# 'Infinite Prime Theorem' – Euclid's Theorem

## Problem:

Prove that there is an infinite number of primes.

## Solution:

The problem was originally called Euclid’s Theorem, named after Euclid, who proved that there were an infinite amount of primes in his book Elements (Book: IX, Proposition: 20) using contradiction. The following is an adaptation of his proof:

Assuming that there is a finite list of primes, let $m$ denote the product of all such primes and be one more than it.

$m=2\times 3\times 5\times 7\times...\times p+1$

It is then shown that $m$ must either be a prime or have prime factors larger than $p$.

Case $m$ is prime:

If $m$ is prime, and is the product of all primes in a finite list plus 1, then it must be larger than the largest prime $p$, meaning that there is a larger prime bigger than the one defined as the largest, and therefore, there cannot be a finite number of prime numbers.

Case $m$ is composite:

If $m$ is composite, it follows that it cannot have a prime factor smaller than $p$:

$m$ cannot be divided by 2 as it is $2(3\times 5\times 7\times...\times p)+1$, or one more than a multiple of 2.

$m$ cannot be divided by 3 as it is $3(2\times 5\times 7\times...\times p)+1$, or one more than a multiple of 3.

$m$ cannot be divided by 5 as it is $5(2\times 3\times 7\times...\times p)+1$, or one more than a multiple of 5.

It follows that $m$ cannot be divided by the assumed largest prime $p$ (in a finite list to comply with contradiction assumption that there is a finite number of primes) as it is $p(2\times 3\times 7\times...)+1$, or one more than a multiple of $p$.

Note by Just C
1 week, 4 days ago

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Nice! I think you meant to say "let $m$ denote the product of all such primes". :)

- 1 week, 4 days ago

Yeah, that was a mistake. Thanks for pointing it out! :)

- 1 week, 4 days ago

No problem!

- 1 week, 4 days ago