Definition:Prime Decomposition

Theorem
Let $$n > 1 \in \mathbb{Z}$$.

Then $$n$$ has a unique factorization of the form:

$$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$$

where $$p_1 < p_2 < \ldots < p_r$$ are distinct primes and $$k_1, k_2, \ldots, k_r$$ are positive integers.

This unique expression is known as the prime decomposition of $$n$$.

Proof
This is just another way of stating the Fundamental Theorem of Arithmetic.