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For elliptic curves given by the equation Ea : y 2 = x 3 + ax, we establish the bestpossible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

Let P be a non-torsion point on the elliptic curve E a : y 2 = x 3 + ax. We show that if a is fourth-power-free and either n > 2 is even or n > 1 is odd with x(P ) < 0 or x(P ) a perfect square, then the n-th element of the elliptic divisibility sequence generated by P always has a primitive divisor.

Abstract. For E b : y 2 = x 3 + b, we establish Lang's conjecture on a lower bound for the canonical height of non-torsion points along with upper and lower bounds for the difference between the canonical and logarithmic height. These results are either best possible or within a small constant of the best possible lower bounds. IntroductionThe canonical height, h (defined in Section 2), on an elliptic curve E defined over a number field K is a measure of the arithmetic complexity of points on the curve. It has many desirable properties. For example, it is a positive definite quadratic form on the lattice E(K)/(torsion), behaving well under the group law on E(K). See [17, Chapter VIII] and [1, Chapter 9] for more information on this height.There is another important, and closely related, height function defined for points on elliptic curves, the absolute logarithmic height (also defined in Section 2). It has a very simple definition which makes it very easy to compute.In this paper, we provide sharp lower bounds for the canonical height as well as bounding the difference between the heights for a well-known and important family of elliptic curves, the Mordell curves defined by E b : y 2 = x 3 + b where b is a sixth-power-free integer (i.e., quasi-minimal Weierstrass equations for all E b /Q).1.1. Lower bounds. Lang's Conjecture proposes a lower bound for the heights of non-torsion points on a curve which varies with the curve. Conjecture 1.1 (Lang's Conjecture). Let E/K be an elliptic curve with minimal discriminant D E/K . There exist constants C 1 > 0 and C 2 , depending only on [K : Q], such that for all nontorsion points P ∈ E(K) we haveSee [13, p. 92] along with the strengthened version in [17, Conjecture VIII.9.9].Such lower bounds have applications to counting the number of integral points on elliptic curves [10], questions involving elliptic divisibility sequences [4,5,23] and several other problems.