On quadratic Diophantine expression

[Bong]

Consider the expression $$x^2 - axy + y^2$$ where $$x$$ and $$y$$ are positive integers. I know that if $$a$$ is an integer, then the expression $$x^2 - axy + y^2$$ can be written as a square of an integer for some $$a$$. However, I wonder if $$a$$ is a rational number such that $$0<a<1$$, is it possible to find such an $$a$$ for which $$x^2 - axy + y^2$$ is a square of an integer? If so, what are the conditions on such an $$a$$?

 

[Bong]

Sorry, the above question should be stated as follows:

Consider the equation $$x^2 - axy + y^2=c^2$$ where $$0<x<y<c$$ are integers. I know that if $$a$$ is an integer, then the equation $$x^2 - axy + y^2=c^2$$ can have integral solution $$x, y,$$ and $$c$$ for some integer $$a$$. However, I wonder if $$a$$ is a rational number such that $$0<a<1$$, is it possible to find such an $$a$$ for which $$x^2 - axy + y^2=c^2$$ has an integral solution? If so, what are the conditions on such an $$a$$?

 

[Denis]

Consider the equation $$x^2 - axy + y^2=c^2$$ where $$0<x<y<c$$ are integers.
I know that if $$a$$ is an integer, then the equation $$x^2 - axy + y^2=c^2$$can have integral solution $$x, y,$$ and $$c$$ for some integer $$a$$.
However, I wonder if $$a$$ is a rational number such that $$0<a<1$$,
is it possible to find such an $$a$$ for which $$x^2 - axy + y^2=c^2$$ has
an integral solution? If so, what are the conditions on such an $$a$$?

Not sure what you're asking...

x=4, y=5, c=6, a = .25

x^2 - axy + y^2 = 4^2 - .25*4*5 + 5^2 = 16 - 5 + 25 = 36
c^2 = 6^2 = 36

That seems to be a solution....is it?

 

[Bong]

Not sure what you're asking...

x=4, y=5, c=6, a = .25

x^2 - axy + y^2 = 4^2 - .25*4*5 + 5^2 = 16 - 5 + 25 = 36
c^2 = 6^2 = 36

That seems to be a solution....is it?

Yes you are right. But the difficult part is to find the necessary and sufficient conditions (if any) for a so that the equation is satisfied. I'm not sure there are ones.

 

[Denis]

Yes you are right. But the difficult part is to find the necessary and sufficient
conditions (if any) for a so that the equation is satisfied. I'm not sure there are ones.

Neither am I.
In case it helps, here's the cases where c>0 and c<10:
x y c a
4 5 6 .250
4 6 7 .125
4 7 8 .036
5 6 7 .400
5 7 8 .286
5 8 9 .200
6 7 8 .500
6 7 9 .095
6 8 9 .396
7 8 9 .571

Equation becomes:
a = (x^2 + y^2 - c^2) / (xy) where c>y>x and 1>a>0