A Note on the Diophantine Equation

Abstract Let q be an odd prime such that q^t + 1= 2c^s, where c, t are positive integers and s= 1, 2. We show that the Diophantine equation x^2 + q^m = c^n has only the positive integer solution (x, m, n) = (c^s - 1, t, 2s) under some conditions. The proof is based on elementary methods and a result concerning the Diophantine equation (x^n - 1) / (x- 1) = y^2 due to Ljunggren. We also verify that when 2 c 30 with c = 12, 24, the Diophantine equation x^2 + (2c- 1) ^m = c^n has only the positive integer solution (x, m, n) = (c- 1, 1, 2).