My instructor provided a proof for the Theorem: The number of primes is infinite.
Proof by Contradiction
Assume finite number of primes this means there is a largest prime say $p$.
Now lets say there is a prime $q$ in which $q | p!+1$. However this means $q > p$, since
we have found $q$ to be greater than the assumed largest prime there is a contradiction
thus there is an infinite number of primes.
From the above proof does $p!$ is p factorial or should it mean p! is the product of
primes ergo $p_1,p_2,…,p_n$ ergo is $p!$ assumed to be the product of primes?
I assume it is the product of primes because most of the proofs
I have seen would say
$p = p_1\times p_2\times…\times p_n$ where $p_1, p_2,…p_n$ are all primes.
Best Answer
You can use either one. The product of all the primes up to $p$ is called the primorial and sometimes written $p\#$. We have $p\#$ divides $p!$ because $p!$ includes all the primes less than $p$ as factors along with other numbers. The statement that any prime dividing $p\#+1$ or $p!+1$ must be greater than $p$ goes through and provides a prime greater than $p$ as required.