Product of Powers of Distinct Primes as Sums of Fibonacci Numbers

Let Fₙ F n be the n -th Fibonacci number. In this paper, we study the Diophantine equation Fₙ+Fₘ=pˣqʸ F n + F m = p x q y in nonnegative integers n≥ 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≤ 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.