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primorial (Definition)

Multiply the first $ n$ consecutive primes, thus:

$\displaystyle \prod_{i = 1}^n p_i$

($ p_i$ is the $ i$th prime number).

This is the primorial of $ n$, or $ n\char93 $. The first few primorials are 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130; these are listed in A002110 of Sloane's OEIS. Sometimes the notation $ n\char93 $ is used to refer to the product of all primes $ p < \pi(n)$, where $ \pi$ is the prime counting function (so then $ 4\char93 = 6$ rather than 210).

Primorials are used in the classic proof that there are infinitely many primes: assuming that there are exactly $ n$ primes and no more, $ n\char93 + 1$ is a number that is not divisible by any of the existing primes but if it is a prime then that contradicts the initial assumption.

If, in reckoning the sieve of Eratosthenes, one strikes out again numbers that have already been struck off, the sequence of the smallest number struck off $ n$ times is precisely the sequence of the primorials.

Any highly composite number (with the exception of 1) can be expressed as a product of primorials in at least one way.



"primorial" is owned by CompositeFan.
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primorial prime (Definition) by PrimeFan
table of the first 100 primorials (Data Structure) by PrimeFan
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Cross-references: highly composite number, sequence, sieve of Eratosthenes, divisible, number, prime counting function, product, OEIS, primes, consecutive
There are 14 references to this entry.

This is version 3 of primorial, born on 2006-06-14, modified 2006-06-15.
Object id is 8036, canonical name is Primorial.
Accessed 1030 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
Pending Errata and Addenda
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highly cototeint numbers congruence conjectures by Lando47 on 2006-12-01 13:07:00
Whit the exception of the small ones, all highly cototient numbres are congruent to -1 modulus a primorial. Furtehrmore, infinitly many primorial primes (of teh -1 variety) are also highly cototient.

But since calculating larger highly cototient numbers requires claculating larger primes, this conjectures can't be proven or disproven empiricly.
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number sign by CompositeFan on 2006-06-14 18:07:19
Help! I can't get the pound key sign # to show up...
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