THE CONTINUING SEARCH FOR LARGE ELITE PRIMES

Author:

CHAUMONT ALAIN1,LEICHT JOHANNES2,MÜLLER TOM2,REINHART ANDREAS2

Affiliation:

1. Laboratoire M.S.M., Université Louis Pasteur, 4, rue B. Pascal, 67000 Strasbourg, France

2. Institut für Cusanus-Forschung an der Universität und der Theologischen, Fakultät Trier, Domfreihof 3, 54290 Trier, Germany

Abstract

A prime number p is called elite if only finitely many Fermat numbers 22n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 1012 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Reference12 articles.

1. �ber Primzahlen, nach denen (fast) alle Fermatschen Zahlen quadratische Nichtreste sind

2. L. Euler, Commentarii Academiae Scientiarum Imperialis Petropolitanae 6 (Imperial Academy of Sciences, St. Petersburg, 1739) pp. 103–107.

3. P. S. Fermat, Oeuvres 2 (Gauthier-Villars, Paris, 1894) p. 309.

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. On the subpartitions of the ordinary partitions;The Ramanujan Journal;2010-04-15

2. On the Fermat Periods of Natural Numbers;J INTEGER SEQ;2010

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