Let's define for a positive integer $n$: $$a(n) = \min \{|n! - 2^m| : m \in \mathbb N \}.$$ Does there exist a good asymptotic lower bound for the values $a(n)$ for large $n$? In particular, is the following conjecture true?:



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> Conjecture: For all positive $k$, $a(n)$ is bigger than a constant multiple of $2^{((\log n)^k)}$. 
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Minimum distance between factorials and powers of 2
Let $2^r$ be the highest power of 2 dividing $n!$. If $m$ minimizes your expression, it's easy to see that $m \geq r$. So your expression, which is non-zero for $n \geq 3$, will be divisible by and hence at least $2^r$. Now your conjecture (and more) follows from (e.g.)
$$r = \sum\_{k=1}^{\infty}{\lfloor\frac{n}{2^k}\rfloor} \geq \frac{n}{2}-1.$$


