On squares in Lucas sequences

Abstract

Let P and Q be non-zero integers. The Lucas sequence {U n (P , Q)} is defined by U 0 = 0, U 1 = 1, U n = P U n−1 − QU n−2 (n 2). The question of when U n (P , Q) can be a perfect square has generated interest in the literature. We show that for n = 2, . . . , 7, U n is a square for infinitely many pairs (P , Q) with gcd(P , Q) = 1; further, for n = 8, . . . , 12, the only non-degenerate sequences where gcd(P , Q) = 1 and U n (P , Q) = , are given by U 8 (1, −4) = 21 2 , U 8 (4, −17) = 620 2 , and U 12 (1, −1) = 12 2 .

Elliptic divisibility sequences, squares and cubes

“…The question of when a term of a Lucas sequence can be square has generated interest in the literature [2,3,18,19]. Similar results concerning cubes were also obtained for specific sequences such as Fibonacci, Lucas and Pell numbers [16,17].…”

“…2 For example, although equation 44 implies that all three α, 2α 2 −2α+1, 3α 2 −3α+1 are C, we only use the fact that the second one is C. 3 The equation x 3 + 2y 3 = 1 has the integer solution (x, y) = (−1, 1), hence, by Theorem 5, Chapter 24 of [14] can not have further solutions with xy = 0. 4 The only solutions are those given by coordinates of the torsion points.…”

“…Note that α can not be 1 by the assumption. The solutions to the last equation are (3,2), (17,12), (99, 70), ... and parameters α corresponding to these solutions are −4, −144, −4900, ... .…”

“…α − 1 = β 3 1 , 2α − 1 = β3 2 and β 1 , β 2 are integers and the only solution of this equation is (β 1 , β 2 ) = (−1, −1) and this solution does not provide desired α.If n ≡ 4,14, 16, 26 (30), then h n = C iff(26) α(−α 2 + 3α − 1) = C,if n ≡ 9, 21 (30), then h n = C iff(27) (2α − 1) 2 (−α 2 + 3α − 1) = C, if n ≡ 12, 18 (30), then h n = C iff (28) (α − 1)(−α 2 + 3α − 1) 2 = C,by (3.8) or equivalently α 2 − 3α+ 1 = C. This last equation leads to equation (5.2). The solutions of this equation do not provide desired α.…”

On the GCD-s of k consecutive terms of Lucas sequences

“…the book of Shorey and Tijdeman [43] and the papers of Bremner and Tzanakis [11][12][13], Luca and Walsh [35], Kiss [26], Brindza, Liptai and Szalay [14], and Luca and Shorey [32][33][34], and the references there.…”