Diophantine triples of Fibonacci numbers
Volume 175 / 2016
Acta Arithmetica 175 (2016), 57-70
MSC: 11D09, 11D45, 11B37, 11J86.
DOI: 10.4064/aa8259-6-2016
Published online: 15 September 2016
Abstract
Let $F_m$ be the $m$th Fibonacci number. We prove that if $F_{2n}F_k+1$ and $F_{2n+2}F_k+1$ are both perfect squares, then $k=2n+4$ for $n\ge 1$, or $k=2n-2$ for $n\ge 2$, except when $n=2$, in which case we can additionally have $k=1$.
