%0 Journal Article %A Refik KESK?N Bahar %A DEM?RTüRK B?T?M %T Fibonacci and Lucas Congruences and Their Applications %D 2011 %R 10.1007/s10114-011-9744-0 %J Acta Mathematica Sinica %P 725-736 %V 27 %N 4 %X 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.   %U https://actamath.cjoe.ac.cn/Jwk_sxxb_en/EN/10.1007/s10114-011-9744-0