Primitive Divisors on Twists of the Fermat Cubic Graham Everest, Patrick Ingram and Shaun Stevens

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u + v = m, with m cube-free, all the terms beyond the first have a primitive divisor. 1. Statement of Main Theorem Let C denote a twist of the Fermat cubic, C : U + V 3 = mW 3 (1) with m a non-zero rational number. If K denotes any field of characteristic zero, the set C(K) of projective K-rational points satisfying (1) forms an elliptic curve. With respect to the usual chord and tangent addition the set C(K) forms a group . The identity of the group is (−1, 1, 0) and the inverse of the point (U, V,W ) is (V, U,W ). Let R ∈ C(Q) denote a non-torsion rational point. Write nR = (Un, Vn,Wn), Un, Vn,Wn ∈ Z in lowest form with gcd(Un, Vn,Wn) = 1. This paper is devoted to proving the following theorem. Theorem 1.1. Let C denote the elliptic curve in (1) with m ∈ Z assumed to be cube-free. Let W = (Wn) denote the sequence obtained as above from R ∈ C(Q), a non-torsion rational point. For all n > 1, the term Wn has a primitive divisor. The term primitive divisor is defined in the following way. Definition 1.2. Let (An) denote a sequence with integer terms. We say an integer d > 1 is a primitive divisor of An if (a) d | An and (b) gcd(d, Am) = 1 for all non-zero terms Am with m < n. 1991 Mathematics Subject Classification. 11G05, 11A41.