Alex Kim

Twin primes: $(3,5)$, $(5,7)$, $(11,13)$, $(17,19)$, $(41,43)$... The Twin Prime Conjecture says there are infinitely many pairs of primes with difference 2.

This problem is all over Twitter and Instagram right now: A square is inscribed in a semicircle of radius $R$. Find the area of the square in terms of $R$.

I gave several AIs this problem and they all failed:

Find ALL integer solutions to:

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.