The Fundamental Theorem of Arithmetic

One of the surprising things about mathematics is its insistence that every assertion needs a justification. Non-mathematicians are often surprised by the extent to which mathematicians enforce this dictum. For example, consider the following result, which is usually called the Fundamental Theorem of Arithmetic.

Theorem: Every integer N > 1 can be written uniquely as a product of finitely many prime numbers.

Most people not only can't figure out where to start looking for a proof of this result, but also don't understand why it needs a proof. Of course it's true; it has been drilled into them since grade school. The whole issue of reducing a fraction to lowest terms doesn't make sense unless you can factor both numerator and denominator as a product of primes, and cancel the common factors.

To a mathematician, however, it is precisely because this result is so basic that it needs to be questioned. Is it so fundamental that it needs to be made an axiom? Or is it a consequence of other, simpler statements?

Another thing to keep in mind when confronted with the statement of a mathematical result is that mathematicians are laconic. They omit needless words. In particular, every word in the statement of a theorem is there for a reason. You won't really understand the statement until you ferret out the reason for the inclusion of each word.

In the case of the present theorem, there are three critical phrases, and you should try to explain why each one is there.

  1. Why does the theorem apply to integers greater than 1?
  2. What is the importance of the word "finitely"?
  3. What is the importance of the word "uniquely"?

One is Not a Prime

The statement of the Fundamental Theorem of Arithmetic only talks about integers greater than one. The main reason for this restriction is that the integer 1 is not a prime number. Oh, it meets the criteria presented in the usual careless definition (only divisible by 1 and itself), but careful mathematicians always explicitly exclude 1 from the list of primes. One reason for this exclusion is the uniqueness part of the Fundamental Theorem of Arithmetic. After all, if you allow 1 to become a member of the august body of primes, then it has the following prime factorizations

1 = 1.1 = 1.1.1 = = ....

Now, having excluded 1 from the list of primes, you certainly can't write 1 as a product of other primes (all of which are bigger than 1). Well, you could if you allowed a product of zero primes, but that seems too much like cheating.

Minus one has a similar problem, but you can handle other negative numbers by factoring out the -1 and then using this theorem. It would be possible to state the theorem to include this case, but it would distort the main meaning so much that it's not worthwhile.

Finite Products

The theorem talks about finite products because it can. Mathematicians are usually careful to distinguish things that can be accomplished in finitely many steps from things that truly require an infinite process. Arithmetic (and, more generally, algebra) are essentially finite mathematics; analysis is essentially the mathematics of the infinite.

Unique Factorization

The main point of the Fundamental Theorem of Mathematics is the uniqueness of factorizations. It is relatively easy to show that some kind of factorization into primes exists; it takes considerably more care to show that there is only one way to factor an integer.

This idea has played an important role in the history of mathematics. In attempting to prove Fermat's Last Theorem, mathematicians of the nineteenth century generalized the notion of numbers to collections of things called algebraic integers. They soon realized that if you could factor algebraic integers uniquely into a product of prime algebraic integers, then you could prove Fermat's Last Theorem. Unfortunately, that approach failed. Unique factorization is a special property of the usual integers, and it rarely holds for algebraic integers.

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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
Department of Biostatistics
University of Texas M.D.Anderson Cancer Center
1515 Holcombe Blvd., Box 447
Houston, TX 77030