Let $p_n\# = \prod_{k=1}^n p_k$ be the $n$-th primorial.

Q1. Given $n$ (in binary) is there an efficient way (polynomial time) to calculate the exact number of digits of the binary representation of $p_n\#$; in other words calculate $m$ such that $2^m \leq p_n\# < 2^{m+1}$

**Edit**: The problem seems harder than I thought, so also the following could help:

Q2. Does the problem become easier if we "drop" a fixed number of low primes, i.e. we want to calculate the exact number of digits of the binary representation of $p_n\#^* = \prod_{k=a}^n p_k$ for some fixed $a$ ? And if we further relax it to $a = \log n$ ?

exactnumber of bits. (I'm not an expert of number theory, but the result could help me to refine a problem in computational complexity) $\endgroup$ – Marzio De Biasi Jan 22 '15 at 20:13