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arXiv · 2602.17903

Product of powers of distinct primes as sums of Fibonacci numbers

Herbert Batte·Florian Luca·Volker Ziegler
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Abstract

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes $(q,p)$ with $q\le \min\{1000,p\}$ such that the above equation has at least two solutions $(x,y)$ (and corresponding $m,n$) in positive integers.

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Herbert Batte, Florian Luca, Volker Ziegler. 2026-02-19. Product of powers of distinct primes as sums of Fibonacci numbers. https://arxiv.org/abs/2602.17903

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