Primitive Divisors on Twists of Fermat's Cubic

Abstract

We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m, with m cube-free, all the terms beyond the first have a primitive divisor. Statement of Main TheoremLet C denote a twist of the Fermat cubic,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 ( This paper is devoted to proving the following theorem.Theorem 1.2. Let C denote the elliptic curve in (1) with m ∈ Z assumed to be cube-free. Let W = (W n ) denote the sequence obtained as above from R ∈ C(Q), a non-torsion rational point. For all n > 1, the term W n has a primitive divisor.The sequence W = (W n ) is a divisibility sequence, which means that, for all m, n ∈ N,In line with recent developments [11, 12, 13, 14, 15, 16, 25, 26, 27] we define the sequence W = (W n ) to be an Elliptic Divisibility Sequence. Admittedly this stretches the definition originally used by Morgan Ward [29] but we believe it is a reasonable name for a divisibility sequence that arises from an elliptic curve.

Lang’s conjecture and sharp height estimates for the elliptic curves $y^{2}=x^{3}+b$

“…In the special case of b = −432m 2 for a cube-free integer m, Jedrzejak [11] proved a sharper result, which was improved by Everest, Ingram and Stevens [4,Lemma 4.3] and further improved very recently by Fujita and Nara [8…”

“…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. …”

“…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.…”

LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

“…See [8, p. 92] along with the strengthened version in [13,Conjecture VIII.9.9]. Such lower bounds have applications to counting the number of integral points on elliptic curves (see [7]), problems involving elliptic divisibility sequences (see [5,6,19]), etc.…”

The uniform primality conjecture for elliptic curves

“…Our starting point is the result stated earlier, that only finitely many terms W n are prime [15,Theorem 4.1]. A related result [13,Theorem 1.2] states that W n has a primitive divisor for all n > 1: results from [13] will be employed in the proofs of our theorems.…”