Fibonacci and Lucas congruences and their applications

On square classes in generalized Lucas sequences

“…If v 2 (n) = v 2 (m), then (V n , V m ) = V d by (2.8), where d = (m, n). Thus V n = V d u 2 and V m = 2V d v 2 or V n = 2V d u 2 and V m = V d v2 for some integers u and v. As a result, it follows that m = 1 and n = 3 by Theorems 3.2 and 3.3. If v 2 (n) = v 2 (m), then (2.8) implies (V n , V m ) = 1 or 2.…”

“…For P = Q = 1, we have classical Fibonacci and Lucas sequences (F n ) and (L n ). In [2], Keskin and Demirtürk showed that if L n = L m x 2 for m ≥ 2, then n = m, and that there is no Lucas number L n such that L n = L 2r L m x 2 , with r ≥ 1 and m > 1. In particular, L n = 3L m x 2 for 1 < m ≤ n. In [3], the authors solved each of the equations L n = 2L m x 2 and L n = 6L m x 2 .…”

Positive integer solutions of the diophantine equation x 2 − L n xy + (−1) n y 2 = ±5 r

“…The proofs of the following three theorems can be found in [1,5] and [8]. The following theorem can be found in [4].…”

Positive integer solutions of the diophantine equations $x^2 -5 F_n xy - 5(-1)^n y^2 = \pm 5^r$

“…These properties are given in several sources such as [1,15,25]. Also one can find the proofs of the following theorems in [1,13]. The following theorem is proved by Cohn in [2].…”