摘要: 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。
Abstract:
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.