The structure of Fibonacci numbers in modular rings

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 10, 2004, Number 1, Pages 12–23
Full paper (PDF, 130 Kb)

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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

An analysis is made of the class and row structures of Fibonacci numbers within the modular ring Z4. It is found that the class structure repeats the pattern ̅04 ̅14 ̅14 ̅24 ̅34 ̅14. Two thirds of the rows in the ring array are even and all are a sum of Fibonacci numbers. Sums of Fibonacci numbers, covering ten, five and three consecutive numbers or number types, had factors of 11, 11 × 31, or 101; (these include specific sets). The Fibonacci number primes all belong to the Class ̅14 and therefore equal a sum of squares. There is only one unique set of squares with no common factors. The factors found for the sums have a link with Fermat and Mersenne numbers.

AMS Classification

  • 06F25
  • 11B39

References

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Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2004). The structure of Fibonacci numbers in modular rings. Notes on Number Theory and Discrete Mathematics, 10(1), 12-23.

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