Primorial
|
n# as a function of n (red dots), compared to n!. Both plots are logarithmic. |
|
pn# as a function of n, plotted logarithmically. |
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 :
2,
6,
30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410.
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).
| 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 |
*
Primorial prime* Harvey Dubner, "Factorial and primorial primes".
J. Recr. Math., 19, 197â€"203, 1987.
*
