Prove that there are no integers $x$ and $y$ such that $ (x/y)^2 = 2$

Question

Prove that there are no $x$ and $y$ such that $(x/y)^2 = 2$

It says that I can assume that $x$ and $y$ are relatively prime so I started the proof as:

• $x$ and $y$ relatively prime
• $\implies$ there are integer $m$ and $n$ s.t $mx + ny = 1$
• $\implies (m/n)^2 = ((1-ny)/(xn))^2$

I don't know what to do after this. Thanks

Answer 1

This is literally the same as asking if $\sqrt 2$ is irrational, and there's a plethora o proofs in this site: Prove that square root of 2 is irrational using the principle of Mathematical Induction

square root of 2 irrational - alternative proof

And many more, use the search option.