The integer solutions of the Diophantine equation

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 ~$x\equiv 1\pmod{2}$~, 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 ~$n\equiv 0 \pmod{9}$~ and ~$n\equiv 4 \pmod{9}$, 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.