ELLIPTIC CURVES CONTAINING SEQUENCES OF CONSECUTIVE CUBES

Celik G. S., SOYDAN G.

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, vol.48, no.7, pp.2163-2174, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 7
  • Publication Date: 2018
  • Doi Number: 10.1216/rmj-2018-48-7-2163
  • Journal Name: ROCKY MOUNTAIN JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2163-2174
  • Keywords: Elliptic curves, rational points, sequences of consecutive cubes, ARITHMETIC PROGRESSIONS
  • Bursa Uludag University Affiliated: Yes

Abstract

Let E be an elliptic curve over Q described by y(2) = x(3)+Kx+L, where K, L is an element of Q. A set of rational points (x(i), y(i)) is an element of E(Q) for i = 1, 2,..., k, is said to be a sequence of consecutive cubes on E if the x-coordinates of the points x(i)'s for i = 1, 2,..., form consecutive cubes. In this note, we show the existence of an infinite family of elliptic curves containing a length-5-term sequence of consecutive cubes. Moreover, these five rational points in E(Q) are linearly independent, and the rank r of E(Q) is at least 5.