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!

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    $\begingroup$ How about “the positive integers under multiplication are the free commutative monoid with the prime numbers the generators”? $\endgroup$ Apr 25, 2022 at 0:29
  • $\begingroup$ That's a nice definition! I guess I should have mentioned that I was trying to go for something that I high schooler could understand. One of the things that always had an issue with was understanding just how general mathematics is. For example, division used to confuse me because, if 5 divided by 1 is 5, and to divide is an action, how can 5 be divided by 1 if nothing happened? But now I've come to appreciate that all-encompassing sense that mathematics requires of us, regardless of how try to describe it. $\endgroup$ Apr 25, 2022 at 0:39
  • $\begingroup$ Yhese matters are discussed further in this answer $\endgroup$ Dec 21, 2022 at 21:21

1 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

  • $\begingroup$ Thank you for the clarification, especially with regard to a set vs multiset. I also like your definition since the only word that might need defining is "multiset." Of the definitions for the FTA, there seemed to be an effort to disregard 1, as they specify numbers greater than 1. I was worried that the theorem was changed or that it was being simplified without regard for the signifigance of the empty product. $\endgroup$ Apr 25, 2022 at 0:47
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    $\begingroup$ Mathematicians are happy with products of zero primes and products of one prime. Students in intro Number Theory courses, not so much. Statements of FTA may be designed with such students in mind. $\endgroup$ Apr 25, 2022 at 1:41

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