Some properties of Fermatian numbers

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 10, 2004, Number 2, Pages 25–33
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Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales, Kensington 1465, &
KvB Institute of Technology, 99 Mount Street, North Sydney, NSW 2065, Australia

Abstract

This paper looks at some basic number theoretic properties of Fermatian numbers. We define the n-th reduced Fermatian number in terms of

\underline{q}_n = \begin{array}{l l} -q^n \underline{q}_{-n} & \quad (n < 0) \\ 1 & \quad (n = 1) \\ 1 + q + q^2 + ... + q^{n-1} & \quad (n > 0) \end{array} .

so that 1n = n, and 1n! = n!, where qn! = qnqn−1…q1.
Some congruence properties and relationships with Bernoulli and Fibonacci numbers are explored. Some aspects of the notation and meaning of the Fermatian numbers are also outlined.

AMS Classification

  • 11B65
  • 11B39
  • 05A30

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Cite this paper

Shannon, A. G. (2004). Some properties of Fermatian numbers. Notes on Number Theory and Discrete Mathematics, 10(2), 25-33.

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