Diophantine equations over the generalized Fibonacci sequences: exploring sums of powers
Roberto Alvarenga, Ana Paula Chaves, Maria Eduarda Ramos, Matheus Silva, Marcos Sosa
TL;DR
The paper extends classical Fibonacci identities by solving Diophantine equations over $k$-generalized Fibonacci numbers. It provides complete classifications for $(F_n^{(k)})^2+(F_{n+d}^{(k)})^2=F_m^{(k)}$ and for $F_n^s+F_{n+d}^s=F_m$, and analyzes sums of $s$-powers with multiple terms, using a toolkit that includes Matveev’s linear forms in logarithms, Legendre’s criterion, Dujella–Pethő’s lemma, and continued fractions, complemented by extensive computational verification. The results yield precise structural constraints: for $k=2$ the two-square case has a finite list of solutions, while for $k\ge 3$ the solutions are restricted to a small set; for $s\ge 3$ the two-term power equations admit only the trivial small-index solution $(n,d)=(1,1)$, with further nonexistence results for longer power-sums under the regime $d+1<n$. The work demonstrates how deep tools from Diophantine approximation, together with computational checks, can fully resolve questions about when Fibonacci-like sequences produce Fibonacci numbers as sums of powers, highlighting the scarcity of such coincidences and offering explicit bounds and finite-verification strategies.
Abstract
Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = 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 \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2.