Elliptic divisibility sequences

Abstract

Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In certain circumstances, the methods prove that only finitely many terms have length below a given bound.

Generalized Greatest Common Divisors, Divisibility Sequences, and Vojta’s Conjecture for Blowups

“…(For further information about elliptic divisibility sequences, including a not-quite-equivalent alternative definition, see [5,6,7,13,15,19,20,21,22,24,25].) These sequences have the property that if m|n, then D mP |D nP , whence their name.…”

“…For example, which such sequences contain infinitely many prime numbers (cf. [7]). This is, of course, a notoriously difficult question, even for the simplest divisibility sequence 2 n − 1.…”

p-adic properties of division polynomials and elliptic divisibility sequences

“…Ward, who made an extensive study of these sequences [24,25], shows that a proper elliptic divisibility sequence W is associated to a (possibly singular) elliptic curve E W and point P W ∈ E W (Q) and that the values of W n are closely related to the values of the division polynomials F n (P W ). More recently, elliptic divisibility sequences have been studied by Shipsey [19], who gives an application to the elliptic curve discrete logarithm problem, and by several other authors [3,4,5,8,9,10,23,22]. (See also [11,13,14] for work on the related Somos sequences.)…”

“…The arithmetic properties of elliptic divisibility sequences were first studied in detail by Morgan Ward [24,25] in the 1940's, and recently there has been a resurgence of interest in their study [3,4,5,8,9,10,19,23,22]. (See also [11,13,14] Remark 6.…”

“…It is easy to check that it suffices that (W n ) satisfy the two relations [24].) Additional material on elliptic divisibility sequences may be found in [4,5,8,9,10,19,23,22,24,25].…”