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 pronicĀ $m$-tuples, triangular $m$-tuples and prove some results related to pronic and triangular triples of balancing numbers.
Keywords
Balancing numbers, Diophantine triples, Linear forms in complex and $p$-adic logarithms
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PDFISSN: 1331-0623 (Print), 1848-8013 (Online)