On the Diophantine equation $((c+1)m^{2}+1)^{x}+(cm^{2}-1)^{y}=(am)^z$

Authors: ELİF KIZILDERE, TAKAFUMI MIYAZAKI, GÖKHAN SOYDAN

Abstract: Suppose that $c$, $m$, and $a$ are positive integers with $a \equiv 11,\,13 \pmod{24}$. In this work, we prove that when $2c+1=a^{2}$, the Diophantine equation in the title has only solution $(x, y, z)=(1,1,2)$ where $m \equiv \pm 1 \pmod{a}$ and $m>a^2$ in positive integers. The main tools of the proofs are elementary methods and Baker's theory.

Keywords: Exponential Diophantine equation, Jacobi symbol, lower bound for linear forms in logarithms

Full Text: PDF