Integral Points in Arithmetic Progression on y2=x(x2−n2)

Bremner, Andrew; Silverman, Joseph H.; Tzanakis, Nikos

doi:10.1006/jnth.1999.2430

Cited by 37 publications

(64 citation statements)

References 20 publications

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“…using the bounds on the difference between naïve and canonical height given in [2]. Under the assumption that B 5m has no primitive divisor, the above and (9) of [5] combine to yield n = 5m 33.…”

Section: Lemmamentioning

confidence: 97%

Elliptic divisibility sequences over certain curves

Ingram

2007

Journal of Number Theory

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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.

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Section: Lemmamentioning

confidence: 97%

Elliptic divisibility sequences over certain curves

Ingram

2007

Journal of Number Theory

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“…The following explicit transformations render the curve E in minimal form: [2,4], [6,0], [6,108] (respectively).…”

Section: Preliminariesmentioning

confidence: 99%

Primitive Divisors on Twists of Fermat's Cubic

Everest¹,

Ingram

Stevens³

2009

LMS J. Comput. Math.

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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.

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“…Let (x, y) ∈ Z 2 be a solution to (1) such that x > n and x ∈ Σ n . Let (s, t) ∈Q 2 be a point on E n such that [2](s, t) = (x, y) (we denote by [2](s, t) the double of the point (s, t) on the elliptic curve E n ). By [18, page 59], we have…”

Section: Proof Of Theoremmentioning

confidence: 99%