On Diophantine, pronic and triangular triples of balancing numbers

Abstract

In this paper, we search for some Diophantine triples of balancing numbers. We prove that, if $(6\pm2)B_nB_k+1$ and $(6\pm2)B_{n+2}B_k+1$ are both perfect squares then $k=n+1$, for any positive integer $n \geq 1$. In addition, we define pronic $m$-tuples, triangular $m$-tuples and prove some results related to pronic and triangular triples of balancing numbers.