# Euclid's Theorem

## Contents

## Theorem

For any finite set of prime numbers, there exists a prime number not in that set.

In the words of Euclid:

*Prime numbers are more than any assigned multitude of prime numbers.*

(*The Elements*: Book $\text{IX}$: Proposition $20$)

### Corollary 1

There are infinitely many prime numbers.

### Corollary 2

There is no largest prime number.

## Proof

Let $\mathbb P$ be a finite set of prime numbers.

Consider the number:

- $\displaystyle n_p = \left({\prod_{p \mathop \in \mathbb P} p}\right) + 1$

Take any $p_j \in \mathbb P$.

We have that:

- $\displaystyle p_j \divides \prod_{p \mathop \in \mathbb P} p$

Hence:

- $\displaystyle \exists q \in \Z: \prod_{p \mathop \in \mathbb P} p = q p_j$

So:

\(\displaystyle n_p\) | \(=\) | \(\displaystyle q p_j + 1\) | $\quad$ Division Theorem | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle n_p\) | \(\perp\) | \(\displaystyle p_j\) | $\quad$ Definition of Coprime Integers | $\quad$ |

So $p_j \nmid n_p$.

There are two possibilities:

- $(1): \quad n_p$ is prime, which is not in $\mathbb P$.

- $(2): \quad n_p$ is composite.

But from Positive Integer Greater than 1 has Prime Divisor, it must be divisible by *some* prime.

That means it is divisible by a prime which is not in $\mathbb P$.

So, in either case, there exists at least one prime which is not in the original set $\mathbb P$ we created.

$\blacksquare$

## Historical Note

This theorem is Proposition $20$ of Book $\text{IX}$ of Euclid's *The Elements*.

## Fallacy

There is a fallacy associated with Euclid's Theorem.

It is often seen to be stated that: the number made by multiplying all the primes together and adding $1$ is not divisible by any members of that set.

So it is not divisible by any primes and *is therefore itself prime*.

That is, sometimes readers think that if $P$ is the product of the first $n$ primes then $P + 1$ is itself prime.

This is not the case.

For example:

- $\left({2 \times 3 \times 5 \times 7 \times 11 \times 13}\right) + 1 = 30\ 031 = 59 \times 509$

both of which are prime, but, take note, not in that list of six primes that were multiplied together to get $30\ 030$ in the first place.

## Also see

## Source of Name

This entry was named for Euclid.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 22 \beta$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Theorem $5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes: Theorem $2$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.19$: The Series $\sum 1/ p_n$ of the Reciprocals of the Primes - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers