From Harvard qualification exam, 1990.
Let $f,g$ be two entire holomorphic functions satisfy the property $$f(z)^{2}=g(z)^{6}-1,\forall z\in \mathbb{C}$$ Prove that $f,g$ are constant functions. Would this be the same if $f,g$ are allowed to be meromorphic functions?
The problem comes with a hint that I should think about the algebraic curve $$y^{2}=x^{6}-1$$but I do not see how they are related. I know this curve is hyperellipitic (from Riemann-Hurwitz or simply the wiki article). But how this help?(this curve should be of genus 2). Taking a short look at the entire function article also seems to be no help. Is the author implying $f,g$ is not best to be treated by classical Riemann Surface applications (as opposed to algebraic geometry ones)?