Irregular primes to one million

Abstract

Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat’s "Last Theorem" and Vandiver’s conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six.