e.g. $12$ can be expressed as a product of its factors: $2 \times 6, 3 \times 4$. The composites can be decomposed further.

$12$ can also be expressed as a sum of its somethings: $1+11, 2+10, 3+9, ...$. The non-$1$'s can be decomposed further.

It seems to me that if primes are the atoms of multiplication, then $1$ is the atom of addition. (BTW and maybe the Fundamental Theorem of Arithmetic should be called the Fundamental Theorem of Multiplication)

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    $\begingroup$ Maybe --- partition? $\endgroup$ – Matti P. Jun 8 '18 at 6:30
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    $\begingroup$ The ways to write a number as sum of numbers (with additional properties perhaps) is investigated under the term partition. $\endgroup$ – Hagen von Eitzen Jun 8 '18 at 6:30
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    $\begingroup$ As others have mentioned, the word you're looking for is partition. This is fundamental to additive number theory. $\endgroup$ – TheSimpliFire Jun 8 '18 at 6:44
  • $\begingroup$ Arithmetic progressions $\endgroup$ – usiro Jun 8 '18 at 23:28
  • $\begingroup$ Just my two cents: I disagree with your BTW. a positive integer $n$ can be broken down into a unique partition of 1's, but that's nowhere near as interesting as the fact that one $n$ can be broken down to a whole bunch of prime factors but $n + 1$ might be the square of a prime. In my opinion. $\endgroup$ – Robert Soupe Jun 9 '18 at 12:11

You are absolutely right, including your BTW. Actually, the notion of prime and irreducible can be defined in any commutative monoid. The two cases you consider correspond to free commutative monoids.

Details. The monoid $(\mathbb{N}-\{0\}, \times)$ is a free commutative monoid over the basis consisting of the prime numbers. That is, every positive integer has a unique decomposition as a product of prime numbers. Similarly, $(\mathbb{N}, +)$ is a free monoid with basis $\{1\}$, the only prime element. Still with addition, $(\mathbb{N}^k, +)$ is a free commutative monoid with a basis consisting of $k$ primes. For instance, the basis of $(\mathbb{N}^3, +)$ is $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$.


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