Generalized Fibonacci and Lucas sequences (U-n) and (V-n) are defined by the recurrence relations Un+1 = PUn +QU(n-1) and Vn+1 = PVn + QV(n-1), n >= 1, with initial conditions U-0 = 0, U-1 = 1 and V-0 = 2, V-1 = P. This ...

where p is prime and a > 1. Assuming the solutions of the Pell equation x(2) -(a(2) - 1) y(2) = 1 are x = x(m) and y = y(m) with m = 2, we prove that the system (0.1) has solutions only when m = 2 or m = 3. In the case of ...

In this paper, we determine when the equation in the title has an infinite number of positive integer solutions x and y when 0 <= n <= 10. Moreover, we give all the positive integer solutions of the same equation for 0 <= n <= 10.

Let P be a nonzero integer and let (U-n) and (V-n) denote Lucas sequences of first and second kind defined by U-0 = 0, U-1 = 1; V-0 = 2, V-1 = P; and Un+1 = PUn + Un-1, Vn+1 = PVn + Vn-1 for n >= 1. In this study, when P ...