Existence of a prime divisor

Statement

Let $n$ be a natural number with $n \ge 2$ . Then there is a prime number $p$ such that $p \mid n$ .

Remarks

One proof takes the smallest divisor $d$ of $n$ with $d > 1$ . Each divisor $e$ of $d$ with $e > 1$ also divides $n$ , so $e = d$ . Thus $d$ is prime. This smallest divisor is the smallest prime factor of $n$ . The result is also true for $n = 0$ , since each prime divides $0$ . It is false for $n = 1$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Divisibility definition
Prime number definition
Existence of a prime divisor statement

Where this leads

12 pages build on this one.

Tag	`0SPBEDVNW1`
URI	`m2://0SPBEDVNW1`
Kind	statement, proposition
Writing	W0 stub
MSC	11A41
Status	This copy of the site has no build data. Run `m2 status 0SPBEDVNW1`.
Uses	Prime number, Divisibility

Existence of a prime divisor

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

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