table of differences between and $n!$ for

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).

$n$	$\sqrt{n!}$	${\lceil \sqrt{n!}\rceil }^2-n!$
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

Title	table of differences between and $n!$ for
Canonical name	TableOfDifferencesBetweenlceilsqrtnrceil2AndNFor0N26
Date of creation	2013-03-22 18:10:20
Last modified on	2013-03-22 18:10:20
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Data Structure
Classification	msc 11A25