On the exponential Diophantine equation (18m2 + 1)x + (7m2 − 1)y = (5m) z

Abstract

Let m be a positive integer. We show that the exponential Diophantine equation (18m2 + 1)x + (7m2 −1)y =(5m)z has only the positive integer solution (x, y, z) = (1, 1, 2) except for m ≡ 23, 47, 63, 87 (mod 120). For m ̸≡ 0(mod 5) we use some elementary methods and linear forms in two logarithms. For m ≡ 0 (mod 5) we apply a result forlinear forms in p-adic logarithms.