# Is there a term analogous to factoring, for addition?

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)

• Maybe --- partition? – Matti P. Jun 8 '18 at 6:30
• The ways to write a number as sum of numbers (with additional properties perhaps) is investigated under the term partition. – Hagen von Eitzen Jun 8 '18 at 6:30
• As others have mentioned, the word you're looking for is partition. This is fundamental to additive number theory. – TheSimpliFire Jun 8 '18 at 6:44
• Arithmetic progressions – usiro Jun 8 '18 at 23:28
• 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. – Robert Soupe Jun 9 '18 at 12:11

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 $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$.