Adequately defining the fundamental theorem of arithmetic.

Question

Adequately defining the fundamental theorem of arithmetic.

So after sifting through the internet, I realized that there are a few ways the fundamental theorem of arithmetic is defined. Paraphrasing, some say it states that every positive integer/non-zero natural number greater than one/ greater than or equal to 2 is the product of a unique factorization made up of primes in no particular order. Others, aware of the confusion of a single number being a product of itself, separate primes and composite numbers, the latter of which are, of course, products of primes.

I've been seeking a definition that not only acknowledges that single numbers can also be products of themselves, but also acknowledges 1 as the empty product. Here's my crack at it:

The fundamental theorem of arithmetic states that every natural number can be uniquely factored as a product of a quantity of primes in no particular order.

Here, I exclude 0 as natural number. I also say "a quantity of primes" so as to allude to the fact that no primes is a quantity of primes, thus acknowledging the empty product 1. I would love to get peoples' thoughts on this! Thank you in advance!

Answer 1

My thoughts.

First, you are overthinking this. The several formulations you propose all make the meaning clear.

If you want to be formal and picky, I'd suggest

Every positive integer is the product of a unique multiset of primes.

You need a multiset rather than a set to allow for powers of primes. That's the technical term for what you were getting at with "quantity". Multisets, like sets, specify no order for their elements.

It's standard that the product of the (nonexistent) elements of an empty set is $1$ and that the product of a set with one element is that element.

See Empty set and empty sum