table of differences between $\lceil \sqrt{n!} \rceil^2$ and $n!$ for $0 < n < 26$
TableOfDifferencesBetweenLceilSqrtnRceil2AndNFor0N26
2008-07-01 19:05:55
2008-07-02 18:17:35
Data Structure
Brocard's problem
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There are only three known solutions to Brocard's problem, and the near misses all seem to occur early on. Notice how, for example, 3! is just 3 shy of a square (compared to 1 shy of a square which is what Brocard's problem asks for). Still, the differences between a factorial and the next higher perfect square don't make for a consistently ascending order sequence. For a few values of $n$, (such as 4, 7, 10, 24, 26, 42, 117, 135) this difference is smaller than the previous difference. In general, however, the difference between a factorial and the next perfect square widens as $n$ gets larger.
The following table gives the square root of $n!$ to six decimal places, and then the difference between the factorial and the next higher square (obtained by taking the ceiling of the square root of $n!$ and squaring that integer).
\begin{tabular}{|r|r|r|}
$n$ & $\sqrt{n!}$ & $\lceil \sqrt{n!} \rceil^2 - n!$ \\
1 & 1.000000 & 0 \\
2 & 1.414214 & 2 \\
3 & 2.449489 & 3 \\
4 & 4.898979 & 1 \\
5 & 10.954451 & 1 \\
6 & 26.832816 & 9 \\
7 & 70.992957 & 1 \\
8 & 200.798406 & 81 \\
9 & 602.395219 & 729 \\
10 & 1904.940944 & 225 \\
11 & 6317.974359 & 324 \\
12 & 21886.105181 & 39169 \\
13 & 78911.474451 & 82944 \\
14 & 295259.701280 & 176400 \\
15 & 1143535.905864 & 215296 \\
16 & 4574143.623456 & 3444736 \\
17 & 18859677.306253 & 26167684 \\
18 & 80014834.285449 & 114349225 \\
19 & 348776576.634429 & 255004929 \\
20 & 1559776268.628498 & 1158920361 \\
21 & 7147792818.185865 & 11638526761 \\
22 & 33526120082.371712 & 42128246889 \\
23 & 160785623545.405884 & 191052974116 \\
24 & 787685471322.938354 & 97216010329 \\
25 & 3938427356614.691406 & 2430400258225 \\
\end{tabular}