Squares in recurrences using elliptic curves

Abstract

Numerous papers have studied the problem of determining upper bounds for the number of integer points on elliptic curves of the form [Formula: see text], and quartic curves of the form [Formula: see text]. Bounds for the number of integer solutions to such quartic equations typically depend on both of the coefficients [Formula: see text]. The purpose of this paper is to examine more closely how the number of integer points on such quartic curves seems to depend almost entirely on the number of prime factors of [Formula: see text]. This is done by focusing in on the problem of bounding the number of squares in certain recurrence sequences. In particular, using some arithmetic on elliptic curves, it is proved that infinitely many such sequences can have four squares, but that a fifth square remains elusive after extensive computation, suggesting that an absolute bound for the number of squares is more than likely, and also suggesting that the above assertion regarding the dependence on the number of prime factors of [Formula: see text].