We want to prove the following result.

**Proposition:** Every integer `N > 1` is a product
of finitely many prime numbers.

**Proof:** We proceed by mathematical
induction. To start the induction, we observe that every prime
number is such a finite product (consisting, in fact, of the product
that only contains itself). We can, therefore, assume inductively
that every integer less than `N` can be written as a product
of finitely many primes. If `N` itself is prime, then we are
done. If `N` is not prime, then by definition it can be
written as a product

`N = AB`

where `1 < A < N` and `1 < B < N`.
Applying the inductive hypothesis to both `A` and
`B`, we can write each of them as a product of finitely many
primes. But then `N` has also been written as such a
product.

A stronger version of this result is known as the Fundamental Theorem of Arithmetic, which asserts that the prime factorization is unique.

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Kevin R. Coombes

Department of Biostatistics

University of Texas M.D.Anderson Cancer Center

1515 Holcombe Blvd., Box 447

Houston, TX 77030