# Thread: integer solutions for a straight line

1. ## integer solutions for a straight line

Hi,

I came across this..

It is well known that the equation Ix + my + n = 0, with rational integral coefficients, has either no solution in rational integers
or an infinite number of solutions.
here http://www.ams.org/journals/tran/194...-0006739-2.pdf

How is it possible to determine whether a straight line has rational integer solutions or not?

$5x^2 - y^2 + 20 = 0$ has integer solutions at (11, 25), (29, 65), (199, 445)......

It appears that there are no integer solutions for $5x^2 - y^2 + 45 = 0$ but I don't know how to go about proving (or disproving) there are no integer solutions.

I'd be grateful if someone could provide any help.

Thanks

3. ## Re: integer solutions for a straight line

Originally Posted by earboth
Thanks earboth, I'll take a peek at my post in a while

btw, there are two corrections to my OP

1) hyperbola, not straight line

2) x and y must be coprime

thanks again