Are the problems PRIMES, FACTORING known to be P-hard? 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 $1 &lt; d &lt; m$?


Is it known whether PRIMES is P-hard? How about FACTORING? 
What are the best known lowerbounds for these problems?
PRIMES is known to be hard for $\mathsf{TC^0}$.  See my paper with Saks and Shparlinski, "A Lower Bound for Primality" in JCSS 62 (2001).  No improvement on that front since then.