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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Disclaimer: This page was last updated on 12 October 2002. It is entirely possible that the information contained herein no longer has any connection with reality (assuming it ever did). Feel free to send constructive comments or inane criticisms to:
Kevin R. Coombes
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University of Texas M.D.Anderson Cancer Center
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