Fermat and Mersenne numbers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 11, 2005, Number 4, Pages 17—24
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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

Fermat numbers (Fn = 22n + 1) and Mersenne numbers (Mm = 2m − 1), m odd, are compared on the basis of integer structure, using the modular rings Z4 and Z6. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2n and the right end digits for Fn result in fewer numbers over a given range than those for Mm. This is shown with two functions, which link the two numbers and show that Fn = (2x2 + y2 − 1 + 1): for primes y = n, but when n > 4, y ≠ n.

AMS Classification

  • 11A41
  • 11A07

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Cite this paper

APA

Leyendekkers, J. V., and Shannon, A. G. (2005). Fermat and Mersenne numbers. Notes on Number Theory and Discrete Mathematics, 11(4), 17-24.

Chicago

Leyendekkers, JV, and AG Shannon. “Fermat and Mersenne Numbers.” Notes on Number Theory and Discrete Mathematics 11, no. 4 (2005): 17-24.

MLA

Leyendekkers, JV, and AG Shannon. “Fermat and Mersenne Numbers.” Notes on Number Theory and Discrete Mathematics 11.4 (2005): 17-24. Print.

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