Primorial

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n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

p_n# 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 (sequence A002110 in OEIS):

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).

1 Table of primorials
2 See also
3 References
4 External links

[edit] 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