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
Full paper (PDF, 118 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

Modular rings are used to analyse integers of the form N = 2^m +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 G_m = G²_m–1 – 2G_m–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

References

1. John H Conway & Richard K Guy, The Book of Numbers. New York: Copernicus,
   1996.
2. L. Euler, Opera Omnia. Leipzig: Teubner, 1911.
3. G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers. 3rd edition. London & New York: Oxford University Press, 1954.
4. Roald Hoffmann. Why Buy That Theory? American Scientist, 91(1) (2003): 9-11.
5. J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Integer Class Properties Associated with an Integer Matrix. Notes on Number Theory & Discrete Mathematics. 1 (2) (1995): 53-59.
6. J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. 3 (2) (1997): 61-74.
7. J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, The Characteristics of Primes and Other Integers within the Modular Ring 4 Z and in Class 14 . Notes on Number Theory & Discrete Mathematics. 4 (1) (1998): 1-17.
8. J.V. Leyendekkers & A.G. Shannon, The Analysis of Twin Primes within 6 Z . Notes on Number Theory & Discrete Mathematics. 7(4) (2001): 115-124.
9. J.V. Leyendekkers & A.G. Shannon, Using Integer Structure to Count the Number of Primes in a Given Interval, Notes on Number Theory & Discrete Mathematics. In press.
10. Piergiorgio Odifreddi. The Mathematical Century. (Translated by Arturo Sangalli;
    foreword by Freeman Dyson.) Princeton: Princeton University Press, 2004.
11. Hans Riesel. Prime Numbers and Computer Methods for Factorization. 2nd edition. Progress in Mathematics, Volume 126. Boston: Birkhäuser, 1994.