Hi,

I came across this..

here http://www.ams.org/journals/tran/194...-0006739-2.pdfIt 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.

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

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

It appears that there are no integer solutions for $\displaystyle 5x^2 - y^2 + 45 = 0$ 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