Can ChatGPT prove that sqrt(2) is irrational?

I asked ChatGPT to prove $\sqrt{2}$ is irrational and it gave a perfect proof:

Assume $\sqrt{2} = \frac{p}{q}$ in lowest terms. Then $2 = \frac{p^2}{q^2}$ so $p^2 = 2q^2$. So $p^2$ is even, hence $p$ is even. Write $p = 2r$. Then $4r^2 = 2q^2$ so $q^2 = 2r^2$. So $q^2$ is even, hence $q$ is even. Contradiction since $p$ and $q$ were in lowest terms.

But heres the thing: this is just Euclids proof from 300 BC. The AI didnt "discover" anything, it just reproduced what it was trained on.

My question: Is AI capable of generating NOVEL mathematical proofs? Or is it just a sophisticated search engine that regurgitates existing mathematics?

What would it take for an AI to genuinely extend mathematical knowledge?