Primorial
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For n ≥ 2, the primorial (n#) is the product of all prime numbers less than or equal to n. For example, 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. The first few primorials are
2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410. Template:OEIS
They grow rapidly.
The idea of multiplying all primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.
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
- Factorial and primorial primes. J. Recr. Math., 19, 1987, 197-203it:Primoriale