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# primorial

The primorial of $n$, or $n\#$, is the product of the first $n$ consecutive primes, thus:

$\prod_{{i=1}}^{n}p_{i}$ |

($p_{i}$ is the $i$th prime number).

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 $n\#$ is used to refer to the product of all primes $p<\pi(n)$, where $\pi$ is the prime counting function (so then $4\#=6$ 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, $n\#+1$ is a number that is not divisible by any of the existing primes, but if that is a prime then it 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.

## Mathematics Subject Classification

11A41*no label found*

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## Comments

## number sign

Help! I can't get the pound key sign # to show up...

## Re: number sign

\#

## Re: number sign

Duh! Of course! Thank you very, very much!

## highly cototeint numbers congruence conjectures

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.

## Re: highly cototeint numbers congruence conjectures

Insightful. The observation 9 mod 10 almost looks a red herring, then.

## Re: highly cototient numbers congruence conjectures

I wouldnt call it a red hering. It gave me the idea to remove the nines and look at teh factorizations of the numbres. Fisrt I noticed that they were all below a multiple of 3, then i noticed they were below a multiple of 6, then 30, then 210. and sure nough, 2309, 4619 and 6929 are listed in A100827. 30029 might not look taht big, but it requires testing some much larger primes.