Let PRIMES (a.k.a. primality testing) be the problem:
Given a natural number $n$, is $n$ a prime number?
Let FACTORING be the problem:
Given natural numbers $n$, $m$ with $1 \leq m \leq n$, does $n$ have a factor $d$ with $1 < d < m$?
Is it known whether PRIMES is P-hard? How about FACTORING? What are the best known lowerbounds for these problems?