity.

The results support the validity of conjectures in algebraic number theory up to high bounds.

Abstract

We compute all irregular primes less than 163,577,356. For all of these primes we verify that the Kummer-Vandiver conjecture holds and that the lambda-invariant is equal to the index of irregularity.