Solution to a pair of linear, two-variable, Diophantine equations with coprime coefficients from balancing and Lucas-balancing numbers

Abstract

Let $B_n$ and $C_n$ be the $n$-th balancing and Lucas-balancing numbers, respectively. We consider the Diophantine equations $ax+by=\frac{1}{2}(a-1)(b-1)$ and $1+ax+by=\frac{1}{2}(a-1)(b-1)$ for $(a,b)$ $\in$ $ \{(B_n,B_{n+1}),(B_{2n-1},B_{2n+1}), (B_n,C_n),(C_n,C_{n+1})\}$ and present the non-negative integer solutions of the Diophantine equations in each case.