In this answer/this blog post by Andrej Bauer, he mentions finding these two short Diophantine equations, which "gave a professional number theorist something to munch on for a couple of weeks":
$$\exists \, a,b,c. \; a^2 - 2 = (a + b) \, b \, c$$
and
$$\exists \, a,b,c. \; a^2 + a - 1 = (a + b) \, b \, c$$
(with $a,b,c \in \mathbb{N}$), but gives no hint about the outcome of the munching.
Do either of these equations have solutions? I have already checked all $a,b,c \le 10^5$.