不定方程 x 2 −kxy+k y 2 +dy=0的正整数解

The Positive Integer Solutions of the Diophantine Equation x 2 −kxy+k y 2 +dy=0

本文研究了在d∈{3,5,7,11,13,17,19},k∈N∗时,不定方程x2−kxy+ky2+dy=0有无穷多个正整数解(x, y)当且仅当d = 3,k = 5, 6, 7;d = 5,k = 5, 7, 9;d = 7,k = 5, 8, 11;d = 11,k = 5, 6, 9, 10, 15;d = 13,k = 5, 11, 17;d = 17,k = 5, 7, 11, 13, 21;d = 19,k = 5, 11, 14, 23。在d为奇素数时,给出了不定方程x2−kxy+ky2+dy=0正整数解的一些必要条件。

In this paper, we study that atd∈{3,5,7,11,13,17,19},k∈N∗, the indefinite equationx2−kxy+ky2+dy=0has infinitely many positive integer solutions (x, y) when and only when d = 3, k = 5, 6, 7;d = 5, k = 5, 7, 9;d = 7, k = 5, 8, 11;d = 11, k = 5, 6, 9, 10, 15;d = 13, k = 5, 11, 17;d = 17, k = 5, 7, 11, 13, 21;d = 19, k = 5, 11, 14, 23. Some necessary conditions for positive integer solutions of the indefinite equationx2−kxy+ky2+dy=0are given when d is an odd prime.