Integer is Expressible as Product of Primes

Theorem

Let $n$ be an integer such that $n > 1$.

Then $n$ can be expressed as the product of one or more primes.

Proof 1

Aiming for a contradiction, suppose this supposition is false.

Let $m$ be the smallest integer which can not be expressed as the product of primes.

As a prime number is trivially a product of primes, $m$ can not itself be prime.

Hence:

$\exists r, s \in \Z: 1 < r < m, 1 < s < m: m = r s$

As $m$ is our least counterexample, both $r$ and $s$ can be expressed as the product of primes.

Say $r = p_1 p_2 \cdots p_k$ and $s = q_1 q_2 \cdots q_l$, where all of $p_1, \ldots, p_k, q_1, \ldots, q_l$ are prime.

Hence $m = r s = p_1 p_2 \cdots p_k q_1 q_2 \cdots q_l$, which is a product of primes.

Hence there is no such counterexample.

$\blacksquare$

Proof 2

If $n$ is prime, the result is immediate.

Let $n$ be composite.

$\exists r, s \in \Z: n = r s, 1 < r < n, 1 < s < n$

This being the case, the set $S_1 = \set {d: d \divides n, 1 < d < n}$ is nonempty, and bounded below by $1$.

By Set of Integers Bounded Below by Integer has Smallest Element, $S_1$ has a smallest element, which we will call $p_1$.

Aiming for a contradiction, suppose $p_1$ is composite.

By Composite Number has Two Divisors Less Than It, there exist $a, b$ such that $a, b \divides p_1$ and $1 < a < p_1, 1 < b < p_1$.

But by Divisor Relation on Positive Integers is Partial Ordering, it follows that $a, b \divides n$ and hence $a, b \in S$.

This contradicts the assertion that $p_1$ is the smallest element of $S_1$.

Thus, $p_1$ is necessarily prime.

We may now write $n = p_1 n_1$, where $n > n_1 > 1$.

If $n_1$ is prime, the proof is complete.

Otherwise, the set $S_2 = \set {d: d \divides n_1, 1 < d < n_1}$ is nonempty, and bounded below by $1$.

By the above argument, the smallest element $p_2$ of $S_2$ is prime.

Thus we may write $n_1 = p_2 n_2$, where $1 < n_2 < n_1$.

This gives us $n = p_1 p_2 n_2$.

If $n_2$ is prime, we are done.

Otherwise, we continue this process.

Since $n > n_1 > n_2 > \cdots > 1$ is a (strictly) decreasing sequence of positive integers, there must be a finite number of $n_i$'s.

That is, we will arrive at some prime number $n_{k - 1}$, which we will call $p_k$.

This results in the prime decomposition $n = p_1 p_2 \cdots p_k$.

$\blacksquare$

Proof 3

The proof proceeds by induction.

For all $n \in \N_{> 1}$, let $\map P n$ be the proposition:

$n$ can be expressed as a product of prime numbers.

First note that if $n$ is prime, the result is immediate.

Basis for the Induction

$\map P 2$ is the case:

$n$ can be expressed as a product of prime numbers.

As $2$ itself is a prime number, and the result is immediate.

This is the basis for the induction.

Induction Hypothesis

Now it needs to be shown that, if $\map P j$ is true, for all $j$ such that $2 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

For all $j \in \N$ such that $2 \le j \le k$, $j$ can be expressed as a product of prime numbers.

from which it is to be shown that:

$k + 1$ can be expressed as a product of prime numbers.

Induction Step

This is the induction step:

If $k + 1$ is prime, then the result is immediate.

Otherwise, $k + 1$ is composite and can be expressed as:

$k + 1 = r s$

where $2 \le r < k + 1$ and $2 \le s < k + 1$

That is, $2 \le r \le k$ and $2 \le s \le k$.

Thus by the induction hypothesis, both $r$ and $s$ can be expressed as a product of primes.

So $k + 1 = r s$ can also be expressed as a product of primes.

So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.

Therefore, for all $n \in \N_{> 1}$:

$n$ can be expressed as a product of prime numbers.