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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 sum and difference of two balancing numbers. Moreover, as a generalization we explore the perfect squares which are 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 $A \geq 3$.
Keywords: sum / two balancing / initial terms / defined recursively / balancing numbers / like sequence / perfect powers / AG_n / geq

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