Publication

Irregular primes and Cyclotomic Invariants to Twelve Million

Mohammad Amin Shokrollahi
2001
Journal paper
Abstract

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to twelve million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes (Shokrollahi (1996)). The latter idea reduces the problem to that of finding zeros of a polynomial over Fp of degree &st (p - 1)/2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives an O(p log 2 p log log p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schönhage- Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for which p-1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer- Vandiver conjecture and for computing the cyclotomic invariants of a prime

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