# Irregular primes to two billion

(with William Hart and Wilson Ong)

To appear in Mathematics of Computation.

arXiv preprint (May 2016).

## 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**(ζ_{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.

## List of irregular pairs

indices.txt.bz2

compressed = 583690021 bytes (557 MB), uncompressed = 1620383972 bytes (1.51 GB)

uncompressed MD5 checksum = `1058f8add5f70c71d578c7cd87e00a7e`

This file contains one line for each odd prime *p* < 2147483648. Each line contains *p* followed by the irregular indices *k* for *p*. For example, the line

1767218027 63562190 274233542 290632386 619227758 902737892 1279901568 1337429618 1603159110 1692877044

means that *p* = 1767218027 has index of irregularity 9, and that *B*_{k} = 0 mod *p* for *k* = 63562190, ..., 1692877044. If a prime is regular, it appears on the line by itself.

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