Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?

$x^2-2$ is irreducible over $\mathbb{Z}$ by reduction since it is irreducible over $\mathbb{F}_3$. Check directly $0^2-2 \equiv 1, 1^2-2 \equiv 2, 2^2-2 \equiv 2 \bmod 3$.

Another alternative: If $z \in \mathbb{Z}$ with $z^2=2$, then the $2$-adic valuation gives $2 \cdot v_2(z)=v_2(2)=1$, contradiction.

This is just the standard proof, rewritten in modular arithmetic:

The key here is that $\gcd(m,n)=1$.

Now, look at $$n^2=2m^2 \pmod{4},$$ in all three cases:

• $m,n$ both odd.
• $m$ even, $n$ odd
• $m$ odd, $n$ even

This proof is pretty artificial though.