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.