A FAMILY OF INTEGER SOMOS SEQUENCES


Gezer B., Capa B., Bizim O.

MATHEMATICAL REPORTS, vol.18, pp.417-435, 2016 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18
  • Publication Date: 2016
  • Journal Name: MATHEMATICAL REPORTS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.417-435
  • Keywords: Somos sequences, elliptic curves, torsion points, ELLIPTIC DIVISIBILITY SEQUENCES, LUCAS SEQUENCES, LAURENT PHENOMENON, PERFECT POWERS, SQUARES, CUBES, FIBONACCI, TORSION, CURVES
  • Bursa Uludag University Affiliated: Yes

Abstract

Somos sequences are sequences of rational numbers defined by a bilinear recurrence relation. Remarkably, although the recurrences describing the Somos sequences are rational, some Somos sequences turn out to have only integer terms. In this paper, a family of Somos 4 sequences is given and it is proved that all Somos 4 sequences associated to Tate normal forms with h(-1) - +/- 1 consist entirely of integers for n >= 0. It is also shown that there are infinitely many squares and infinitely many cubes in Somos 4 sequences associated to Tate normal forms.