Natural Number is Prime or has Prime Factor

Theorem

Let $a$ be a natural number greater than $1$.

Then either:

$a$ is a prime number

or:

there exists a prime number $p \ne a$ such that $p \divides a$

where $\divides$ denotes is a divisor of.

In the words of Euclid:

Any number either is prime or is measured by some prime number.

(The Elements: Book $\text{VII}$: Proposition $32$)

Proof

By definition of composite number $a$ is either prime or composite.

Let $a$ be prime.

Then the statement of the result is fulfilled.

Let $a$ be composite.

Then by Proposition $31$ of Book $\text{VII} $: Composite Number has Prime Factor:

$\exists p: p \divides a$

where $p$ is a prime number.

The result follows by Proof by Cases.

$\blacksquare$

Historical Note

This proof is Proposition $32$ of Book $\text{VII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VII}$. Propositions