Perfect powers in elliptic divisibility sequences

Abstract

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations. 1 2 JONATHAN REYNOLDS singular point and so not define an elliptic curve. Via the work of Everest [21, 23], Ingram [29], Silverman [31, 41] et al, it has now become conventional to use the following Definition 1.1. Let E/Q be an elliptic curve and (1.1) a Weierstrass equation for E. Let P ∈ E(Q) be a non-torsion point. For m ∈ N denote B mP , as in (1.2), by B m . The sequence (B m ) is an elliptic divisibility sequence.In the current paper, we are interested in analogues for elliptic divisibility sequences (in the sense of Definition 1.1) to the result for Fibonacci numbers. There are certainly perfect powers in some elliptic divisibility sequences. For example, 3) gives B m = 1 for m = 1, 2, 3, 4, 7 and B 12 = 2 7 . However, the following theorem shows that one can often prove that there are only finitely many perfect powers in such sequences.Theorem 1.2. Let (B m ) be an elliptic divisibility sequence whose first term is divisible by 2 or 3. There are finitely many perfect powers in (B m ). Moreover, if B m = z l for some integer z and prime l then l can be bounded explicitly in terms of E and P.The proof of Theorem 1.2 combines a recent Frey-Hellegouarch construction for Klein forms by Bennett and Dahmen [3] with a primitive divisor result due to Silverman [41]. The method of proof is so flexible that it also allows one in certain concrete cases to completely determine the set of all perfect power terms, as was done for the Fibonacci sequence (see Proposition 1.5 and Example 1.9 below). The condition that only 2 or 3 divides the first term is because higher primes, such as 5, do not give a Klein form as in Defintion 3.7.Let E/Q be an elliptic curve and (1.1) a Weierstrass equation for E. Siegel [40] proved that there are finitely many P ∈ E(Q) with B P = 1. In [24] it is shown that for fixed l > 1, there are finitely many P ∈ E(Q) with B P = z l for some z ∈ Z. Since their denominator is a perfect power, perhaps it is reasonable to give the following Definition 1.3. Call P ∈ E(Q) power integral if B P (as in (1.2)) is equal to a perfect power.Note that 1 is a perfect power and so power integral points can be thought of as a generalization of the integral points. A lot of work has been done to make Siegel's theorem effective [2,11,25,28,34] and there are many techniques which can find all of the integral points for large classes of elliptic curves [26,35,46, 47]. For certain curves we are now able to find all of the power integral points.1.1. Mordell curves. Theorem 1.2 can be strengthened considerably for Mordell curves.Theorem 1.4. Let D ...

Sequences associated to elliptic curves with non-cyclic torsion subgroup

“…The problem of determining square terms in linear recurrence sequences is of great interest in the literature [3,4,[19][20][21]. The authors have also considered the square terms in elliptic divisibility sequences [8,10], see also [16,18]. In [8,Section 5], the author determined square and cube terms in the sequence (F n (P )) n≥0 associated to a normal form of an elliptic curve with cyclic torsion subgroup.…”

Perfect Powers in Products of Terms of Elliptic Divisibility Sequences

“…As in the case of linear recurrences, listing the elements of P ℓ (B) is a highly non-trivial problem. A paper of Reynolds [15] explains a procedure to find every perfect power in the sequence when B 1 is divisible by 2 or 3. There are more explicit results for square and cube terms by Bizim and Gezer [1,2].…”

A height inequality for rational points on elliptic curves implied by the abc-conjecture

“…See for instance [EEW01] or [Rey12] and the references therein. In this paper we derive strong conditions on the denominators d P from the uniform abc-conjecture over number fields (see Conjecture 2.2 or [GS00]).…”

“…If E/Q is an elliptic curve in Weierstraß form and P ∈ E(Q) \ {O}, where O is the point at infinity, then it is well known that we can write The structure of the denominators d P has been studied, for instance, by Everest-Reynolds-Stevens [ERS07], and has recently received increasing attention in the context of elliptic divisibility sequences first studied by Ward [War48], see for instance [EEW01] or [Rey12b] and the references therein. In this paper we make the following conjecture, where rad(n) denotes the product of distinct prime divisors of an integer n.…”