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Abstract:

Abstract. Elliptic curves and cyclotomy are the current main streams for primality proving. Both rely on relatively simple ideas derived from the Lucas-- Pocklington lemma and use less simple theories. They have been studied and implemented during the last decade. We make a first attempt towards combining methods specific for the two approaches, based on the new concept of dual elliptic primes. Two primes m;n are dual elliptic if there is an elliptic curve mod n having m points and, reciprocally, a curve mod m having n points. Dual elliptic pseudoprimes are two, e.g. strong pseudoprimes, which have additionally the dual elliptic property. It can be proved that, not only are dual elliptic pseudoprimes close in absolute value, but, if they are composite, then their least prime factors are also dual elliptic pseudoprimes and accordingly close. This fact allows a sensible improvement of the final superpolynomial trial division stage of the cyclotomy primality proof [19]. An implementation has been made, by adapting the ECPP [10] implementaion of elliptic curves and the CYCLOPROV cyclotomy program, written, respectively, by the two authors. The implementation has confirmed the theoretical espectations. With this result we improve upon an earlier method of Lenstra [?], which required factored parts sj(n t

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