  We provide a method, using essentially elementary arithmetic, to solve the
exponential diophantine equation 7x^2 + 59y^2 = 3^m, which leads to a simple
algorithm, with no use of radicals or complex numbers, that generates all the
(infinitely many) primitive and positive solutions of this equation. The method
can be generalized to solve a class of exponential equations of the form ax^2 +
by^2 = ck^z.
