Fibonacci and Lucas Congruences and Their Applications
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Acta Mathematica Sinica, English Series  2011, Vol. 27 Issue (4): 725-736    DOI: 10.1007/s10114-011-9744-0
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Fibonacci and Lucas Congruences and Their Applications
Department of Mathematics, Faculty of Science and Arts, Sakarya University, 54187 Sakarya, Turkey
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Abstract In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L2mn+k ≡ (-1)(m+1)n Lk (mod Lm), F2mn+k ≡ (-1)(m+1)n Fk (mod Lm), L2mn+k ≡ (-1)mn L2mn+k(mod Fm) and F2mn+k ≡ (-1)mn Fk (mod Fm). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number Ln such that Ln = L2ktLmx2 for m > 1 and k ≥ 1. Moreover it is proved that Ln = LmLr is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.  
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Refik KESK?N Bahar
Key wordsFibonacci numbers   Lucas numbers   congruences     
Received: 2009-12-17;
Cite this article:   
Refik KESK?N Bahar,DEM?RTüRK B?T?M. Fibonacci and Lucas Congruences and Their Applications[J]. Acta Mathematica Sinica, English Series, 2011, 27(4): 725-736.
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