Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits for computing the permanent of n by n matrices having size polynomial in n, then τ(n!) is polynomially bounded in logn. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials $\prod_{k=1}^n (X-k)$ and the Taylor approximations and of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in logn (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.