Lucas and fibonacci numbers and some diophantine Equations

Abstract

The Lucas numbers υn and the Fibonacci numbers υn are defined by υ1 = 1, υ2= 3, υn+2 = υn+1 + υn and u1 = u2 = 1, un+2 = un+1 + un for all integers n. The elementary properties of these numbers are easily established; see for example [2].However, despite the ease with which many such properties are proved, there are a number of more difficult questions connected with these numbers, of which some are as yet unanswered. Among these there is the well-known conjecture that un is a perfect square only if n = 0, ± 1, 2 or 12. This conjecture was proved correct in [1]. The object of this paper is to prove similar results for υn, ½un and ½υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations.