階乗素数 (Factorial prime) とは\(n!\pm1\)の形で表される素数のことである[1]。
なお、名称が類似しているが別物の素数として素数階乗素数 \(p_{n}\#\pm1\) がある。ここで\(p_{n}\#\)は素数階乗を意味する。
\(n!+1\)の形式での一覧[]
| \(n!+1\) | 大きさ |
|---|---|
| \(0!+1\) | \(2\) |
| \(1!+1\) | \(2\) |
| \(2!+1\) | \(3\) |
| \(3!+1\) | \(7\) |
| \(11!+1\) | \(39916801\) |
| \(27!+1\) | \(\sim1.0889\times10^{28}\) |
| \(37!+1\) | \(\sim1.3734\times10^{43}\) |
| \(41!+1\) | \(\sim3.3453\times10^{49}\) |
| \(73!+1\) | \(\sim4.4701\times10^{105}\) |
| \(77!+1\) | \(\sim1.4518\times10^{113}\) |
| \(116!+1\) | \(\sim3.3931\times10^{190}\) |
| \(154!+1\) | \(\sim3.0898\times10^{271}\) |
| \(320!+1\) | \(\sim2.1161\times10^{664}\) |
| \(340!+1\) | \(\sim5.1009\times10^{714}\) |
| \(399!+1\) | \(\sim1.6009\times10^{866}\) |
| \(427!+1\) | \(\sim2.9064\times10^{939}\) |
| \(872!+1\) | \(\sim1.9723\times10^{2187}\) |
| \(1477!+1\) | \(\sim5.0798\times10^{4041}\) |
| \(6380!+1\) | \(\sim1.8140\times10^{21506}\) |
| \(26951!+1\) | \(\sim2.3308\times10^{107706}\) |
| \(110059!+1\) | \(\sim1.4835\times10^{507081}\) |
| \(150209!+1\) | \(\sim2.3458\times10^{712354}\) |
| \(288465!+1\) | \(\sim1.3654\times10^{1449770}\) |
| \(308084!+1\) | \(\sim1.2791\times10^{1557175}\) |
| \(422429!+1\) | \(\sim1.4081\times10^{2193026}\) |
\(n!-1\)の形式での一覧[]
| \(n!-1\) | 大きさ |
|---|---|
| \(3!-1\) | \(5\) |
| \(4!-1\) | \(23\) |
| \(6!-1\) | \(719\) |
| \(7!-1\) | \(5039\) |
| \(12!-1\) | \(479001599\) |
| \(14!-1\) | \(87178291199\) |
| \(30!-1\) | \(\sim2.6525\times10^{32}\) |
| \(32!-1\) | \(\sim2.6313\times10^{35}\) |
| \(33!-1\) | \(\sim8.6833\times10^{36}\) |
| \(38!-1\) | \(\sim5.2302\times10^{44}\) |
| \(94!-1\) | \(\sim1.0874\times10^{146}\) |
| \(166!-1\) | \(\sim9.0037\times10^{297}\) |
| \(324!-1\) | \(\sim2.2890\times10^{674}\) |
| \(379!-1\) | \(\sim2.4840\times10^{814}\) |
| \(469!-1\) | \(\sim6.7718\times10^{1050}\) |
| \(546!-1\) | \(\sim1.4130\times10^{1259}\) |
| \(974!-1\) | \(\sim5.5848\times10^{2489}\) |
| \(1963!-1\) | \(\sim3.3733\times10^{5613}\) |
| \(3507!-1\) | \(\sim1.5508\times10^{10911}\) |
| \(3610!-1\) | \(\sim9.4574\times10^{11276}\) |
| \(6917!-1\) | \(\sim1.0360\times10^{23559}\) |
| \(21480!-1\) | \(\sim1.0373\times10^{83726}\) |
| \(34790!-1\) | \(\sim5.8126\times10^{142890}\) |
| \(94550!-1\) | \(\sim1.3855\times10^{429389}\) |
| \(103040!-1\) | \(\sim2.1133\times10^{471793}\) |
| \(147855!-1\) | \(\sim3.0024\times10^{700176}\) |
| \(208003!-1\) | \(\sim8.5854\times10^{1015842}\) |
特殊な値[]
\(n!\pm1\)が共に素数となるのは、知られているのは\(3!\pm1=5,\ 7\)のみである。
\(n!+1\)と\((n+1)!+1\)、あるいは\(n!-1\)と\((n+1)!-1\)が共に素数となるのは\(n\)が小さい値ならばしばしば見られる。ただし大きな例として\(32!-1,\ 33!-1\)は共に素数である。
出典[]
- ↑ "Factorial Prime". Wolfram MathWorld.
- ↑ 2.0 2.1 "Factorial primes". The Prime Pages.
- ↑ "A002981: Numbers k such that k! + 1 is prime". On-Line Encyclopedia of Integer Sequences.
- ↑ "A002982: Numbers k such that k! - 1 is prime". On-Line Encyclopedia of Integer Sequences.