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 attributed to Harvey Dubner. The first few primorials are

2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410. (sequence A002110 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