On the Exponential Diophantine Equation x2+4n = y13

In this paper, we studied the Diophantine equation x(2) + 4(n) = y(13). By using the elementary method and algebraic number theory, we obtain the following conclusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 2(12n)-1/13 contains some square factors. (ii) Let x be an even number, when (n equivalent to 0(mod13)), that is n = 13k (k >= 1), all integer solutions for the equation are (x, y) = (0, 4(k)); when (n equivalent to 6(mod13)), that is n = 13k + 6 (k >= 0), all integer solutions are (x, y) = (+/- 2(13k+6),2(2k+1)); when n equivalent to 1, 2, 3, 4, 5, 7, 8, 9,10, 11, 12(mod 13) the equation has no integer solution.