Perfect powers in elliptic divisibility sequences

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-08-14 DOI:10.1112/blms.13135

Maryam Nowroozi, Samir Siksek

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Abstract

Let $E / Q$ be an elliptic curve given by an integral Weierstrass equation. Let $P \in E (Q)$ be a point of infinite order, and let ${(B_{n})}_{n ⩾ 1}$ be the elliptic divisibility sequence generated by $P$ . This paper is concerned with a question posed in 2007 by Everest, Reynolds and Stevens: does ${(B_{n})}_{n ⩾ 1}$ contain only finitely many perfect powers? We answer this question positively under the following three additional assumptions:

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椭圆可除序列中的完全幂

设 E / Q $E/\mathbb {Q}$ 是由韦尔斯特拉斯积分方程给出的椭圆曲线。让 P ∈ E ( Q ) $P \in E(\mathbb {Q})$ 是一个无穷阶点，让 ( B n ) n ⩾ 1 $(B_n)_{n\geqslant 1}$ 是由 P $P$ 产生的椭圆可分序列。本文关注 Everest, Reynolds 和 Stevens 于 2007 年提出的一个问题：( B n ) n ⩾ 1 $(B_n)_{n \geqslant 1}$ 是否只包含有限多个完全幂？在以下三个附加假设下，我们可以肯定地回答这个问题：

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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