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Hometable of differences between $\lceil \sqrt{n!} \rceil^2$ and $n!$ for $0 < n < 26$

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# table of differences between $\lceil\sqrt{n!}\rceil^{2}$ and $n!$ for $0<n<26$

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 |

## Mathematics Subject Classification

11A25*no label found*

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