Product of Powers of Distinct Primes as Sums of Fibonacci Numbers

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