素数階乗素数 (Primorial prime) とは、\(p_{n}\#\pm1\)の形で表される素数である。ここで\(p_{n}\#\)は素数階乗であり、\(n\)番目までの素数の積である。また\(p_{0}\#=1\)である。素数階乗素数が無数に存在するかどうかは未解決問題である[1]。
なお、素数が無限にあるというユークリッドの定理で、\(p_{n}\#+1\)に相当する数の仮定がされたことから、\(p_{n}\#+1\)形式の数はユークリッド数 (Euclid number) と呼ばれることがある[2][3]。また、\(p_{n}\#-1\)形式の数はクンマー数 (Kummer number) と呼ばれることがある[4]。よってユークリッド数な素数はユークリッド素数、クンマー数な素数はクンマー素数と命名できると考えられる。
\(p_{n}\#+1\)形式の素数の一覧[]
| \(n\)[5] | \(p_{n}\)[6] | \(p_{n}\#+1\) |
|---|---|---|
| \(0\) | \(p_{0}\#+1=2\) | |
| \(1\) | \(2\) | \(p_{1}\#+1=3\) |
| \(2\) | \(3\) | \(p_{2}\#+1=7\) |
| \(3\) | \(5\) | \(p_{3}\#+1=31\) |
| \(4\) | \(7\) | \(p_{4}\#+1=211\) |
| \(5\) | \(11\) | \(p_{5}\#+1=2311\) |
| \(11\) | \(31\) | \(p_{11}\#+1=200560490131\) |
| \(75\) | \(379\) | \(p_{75}\#+1\approx1.71962\times10^{153}\) |
| \(171\) | \(1019\) | \(p_{171}\#+1\approx2.04041\times10^{424}\) |
| \(172\) | \(1021\) | \(p_{172}\#+1\approx2.08326\times10^{427}\) |
| \(384\) | \(2657\) | \(p_{384}\#+1\approx7.82447\times10^{1114}\) |
| \(457\) | \(3229\) | \(p_{457}\#+1\approx6.89481\times10^{1367}\) |
| \(616\) | \(4547\) | \(p_{616}\#+1\approx1.31957\times10^{1938}\) |
| \(643\) | \(4787\) | \(p_{643}\#+1\approx1.50348\times10^{2037}\) |
| \(1391\) | \(11549\) | \(p_{1391}\#+1\approx7.78673\times10^{4950}\) |
| \(1613\) | \(13649\) | \(p_{1613}\#+1\approx1.03469\times10^{5861}\) |
| \(2122\) | \(18523\) | \(p_{2122}\#+1\approx2.10914\times10^{8001}\) |
| \(2647\) | \(23801\) | \(p_{2647}\#+1\approx1.59416\times10^{10272}\) |
| \(2673\) | \(24029\) | \(p_{2673}\#+1\approx1.12198\times10^{10386}\) |
| \(4413\) | \(42209\) | \(p_{4413}\#+1\approx5.88089\times10^{18240}\) |
| \(13494\) | \(145823\) | \(p_{13494}\#+1\approx1.86763\times10^{63141}\) |
| \(31260\) | \(366439\) | \(p_{31260}\#+1\approx2.11710\times10^{158935}\) |
| \(33237\) | \(392113\) | \(p_{33237}\#+1\approx1.04396\times10^{169965}\) |
\(p_{n}\#-1\)形式の素数の一覧[]
| \(n\)[7] | \(p_{n}\)[8] | \(p_{n}\#-1\) |
|---|---|---|
| \(2\) | \(3\) | \(p_{2}\#-1=5\) |
| \(3\) | \(5\) | \(p_{3}\#-1=29\) |
| \(5\) | \(11\) | \(p_{5}\#-1=2309\) |
| \(6\) | \(13\) | \(p_{6}\#-1=30029\) |
| \(13\) | \(41\) | \(p_{13}\#-1=304250263527209\) |
| \(24\) | \(89\) | \(p_{24}\#-1\approx2.37687\times10^{34}\) |
| \(66\) | \(317\) | \(p_{66}\#-1\approx1.93614\times10^{130}\) |
| \(68\) | \(337\) | \(p_{68}\#-1\approx2.15971\times10^{135}\) |
| \(167\) | \(991\) | \(p_{167}\#-1\approx1.96493\times10^{412}\) |
| \(287\) | \(1873\) | \(p_{287}\#-1\approx7.14887\times10^{789}\) |
| \(310\) | \(2053\) | \(p_{310}\#-1\approx4.04764\times10^{865}\) |
| \(352\) | \(2377\) | \(p_{352}\#-1\approx1.33725\times10^{1006}\) |
| \(564\) | \(4093\) | \(p_{564}\#-1\approx1.20392\times10^{1749}\) |
| \(590\) | \(4297\) | \(p_{590}\#-1\approx1.99837\times10^{1843}\) |
| \(620\) | \(4583\) | \(p_{620}\#-1\approx5.73047\times10^{1952}\) |
| \(849\) | \(6569\) | \(p_{849}\#-1\approx1.16321\times10^{2810}\) |
| \(1552\) | \(13033\) | \(p_{1552}\#-1\approx2.44956\times10^{5609}\) |
| \(1849\) | \(15877\) | \(p_{1849}\#-1\approx8.29041\times10^{6844}\) |
| \(67132\) | \(843301\) | \(p_{67132}\#-1\approx7.44251\times10^{365850}\) |
| \(85586\) | \(1098133\) | \(p_{85586}\#-1\approx1.04416\times10^{476310}\) |
| \(234725\) | \(3267113\) | \(p_{234725}\#-1\approx2.64328\times10^{1418397}\) |
出典[]
- ↑ Eric Weisstein. "Primorial Prime". Wolfram MathWorld.
- ↑ Eric Weisstein. "Euclid Number". Wolfram MathWorld.
- ↑ "A006862: Euclid numbers: 1 + product of the first n primes". On-Line Encyclopedia of Integer Sequences.
- ↑ "A057588: Kummer numbers: -1 + product of first n consecutive primes". On-Line Encyclopedia of Integer Sequences.
- ↑ "A014545: Primorial plus 1 prime indices: n such that n-th Euclid number A006862(n) = 1 + (Product of first n primes) is prime". The On-Line Encyclopedia of Integer Sequences.
- ↑ "A014545: Primorial plus 1 primes: primes p such that 1 + product of primes up to p is prime". The On-Line Encyclopedia of Integer Sequences.
- ↑ "A057704: Primorial - 1 prime indices: integers m such that the m-th primorial minus 1 is prime". The On-Line Encyclopedia of Integer Sequences.
- ↑ "A006794: Primorial -1 primes: primes p such that -1 + product of primes up to p is prime". The On-Line Encyclopedia of Integer Sequences.
関連項目[]
外部リンク[]
- "Primorial". The PrimePages. (素数階乗の表記が異なる事に注意。\(n\#\)は\(2\)から\(n\)までの素数の積。)