A parametric family of ternary purely exponential Diophantine equation $A^x + B^y = C^z$

Let a, b, c be fixed positive integers such that $a + b = c^2 , 2 ∤ c and (b/p) ̸= 1$ for every prime divisor p of c , where (b/p) is the Legendre symbol. Further let m be a positive integer with m > 1. In this paper, using the Baker method, we prove that if m > $max{10^8 , c^2 }$, then the equation $(am^2 + 1)x + (bm^2 − 1)^y = (cm)^z$ has only one positive integer solution (x, y, z) = (1, 1, 2).

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