The number of digits in $n!$ grows as $n\log n$, and some roughly fixed fraction of those are zeroes, so $n!$ will tend to have fewer than $n$ zeroes for small $n$ and greater than $n$ zeroes for large $n$. Can anyone split the difference and find the exact solution(s)? If this is infeasible in base $10$, can you do it for smaller bases?

  • $\begingroup$ Are you talking about trailing zeros only or total number of zeros? $\endgroup$ – Jaideep Khare Mar 15 '17 at 19:21
  • $\begingroup$ Total number of zeroes. $\endgroup$ – mjqxxxx Mar 15 '17 at 19:37

Heuristically: we have $\approx \frac n4$ trailing zeroes at the end of $n!$ (as Jaideep Khare's answer notes); by Stirling's approximation there are (very) roughly $\log_{10}\left(\left(\frac ne\right)^n\right)\approx n\log_{10}n-.43n$ total digits in $n!$. Of the $n\log_{10}n-.68n$ remaining digits (i.e., the digits that aren't part of the trailing zeroes) we expect $\frac1{10}$ to be zero, so to have $n$ total zeroes we want $\frac1{10}(n\log_{10}n-.68n)=\frac34n$, or $(\frac{30}4+.68)n=n\log_{10}n$, or $\log_{10}n\approx8.18$; thus, looking for $n$ between about $10^7$ and $10^9$ — which should be fairly computer-accessible — should turn up potential candidates.

(If I were to do it, I'd test a few numbers in the general vicinity to make sure that the $\frac1{10}$ heuristic is roughly accurate, then use binary search to narrow down to a smaller plausible range where I could test individual cases. Just computing $n!$ for numbers in this range is going to be challenging; doing it by linear multiplication will take $\Omega(n^2)$ time which is barely on the edge of feasible, so you might need to look at approaches based around the prime factorization and/or CRT-based approaches. Strangely, I can't find any information on the web about computing ultra-large factorials.)

  • $\begingroup$ Thanks, Steven. For a square-free base $b$ you've got $n/(p-1)$ trailing zeroes (where $p$ is the largest prime dividing the base), and $n\log_{b}(n/e)$ total digits, of which $1/b$ of the non-trailing digits are zeroes. So we expect solutions where $n \approx e b^{(b(p-2) + 1)/(p-1)}$... for $(b,p)=(10,5)$, that's $n\approx 10^{31/4} e$. Some bases I can check manually: $(b,p)=(5,5)$ should have solutions for $n\approx 5^4 e\approx 1700$ (and has them from $n=1569$ to $n=2091$); and $(b,p)=(6,3)$ should have them for $n\approx 6^{7/2}e \approx 1440$ (and has them from $n=870$ to $n=2200$). $\endgroup$ – mjqxxxx Mar 16 '17 at 0:03
  • $\begingroup$ @mjqxxxx Out of curiosity, near the cores of those intervals what are the approximate densities of 'matches'? I haven't worked through the heuristic deeply enough to have a sense of what the standard deviation would be (and it would be a chunk of work from there to translate down to an expected number of solutions) $\endgroup$ – Steven Stadnicki Mar 16 '17 at 3:20
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    $\begingroup$ The number of non-trailing zeroes should be Poisson-distributed with mean (and variance) $\Theta(n)$, so the standard deviation is $\Theta(\sqrt{n})$, and the maximum probability of a solution at the core is $\Theta(n^{-1/2})$. They're maybe $50$ apart for the $b=5$ and $b=6$ cases, which is in the ballpark. So unfortunately for the decimal case you might have to try $10^4$ guesses before hitting a solution. $\endgroup$ – mjqxxxx Mar 16 '17 at 3:44
  • $\begingroup$ Just tested this for one more tractable case: $(b,p)=(7,7)$, where we expect solutions at $n\approx 7^6 e \approx 320000$. Starting at $316000$ and checking every possibility, I found solutions at $n=316464$, $n=316940$, $n=317354$, and $n=317989$. $\endgroup$ – mjqxxxx Mar 17 '17 at 5:20
  • $\begingroup$ Last comment :). For $b=8$, there are asymptotically $n/3$ trailing zeroes (exercise for the reader), so solutions are expected at $n\approx 8^{17/3} e \approx 356000$. Starting searching at $354000$, the first solution found is at $n=364803$ (a surprising number of misses). That is: the number $364803! \approx 10^{1870627}$ has exactly $364803$ zeroes when written in octal. $\endgroup$ – mjqxxxx Mar 17 '17 at 23:24

For all bases, the only time $n!$ has exactly $n$ zeros is $n=0$.

$$0! = 1$$

To see why, we are essentially counting the number of 5's in the prime factorization (2's are plentiful), and $n!$ has exactly $\sum_{i=1}^\infty \left\lfloor\frac{n}{5^i}\right\rfloor$ in its factorization (Landau 1900). Thus, we solve

$$ n = \sum_{i=1}^\infty \left\lfloor\frac{n}{5^i}\right\rfloor $$ which is only true for $n = 0$. It's the exact same story for smaller bases.

  • $\begingroup$ If i get your trail of thought you are considering only trailing $0$'s aren't you? $\endgroup$ – Andreas Ch. Mar 15 '17 at 19:15
  • $\begingroup$ yes, just counting the trailing 0s. $\endgroup$ – RationalHusky Mar 15 '17 at 19:17

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