Pythagorean triples containing generalized lucas numbers

Abstract

Let P and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U0(P,Q) = 0;U1(P,Q) = 1, and Un+1(P,Q) = PUn(P,Q)+QUn-1(P,Q) for n ≥ 1 and V0(P,Q) = 2, V1(P,Q) = P, and Vn+1(P,Q) = PVn(P,Q) + QVn-1(P;Q) for n ≥ 1; respectively. In this paper, we assume that P and Q are relatively prime odd positive integers and P2 + 4Q > 0: We determine all indices n such that Un = (P2 + 4Q)x2: Moreover, we determine all indices n such that (P2 +4Q)UNn = x2: As a result, we show that the equation V2 n (P; 1)+V2 n+1(P; 1) = x2 has solution only for n = 2; P = 1; x = 5 and V 2 n+1(P;-1) = V 2 n (P;-1) + x2 has no solutions. Moreover, we solve some Diophantine equations. ©TÜBITAK.