Abstract: his paper proves that the Diophantine equation ~x^2+4^n=y^9~ has no integer solution by using the method of algebraic number theory, where ~x\equiv 1\pmod2~, and further shows that the Diophantine equation ~x^2+4^n=y^9~(n=6,7,8) has no integer solution. Then it shows that the Diophantine equation ~x^2+4^n=y^9~ has integer solution only when ~n\equiv 0 \pmod9~ and ~n\equiv 4 \pmod9, say, the Diophantine equation ~x^2+4^n=y^9~ has integer solutions ~(x,y)=(0,4^m)~ when n=9m, and the Diophantine equation ~x^2+4^n=y^9~ has integer solutions ~(x,y)=(\pm16\times2^9m,2\times4^m)~ when n=9m+4, where n\in N. Furthermore, based on the results of k=5,9, the paper proposes a conjecture about the integer solutions of the Diophantine equation ~x^2+4^n=y^k for further research, where k is odd.