Primorial
Encyclopedia : P : PR : PRI : Primorial
For n ≥ 2, the primorial (n#) is the product of all prime numbers less than or equal to n. For example, 7# = 210 is a primorial which is the product of the first four primes multiplied together (2·3·5·7). The name is attributed to Harvey Dubner and is a portmanteau of prime and factorial. The first few primorials are (sequence in OEIS):
- 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410.
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials (e.g. 360 = 2·6·30).
Table of primorials
| p | p# |
|---|---|
| 2 | 2 |
| 3 | 6 |
| 5 | 30 |
| 7 | 210 |
| 11 | 2310 |
| 13 | 30030 |
| 17 | 510510 |
| 19 | 9699690 |
| 23 | 223092870 |
| 29 | 6469693230 |
| 31 | 200560490130 |
| 37 | 7420738134810 |
| 41 | 304250263527210 |
| 43 | 13082761331670030 |
| 47 | 614889782588491410 |
| 53 | 32589158477190044730 |
| 59 | 1922760350154212639070 |
| 61 | 117288381359406970983270 |
| 67 | 7858321551080267055879090 |
| 71 | 557940830126698960967415390 |
| 73 | 40729680599249024150621323470 |
| 79 | 3217644767340672907899084554130 |
| 83 | 267064515689275851355624017992790 |
| 89 | 23768741896345550770650537601358310 |
| 97 | 2305567963945518424753102147331756070 |
See also
References
- Harvey Dubner, "Factorial and primorial primes". J. Recr. Math., 19, 197–203, 1987.
External links
- , [Primorial] at MathWorld.
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.