Primorials
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 x 3 x 5 x 7). The name is atttributed to Harvey Dubner. The first few primorials are
2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410. (sequence in OEIS)
They grow rapidly.
The idea of multiplying all primes occurs in the 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 = 2x6x30).
Table of primorials
p: p# (p prime)
--- ------------
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
References
- Factorial and primorial primes. J. Recr. Math., 19, 1987, 197-203
