Vol. LXXIX, 2 (2010)
p. 199 - 208

Trigonometric expressions for Fibonacci and Lucas numbers

B. Sury

Received: April 24, 2009;   Revised: April 21, 2010;   Accepted: June 6, 2010

Abstract.   The amount of literature bears witness to the ubiquity of the Fibonacci numbers and the Lucas numbers. Not only are these numbers popular in expository literature because of their beautiful properties, but also the fact that they `occur in nature' adds to their fascination. Our purpose is to use a certain polynomial identity to express these numbers in terms of trigonometric functions. It is interesting that these expressions provide natural proofs of old and new divisibility properties for the Fibonacci numbers. One can recover naturally some divisibility properties and discover/observe some others which seem to be new. There are some fascinating open questions about the periodicity of the Fibonacci sequences modulo primes and we shall also prove some partial results on this.

AMS Subject classification: Primary:  11B39  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

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