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The primorial, a portmanteau of prime and factorial, is formally defined as

$p_n \# = \prod^{n}_{i = 1} p_i$

where $$p_n$$ is the nth prime.

Another slightly more complex definition, which expands the domain of the function beyond prime numbers, is

$n \# = \prod^{\pi (n)}_{i = 1} p_i$

where $$p_n$$ is the nth prime and $$\pi (n)$$ is the prime counting function.

Using either definition, the primorial of n can be informally defined as "the product of all prime numbers up to n, inclusive." For example, $$16 \# = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 = 30,030$$.

The primorial's relationship to the Chebyshev function $$\theta (x)$$ gives it the property

$\lim_{n\rightarrow\infty} \sqrt[p_n]{p_n \#} = e$

where e is the mathematical constant.

The sequence of primorials goes:

1, 2, 6, 30, 210, 2,310, 30,030, 510,510, ... (OEIS A002110)

## Euclid's theorem

The primorial can be used to prove that there are infinitely many primes (Euclid's theorem). If there was a largest prime $$P$$, then $$P \# + 1$$ and $$P \# - 1$$ would also be prime, which is a contradiction. Outside of the proof by contradiction, $$p \# + 1$$ or $$p \# - 1$$ is not always prime. Such primes are called primorial primes.