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）nLk（modLm...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）nLk（modLm）, F2mn＋k ≡（-1）（m＋1）nFk （modLm）, L2mn＋k ≡ （-1）mn Lk（mod Fm） and F2mn＋k≡ （-1）mn Fk （mod Fm）. By the achieved identities, divisibility properties of Fibonacci and Lueas 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.展开更多
文摘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）nLk（modLm）, F2mn＋k ≡（-1）（m＋1）nFk （modLm）, L2mn＋k ≡ （-1）mn Lk（mod Fm） and F2mn＋k≡ （-1）mn Fk （mod Fm）. By the achieved identities, divisibility properties of Fibonacci and Lueas 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.
