# Thread: Rational solutions to an equation

1. ## Rational solutions to an equation

Prove that the equation $x^{2} + y^{2} = 7$ has no rational solutions.

I tried to go about this using the fact that for any integer $a$, $a^{2} \equiv 0(mod 4)$ or $a^{2} \equiv 1(mod4)$, but when I supposed that there existed a rational solution $(\frac{p}{q},\frac{m}{n})$ and arrived to the equation $(pn)^{2} + (qm)^{2} = 7(qn)^{2}$, I can't draw the conclusion I want. How do I go about this? Any help would be appreciated.

2. ## Re: Rational solutions to an equation

I used $mod7$ instead, and I reached the conclusion that $pn, qm, qn$ are all divisible by $7$. I know this in turn means the fractions assumed to be solutions are irreducible but I don't know how to prove that exactly.