ON DIVISIBILITY OF GENERALIZED FIBONACCI NUMBERS

摘要

It is well-known that p divides some Fibonacci numbers Fn for any prime number p. Moreover, it is also known that any Lucas number Ln cannot be divided by 5. Let p be a prime number and d(p) be the smallest positive integer n for which p | Fn. In this article, we consider the generalized Fibonacci sequence {Gn}, which satisfies the Fibonacci recurrence relation, but with arbitrary initial conditions. We define an equivalence relation among the sequences {Gn} and give all equivalence classes {Gn}, whose representatives {Gn} satisfy p - Gn for any n 2 N. From the result, we know that if p ±1 (mod 5), then there are infinitely many generalized Fibonacci sequences {Gn} that satisfy p - Gn for any n 2 N, and if p ±2 (mod 5) and d(p) = p+1, then for any generalized Fibonacci sequences {Gn}, we have p|Gn for some n 2 N.