不定方程x^(3)±4913=34y^(2)的整数解

Interger solutions to the indefinite equation x^(3)±4913=34y^(2)

运用同余式的性质研究了不定方程x^(3)±4913=34y^(2)的整数解问题,并得到了不定方程x^(3)+4913=34y^(2)仅有正整数解(x,y)=(17,17),(391,1326),不定方程x^(3)-4913=34y^(2)仅有整数解(x,y)=(17,0)的结果。研究结果为解决x^(3)±a3=Dy^(2)这类不定方程的整数解问题奠定了一定的基础。

The integer solutions of the indefinite equation x^(3)±4913=34y^(2)are studied by using the properties of congruence,and it was obtained that the indefinite equation x^(3)+4913=34y^(2)has only the positive integer solutions of(x,y)=(17,17),(391,1326)and the indefinite equation x^(3)-4913=34y^(2)has only the integer solution of(x,y)=(17,0).Therefore the integer solutions to the indefinite equation x^(3)±4913=34y^(2)are completely solved,which lays a certain foundation for solving integer solutions to this type of indefinite equations x^(3)±a3=Dy^(2).