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
Full paper (PDF, 114 Kb)

Details

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 (F_n = 2²ⁿ + 1) and Mersenne numbers (M_m = 2^m − 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 F_n result in fewer numbers over a given range than those for M_m. This is shown with two functions, which link the two numbers and show that F_n = (2^{x2 + y2 − 1} + 1): for primes y = n, but when n > 4, y ≠ n.

AMS Classification

• 11A41
• 11A07

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