Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2007 / B / 5 / iii

No prime works. The P-adic valuation induces a well-defined homomorphismThe class of $p$ maps to one, whereas the class of $p^p$ maps to zero. They therefore cannot be equal. Equivalently, $p$ cannot be a $p$th power because its valuation is not divisible by $p$.