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 (18m(2) + 1)(x) + (7m(2) - 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 not equivalent to 0 (mod 5) we use some elementary methods and linear forms in two logarithms. For m 0 (mod 5) we apply a result for linear forms in p-adic logarithms.