关于不定方程x2+4n=y9的整数解

该文首先应用代数数论的方法证明了不定方程~$x{^2}+4{^n}=y{^9}$~在~$xequiv 1 pmod{2}$ 时无整数解, 再证明不定方程~$x{^2}+4{^n}=y{^9}$~在~$n in{6, 7, 8}$~ 时均无整数解, 进而证明不定方程~$x{^2}+4{^n}=y{^9}$~仅当~$nequiv 0 pmod{9}$~和~$nequiv 4 pmod{9}$ 时有整数解, 且当~$n=9m$~时, 其整数解为~$(x,y)=(0,4{^m})$; 当~$n=9m+4$~时, 其整数解为~$(x,y)=(pm16times2{^{9m}},2times4{^m}),$~ 这里的~$m$~为非负整数. 进一步, 根据~$k=5,9$ 的结论, 文章提出了一个关于不定方程~$x{^2}+4{^n}=y{^k}$ $(k$ 为奇数$)$ 的整数解的猜想, 以供后续研究.

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 ~$xequiv 1pmod{2}$~, 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 ~$nequiv 0 pmod{9}$~ and ~$nequiv 4 pmod{9}$, 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 ~$(x,y)=(pm16times2{^{9m}},2times4{^m})$~ when $n=9m+4$, where $nin N.$ 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.