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.

- Why does the theorem apply to integers greater
than
`1`? - What is the importance of the word "finitely"?
- What is the importance of the word "uniquely"?

`1 = 1.1 = 1.1.1 = 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.

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.

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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Kevin R. Coombes

Department of Biostatistics

University of Texas M.D.Anderson Cancer Center

1515 Holcombe Blvd., Box 447

Houston, TX 77030