Exploring Sums of Squares and Cubes of Fibonacci Numbers in Diophantine Equations

This study investigates numbers that are powers of two and can be expressed as the sum of the squares or cubes of any two Fibonacci numbers, utilizing Matveev's theorem and the Dujella-Pethő reduction lemma. More precisely, all the solutions to the Diophantine equation Fxm + Fxn =2a with x 3 , where m, n, and a are positive integers, are presented herein.