The actual problem is: "Does x^2+y^2=3 have any rational points? If so, find a way to describe all of them. If not, prove it."
The Attempt at a Solution
I found a book on Google Books (can't find it again) that said that this circle has no rational points, so I have been working at proving this. If this book is wrong, I would only need to find one rational point to prove that the circle has infinite rational points, as the center is rational and any line through that single rational point with a rational slope would give a rational point wherever it intersects the circle.
Anyway, to prove that there are no rational points:
First, to test my theory that there are no rational points, I wrote a program. I chose a bunch of random rational x values, solved for y, and tested the values for rationality. I found nothing rational, so that was inconclusive.
I also tried a proof by contradiction. The definition of a rational point is that both values have to be rational and the definition of rational is that the number can be formed by m/n where m and n are both integers. So I tried plugging into an equation:
(x/a)^2+(y/b)^2=3 and simplifying, but this didn't seem to get me anywhere. A proof by contradiction would require an impossibility but I kept getting functions no matter how much I algebra'd it.
Next I tried a proof by induction. My theory was that I could plug in x=n=0 and at each inductive step plug in n=n+lim(k->inf)(1/k), but obviously this doesn't work (for too many reasons). Also, I would need some function or method to test if the value was rational, which I just don't see. Since theoretically it is impossible to get a step of any size and still test all the points on the circle, I have thrown away the possibility of a proof by induction. (I'm really bad at them though, so I might be missing something).
I don't really know where to go from here. Obviously this is open ended and it is quite possible that my theory and that book I found were wrong and there is a rational point on this circle. If anybody could point me in the right direction, I would appreciate it. Thank you!