关于不定方程x^2+64=4y^n(n=5,9)的解

The solution on Diophantine equation x^2+64=4y^n(n=5,9)

利用代数数论整数环的唯一分解性,研究了不定方程x^2+64=4y^n(n=5,9)的整数解问题,并证明了当n=5时,该方程仅有整数解(x,y)=(±8,2);当n=9时,该方程无整数解。

The problem of integer solution to the Diophantine equation x^2 ＋ 64 = 4 y^n（n = 5, 9） is discussed by using the methods of algebraic number theory. The Diophantine equation has integer solution（x, y） =（±8, 2） when n = 5, and the Diophantine equation has no integer solution when n=9.