Publication: On the diophantine equation Σj=1k jFjp = Fnq
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Date
2018-01-01
Authors
Soydan, Gökhan
Authors
Nemeth, Laszlo
Szalay, Laszlo
Journal Title
Journal ISSN
Volume Title
Publisher
Masaryk Univ, Fac Science
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.
Description
Keywords
Consecutive fibonacci numbers, Powers, Sequence, Sum, Fibonacci sequence, Diophantine equation, Science & technology, Physical sciences, Mathematics