Abstract. In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the "number field sieve", which was invented by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime factors, not just integers of a special form like the ninth Fermat number. Under reasonable heuristic assumptions, the analysis predicts that the time needed by the general number field sieve to factor n is exp((c+o(l))(logn)1/3(loglogn)2/3) (for n-*oo), where c=(64/9)l/3 = 1.9223. This is asymptotically faster than all other known factoring algorithms, such äs the quadratic sieve and the elliptic curve method. There does not yet exist an Implementation of the number field sieve for general integers, so that a practical comparison cannot yet be made.
Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 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 < (p − 1)/2 among the quadratic nonresidues mod p. Use of fast polynomial gcdalgorithms 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.
The Madelung constant-essentially the Coulomb energy density of a crystal-is usually calculated via Ewald error function expansions or, for the simpler cubic structures, by the 'cosech' series of modern vintage. By considering generalised functional equations for multidimensional zeta functions, we provide explicit expansions for the spatial potential and energy density of three-dimensional periodic structures. These formulae, involving only elementary functions, are suitable for systematic calculation of Madelung constants of arbitrary point-charge crystals. We indicate how zeta function relations may be used for dimensional reduction of certain multiple sums arising in the special cubic cases.
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