Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers

2025/01/15 by Roberto Alvarenga, Ana Paula Chaves, Alvarenga, Roberto +6
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2501.08899

openalex publication_date 2025/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (F_n)_n be the classical Fibonacci sequence. It is well-known that it satisfies F_n² + F_n+1² = F_2n+1. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_n^(k))² + (F_n+d^(k))² = F_m^(k) over the k-generalized Fibonacci numbers for every k ≥ 2, generalizing Chaves and Marques. Next, we solve F_n^s + F_n+d^s = F_m over the Fibonacci numbers for every s ≥ 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_n^s + ⋯ + F_n+d^s = F_m for d+1 < n and s ≥ 2.

Related

• On Diophantine equations involving sums of Fibonacci numbers and powers of 2
• Powers of two as sum of two generalized Fibonacci numbers
• The proof of a conjecture concerning the intersection of k-generalized\n Fibonacci sequences
• On the Diophantine equation ∑ _k=1⁵F_nk=2^a
• On the diophantaine equations J_N +J_M =F_A & F_N +F_M =J_A