Pythagorean triples containing generalized Lucas numbers

Abstract

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