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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.

Dando18
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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.

Ross Millikan
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  • "because p! includes all the primes less than p as factors along with other numbers." Doesn't p! include all the primes up to p including p? –  Jun 19 '17 at 18:23
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    Yes it does @DanielTellier, but is not a matter of significance. the answer is fairly well written. +1 for primorial and making it clear that the question isn't actually relevant whether it is product of primes upto p or product of natural number upto p :-( – Vidyanshu Mishra Jun 19 '17 at 18:44
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The usual proof works with the product of the primes up to $p$, but this argument with the factorial $p!$ works just as well.

Ethan Bolker
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The factorial and the primorial are different things. Given a positive integer $n$, the factorial $n!$ is divisible by each integer from $1$ to $n$, whether it's prime or not. e.g., $5! = 1 \times 2 \times 3 \times 4 \times 5 = 120.$

What is $5$ primorial? There is some disagreement: it could be $2 \times 3 \times 5 = 30$, or it could be $2 \times 3 \times 5 \times 7 \times 11 = 2310$ (the product of the first five primes).

But if we are agreed on the definitions, we can use either factorials or primorials. The important thing is that we have some number divisible by lots of distinct primes, and one which is divisible by none of those (add or subtract $1$ from the former to obtain the latter).

The factorials have a slight disadvantage in that they get larger quicker. Compare $29! = 8841761993739701954543616000000$ to the product of the primes from $2$ to $29$, which is $6469693230$.

Now notice that $6469693231 = 331 \times 571 \times 34231$, while the least prime factor of $8841761993739701954543616000001$ is $14557$.

Mr. Brooks
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  • The size of the product cannot be a disadvantage in such a proof since it plays no role. But it would come into play if one interpreted the proof constructively to obtain an algorithm generating an infinite sequence of primes. – Bill Dubuque Jun 19 '17 at 23:46
  • @BillDubuque I feel like the factorials have a disadvantage both from a pedagogical perspective and from the perspective of elegance. A handheld scientific calculator can go farther with the primorials than with the factorials before switching to scientific notation. And from the standpoint of elegance, the factorials have a lot of repeated prime factors, while the primorials have only distinct prime factors. – Mr. Brooks Jun 20 '17 at 21:29
  • But the proof does not involve calculations with specific numbers. – Bill Dubuque Jun 20 '17 at 21:38
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    It sure doesn't, which is why it's most often stated without reference to either factorials or primorials. If you have to choose one or the other, choose primorials. With the factorials you might confuse people like young Daniel. – Mr. Brooks Jun 20 '17 at 21:49
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Proof that primes are infinite using factorial. $1 +$ the factorial of a number is a number that is not divisible by the numbers that make up the factorial with the exception of $1$. At this point two cases are possible: $n! +1$ is prime or $n! +1$ is the product of primes that are not part of $n!$ of which the smallest is at least $n + 1$. Reasons why prime numbers are infinite.

Mohsen Shahriari
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Davide Rotondo
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