The elliptic logarithm method for the determination of all integral solutions of a given elliptic equation is discussed for equations with associated elliptic curve of moderately large rank. Major attention is given to the question of optimizing the choice of Mordell-Weil basis for the curves in question. A speculative argument suggests that for any curve of rank larger then 8 the calculations involved are unlikely to be feasible. The arguments are illustrated by examples of curves of rank 5, 6, 7, and 8, taken from the literature.

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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 .

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