On quadratic Diophantine expression

Bong

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Consider the expression \(\displaystyle x^2 - axy + y^2\) where \(\displaystyle x\) and \(\displaystyle y\) are positive integers. I know that if \(\displaystyle a\) is an integer, then the expression \(\displaystyle x^2 - axy + y^2\) can be written as a square of an integer for some \(\displaystyle a\). 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 \(\displaystyle x^2 - axy + y^2\) is a square of an integer? If so, what are the conditions on such an \(\displaystyle a\)?
 

Bong

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Sorry, the above question should be stated as follows:

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 \(\displaystyle a\)?
 

Denis

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Feb 17, 2004
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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...

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

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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

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Feb 17, 2004
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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
 
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