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Mathematics of Computation

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Irregular primes to two billion

Authors: William Hart, David Harvey and Wilson Ong
Journal: Math. Comp. 86 (2017), 3031-3049
MSC (2010): Primary 11R18, 11Y40
Published electronically: March 3, 2017
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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 $ \mathbf {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.

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Additional Information

William Hart
Affiliation: Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany

David Harvey
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia

Wilson Ong
Affiliation: University of Cambridge, Department of Engineering, Information Engineering Division, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom

Received by editor(s): May 29, 2016
Published electronically: March 3, 2017
Article copyright: © Copyright 2017 by the authors

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