Elliptic divisibility sequences over certain curves

Abstract

Let n 5 be an integer. We provide an effective method for finding all elliptic curves in short Weierstrass form E/Q with j (E) ∈ {0, 1728} and all P ∈ E(Q) such that the nth term in the elliptic divisibility sequence defined by P over E fails to have a primitive divisor. In particular, we improve recent results of Everest, Mclaren, and Ward on the Zsigmondy bounds of elliptic divisibility sequences associated with congruent number curves.

Algebraic Divisibility Sequences Over Function Fields

“…https://doi.org/10.1017/S1446788712000092 [21] Algebraic divisibility sequences 119 Theorem 1.5 provides examples of elliptic divisibility sequences such that for infinitely many indices n, the support of D nP is exactly the Gal(K/K)-orbit of the single point γ n . The example of Lucas sequences with finitely many irreducible terms (2.1) suggests that the same should be true for some EDSs.…”

Primitive Divisors on Twists of Fermat's Cubic

“…This suggests that a proof that Z(B) is uniformly bounded above seems to require a result as strong as Lang's conjecture. The following example appeared in [14].…”

“…The method proceeds in a pincer movement, somewhat similar to that in the two papers [12] and [14]. These papers used a good lower bound for the canonical height of a rational point which were obtained in [4].…”

“…The proof of Theorem 1.2 exploits both the denominator sequence B = (B n ) in (5), as in [12,14], and the numerator sequence A = (A n ). The latter may have independent interest, not least because it is a rare example where a primitive divisor result may be obtained, Theorem 2.3 below, for a sequence which is not a divisibility sequence.…”

“…Instead, a weaker kind of divisibility relation holds, as detailed in Lemma 3.2. In principle, techniques analogous to those used in [14] and Section 5 can be used here to reduce the bound stated in Theorem 2.3. In practice, however, the computations involved for some cases are beyond our current capabilities.…”