ON THE DIOPHANTINE EQUATION Sigma(k)(j=1) jF(j)(p) = F-n(q)

SOYDAN G., Nemeth L., Szalay L.

ARCHIVUM MATHEMATICUM, vol.54, no.3, pp.177-188, 2018 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 54 Issue: 3
  • Publication Date: 2018
  • Doi Number: 10.5817/am2018-3-177
  • Journal Name: ARCHIVUM MATHEMATICUM
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Page Numbers: pp.177-188
  • Keywords: Fibonacci sequence, Diophantine equation, CONSECUTIVE FIBONACCI NUMBERS, POWERS, SEQUENCE, SUM
  • Bursa Uludag University Affiliated: Yes

Abstract

Let F-n denote the nth term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F-1(p) + 2F(2)(p) + . . . + kF(k)(p) = F-n(q) in the positive integers k and n, where p and q are given positive integers. A complete solution is given if the exponents are included in the set {1, 2}. Based on the specific cases we could solve, and a computer search with p, q, k <= 100 we conjecture that beside the trivial solutions only F-8 = F-1 + 2F(2 )+ 3F(3 )+ 4F(4), F-4(2 )= F-1 + 2F(2) + 3F(3), and F-4(3) = F-1(3)+ 2F(2)(3 )+ 3F(3)(3) satisfy the title equation.