This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even partial answers to it may turn out useful and instructive for a number of users (certainly including me).

A trivial remark is that, in view of Stirling's formula, this question can be restated as follows: Is the sequence $((n+1/2)\log n-n\log e\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$? It would be surprising to me if the answer here is no or if it depends on the base of the logarithm.

equidistributed? $\endgroup$ – Mateusz Kwaśnicki Mar 27 '18 at 22:32A note on$N!$ inMathematics Magazine, which I discussed last week in my answer to Short papers for undergraduate course on reading scholarly math. $\endgroup$ – Dave L Renfro Mar 29 '18 at 18:56