Perfect squares in the sum and difference of balancing-like numbers

Abstract

In this study, we deal with the existence of perfect powers which are the sum and difference of two balancing numbers. Moreover, as a generalization we explore the perfect squares which are the sum and difference of two balancing-like numbers, where balancing-like sequence is defined recursively as G(n+1) = AG(n)-G(n-1) with initial terms G(0) = 0, G(1) = 1 for A >= 3.