Here, we show that there is no integer $s\ge 3$ such that the sum of $s$th powers of two consecutive Fibonacci numbers is a Fibonacci number.
"An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers." Proc. Japan Acad. Ser. A Math. Sci. 87 (4) 45 - 50, April 2011. https://doi.org/10.3792/pjaa.87.45