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

Not sure what you're asking...Consider the equation \(\displaystyle x^2 - axy + y^2=c^2\) where \(\displaystyle 0<x<y<c\) are integers.

I know that if \(\displaystyle a\) is an integer, then the equation \(\displaystyle x^2 - axy + y^2=c^2\)

can have integral solution \(\displaystyle x, y, \) and \(\displaystyle c\) for some integer \(\displaystyle a\).

However, I wonder if \(\displaystyle a\) is a rational number such that \(\displaystyle 0<a<1\),

is it possible to find such an \(\displaystyle a\) for which \(\displaystyle x^2 - axy + y^2=c^2\) has

an integral solution? If so, what are the conditions on such an \(\displaystyle a\)?

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.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?

Neither am I.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.

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