Consider the expression [math]x^2 - axy + y^2[/math] where [math]x[/math] and [math]y[/math] are positive integers. I know that if [math]a[/math] is an integer, then the expression [math]x^2 - axy + y^2[/math] can be written as a square of an integer for some [math]a[/math]. However, I wonder if [math]a[/math] is a rational number such that [math]0<a<1[/math], is it possible to find such an [math]a[/math] for which [math]x^2 - axy + y^2[/math] is a square of an integer? If so, what are the conditions on such an [math]a[/math]?
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On quadratic Diophantine expression
- Thread starter Bong
- Start date
Sorry, the above question should be stated as follows:
Consider the equation [math]x^2 - axy + y^2=c^2[/math] where [math]0<x<y<c[/math] are integers. I know that if [math]a[/math] is an integer, then the equation [math]x^2 - axy + y^2=c^2[/math] can have integral solution [math]x, y, [/math] and [math]c[/math] for some integer [math]a[/math]. However, I wonder if [math]a[/math] is a rational number such that [math]0<a<1[/math], is it possible to find such an [math]a[/math] for which [math]x^2 - axy + y^2=c^2[/math] has an integral solution? If so, what are the conditions on such an [math]a[/math]?
Consider the equation [math]x^2 - axy + y^2=c^2[/math] where [math]0<x<y<c[/math] are integers. I know that if [math]a[/math] is an integer, then the equation [math]x^2 - axy + y^2=c^2[/math] can have integral solution [math]x, y, [/math] and [math]c[/math] for some integer [math]a[/math]. However, I wonder if [math]a[/math] is a rational number such that [math]0<a<1[/math], is it possible to find such an [math]a[/math] for which [math]x^2 - axy + y^2=c^2[/math] has an integral solution? If so, what are the conditions on such an [math]a[/math]?
Not sure what you're asking...Consider the equation [math]x^2 - axy + y^2=c^2[/math] where [math]0<x<y<c[/math] are integers.
I know that if [math]a[/math] is an integer, then the equation [math]x^2 - axy + y^2=c^2[/math]can have integral solution [math]x, y, [/math] and [math]c[/math] for some integer [math]a[/math].
However, I wonder if [math]a[/math] is a rational number such that [math]0<a<1[/math],
is it possible to find such an [math]a[/math] for which [math]x^2 - axy + y^2=c^2[/math] has
an integral solution? If so, what are the conditions on such an [math]a[/math]?
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