'Infinite Prime Theorem' – Euclid's Theorem - Just C

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 $p$ .

Therefore, $m$ can only be the product of two or more primes larger than $p$ , and therefore contradicts the assumption that $p$ is the largest prime in a finite list.

Therefore, there is an infinite number of primes.

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

David Stiff - 2 months, 1 week ago

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

Just C - 2 months, 1 week ago

No problem!

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$