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Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation ${(4m²+1)}^x+{(5m²-1)}^y={\left(3m\right)}^z$ has only the positive integer solution (x,y,z) = (1,1,2).