Irregular primes to two billion

(with William Hart and Wilson Ong)

To appear in Mathematics of Computation.

arXiv preprint (May 2016).


We compute all irregular primes less than 231 = 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 Qp) 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

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 Bk = 0 mod p for k = 63562190, ..., 1692877044. If a prime is regular, it appears on the line by itself.

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