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!