Recurrence Sequences

Prime Divisors of Sequences Associated to Elliptic Curves

“…Although a division is needed in the definition of the later terms, they do all turn out to be integral. For an introduction to Somos sequences, as well as links to other areas of mathematics, consult [6], [7], [8], [10], [13], [14]. If P = (x, y) and Q are rational points with Q + nP never the point at infinity, continue to write Q + nP = (x n , y n ).…”

Irrationality of Power Series for Various Number Theoretic Functions

“…The following (and more general) bound on the number of zeros of a linear recurrence sequence over a finite field can be found in Section 7.1 of [3], in Chapter 8 of [7], and in Section 7.1 of [9]. Lemma 6.…”

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].…”