$ \newcommand{\F}{{\mathbb F}}$ $ \newcommand{\R}{{\mathbb R}}$ $ \renewcommand{\phi}{\varphi}$ Let $ p\ge 5$ be a prime. As an easy exercise, if the functions $ \phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $ \phi_1(x)+\phi_2(y)=\phi_3(x+y)$ for all pairs $ (x,y)\in\F_p^2$ , then they are constant functions. Given that $ \phi_1,\phi_2,\phi_3\colon\F_p\to\R$ are non-constant, what is the largest possible number of pairs $ (x,y)\in\F_p^2$ satisfyingRead more