Fermat’s theorem on binary powers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 11, 2005, Number 2, Pages 13—22
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Authors and affiliations

J. V. Leyendekkers
The University of Sydney, 2006 Australia

A. G. Shannon
Warrane College, Kensington, NSW 1465,
& KvB Institute of Technology, North Sydney, NSW 2060, Australia

Abstract

Modular rings are used to analyse integers of the form N = 2m +1. When m is odd, the integer structure prevents the formation of primes. When m is even, N ‘commonly’ has a right-end-digit of 5 and so is not a prime then. However, a sequence defined by m = 4 + 4q, q = 0, 1, 2, 3, can generate some primes as the right-end-digit is 7. Elements of this sequence satisfy the non-linear recurrence relation Gm = G2m–1 – 2Gm–1 + 2. Fermat numbers, where m = 2n satisfy this recurrence relation. However, in this case, the integer structure reveals that primes are limited to n < 5.

AMS Classification

  • 11A41
  • 11A07

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Cite this paper

APA

Leyendekkers, J. V., and Shannon, A. G. (2005). Fermat’s theorem on binary powers. Notes on Number Theory and Discrete Mathematics, 11(2), 13-22.

Chicago

Leyendekkers, JV, and AG Shannon. “Fermat’s Theorem on Binary Powers.” Notes on Number Theory and Discrete Mathematics 11, no. 2 (2005): 13-22.

MLA

Leyendekkers, JV, and AG Shannon. “Fermat’s Theorem on Binary Powers.” Notes on Number Theory and Discrete Mathematics 11.2 (2005): 13-22. Print.

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