A note on the exponential Diophantine equation \((44m^2+1)^x+(5m^2-1)^y=(7m)^z\) (Q6546707)

Let \( m \) be a positive integer. In the paper under review, the author studies the exponential Diophantine equation \N\[\N(44m^2+1)^x+(5m^2-1)^y=(7m)^x,\tag{1}\N\]\Nin positive integers \( (x,y,z) \) under some conditions on \( m \). The main result of the author is the following.\N\NTheorem 1. Let \( m \) be a positive integer. When \( m \) is odd, suppose that \( m\equiv 2\pmod 5\) or \( m\equiv 0,\pm 1, \pm 3\pmod 7\). Then, the Diophantine equation (1) has only the positive integer solution \( (x,y,z)=(1,1,2) \). \N\NThe proof of Theorem 1 is based on elementary methods in number theory, Baker's method and linear forms in \(p\)-adic logarithms.