Diophantine方程x^2+axy-y^2=b的可解性

The Solvability of the Diophantine Equation x^2+axy-y^2=b

设a,b分别是给定的正整数和非零整数.运用非平方正整数平方根的简单连分数性质,证明了如果2|a且1<|b|<a/2,或2|a且4<|b|<a/4,则方程x^2+axy-y^2=b没有适合gcd(x,y)=1的正整数解(x,y).

Let aand b be a fixed positive integer and a fixed nonzero integer respectively.Using certain properties of simple continued fractions and some results about Pell equation,we prove that either if 2｜a and 1〈｜b｜〈a/2 or 2｜a and 4〈｜b｜〈a/4,then the equation x2＋axy-y2=b has no positive integer solutions（x,y）with gcd（x,y）=1.