Section 1.5
Factorization
We will show how integers are built out of primes.
Theorem
a = p1
···
ps,
Every positive integer a > 1 can be written as the product of finitely many primes:
where s is a positive integer and each pi is a prime.
Up to the order of the factors, this
factorization is unique.
For a positive integer a, we denote the number of times that
the prime p occurs in its factorization by
ordp(a). The factorization in primes of
a can be written as
a = 
p
prime pordp(a).
Note that only a finite number of factors is distinct from 1.
By definition a product that has the empty set as index set (the empty product) is 1. So, using this convention, we could have included this case a = 1 in the theorem.