Number Theory 17

the outcome of the prime factorization might depend on how we had proceeded to compute

it? Perhaps. If we had started with a slightly different product at the first step, might we

have found ourselves ultimately with a different prime factorization in the end?

The number 11543, for example, can be factored as 7 ·17 ·97, and these factors are each

prime. Do we know that there is not also some other way to factor it into primes? For

a comparatively small number like this, we might hope to try out all the other candidates

and be convinced this way. That method is actually impractical, even for numbers such as

11543, and for much larger numbers like 653465345453435463456534655534354652, it

is downright infeasible. Do we actually know that this number has a unique prime factor-

ization?

When factoring numbers, to be sure, we routinely refer to the prime factorization. But

perhaps we should be saying merely that we have a prime factorization? Can there be more

than one? On my view, this is a serious question, more difficult than it might initially ap-

pear. The fact of the matter is that the usual naive treatment of prime factorization simply

does not touch on the question of uniqueness. We have become familiar with the unique-

ness of prime factorizations largely by observing many instances of prime factorization,

without ever encountering a situation where a number admits more than one factorization.

Does this experience constitute proof? No, of course not.

Meanwhile, prime factorizations are indeed unique, and this fact is the first deep theorem

of number theory, known as the fundamental theorem of arithmetic:

Theorem 9 (Fundamental theorem of arithmetic). Every positive integer can be ex-

pressed as a product of primes, and furthermore, this factorization is unique up to a rear-

rangement of these prime factors.

We shall prove this theorem presently, but before doing so, let me remark on a certain

issue arising in the statement of this theorem and a habit that mathematicians have. Namely,

the theorem says that every positive integer is a product of primes; but is this right? What

about the number 5? Yes, it is prime, but is it a product of primes?

Mathematicians will say yes, 5 is a product of primes, a product consisting of just one

factor, the number 5 itself. You might reply, that is not a product at all! Should not the the-

orem say, “Every positive integer is either prime or a product of primes”? Mathematicians

will insist, nevertheless, that it is a good idea to consider 5 and the other prime numbers as

products of primes, degenerate products if you will, products consisting of just one factor.

The reason is that our mathematical theories often become more robust when we incorpo-

rate the trivial or degenerate instances of a phenomenon into the fundamental definitions.

Every square counts also as a rectangle; every equilateral triangle is also isosceles. This

practice often leads to a smoother mathematical analysis in the end. A rectangle is pre-

cisely a quadrilateral with four right angles, for example, but this would not be true if we

did not count squares also as rectangles. In the same way, we allow ourselves to refer to

the product of just one number, without being multiplied by any other number.