Existence of prime factorization

Builds on Prime factorization Existence of a prime divisor

Statement

Each positive integer $n$ has a prime factorization. That is, there are prime numbers $p_1, \dots, p_k$ , with $k \ge 0$ , such that $n = p_1 p_2 \cdots p_k$ .

Remarks

The proof uses strong induction on $n$ . If $n = 1$ , take $k = 0$ . If $n$ is prime, take $k = 1$ . Otherwise $n = a b$ with $1 < a, b < n$ , and we join the factorizations of $a$ and $b$ . The fundamental theorem of arithmetic adds that the factorization is unique, up to the order of the factors.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Divisibility definition
Prime number definition
Existence of a prime divisor statement
Prime factorization definition
Existence of prime factorization statement

Where this leads

7 pages build on this one.

Tag	`020K0X7HVR`
URI	`m2://020K0X7HVR`
Kind	statement, theorem
Writing	W0 stub
MSC	11A51
Status	This copy of the site has no build data. Run `m2 status 020K0X7HVR`.
Uses	Prime factorization, Existence of a prime divisor

Existence of prime factorization

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

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