Smallest prime factor

Builds on Prime number Existence of a prime divisor

Definition

Let $n$ be a natural number with $n \ge 2$ . The smallest prime factor of $n$ is the smallest prime number $p$ such that $p \mid n$ . It exists by the existence of a prime divisor.

Remarks

The smallest prime factor of $n$ is also the smallest divisor $d$ of $n$ with $d > 1$ . The number $n$ is prime if and only if its smallest prime factor is $n$ . If $n$ is composite, its smallest prime factor $p$ satisfies $p^2 \le n$ . See a composite number has a prime factor at most its square root. For example, the smallest prime factor of $91 = 7 \cdot 13$ is $7$ .

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Divisibility definition
Prime number definition
Existence of a prime divisor statement
Smallest prime factor definition

Where this leads

1 page builds on this one.

A composite number has a prime factor at most its square root

Tag	`0S2SYHTWPY`
URI	`m2://0S2SYHTWPY`
Aliases	`least-prime-factor`, `minfac`
Kind	definition
Writing	W0 stub
MSC	11A41
Defines	Smallest prime factor
Uses	Prime number, Existence of a prime divisor

Smallest prime factor

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

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