What is the analogue of prime numbers in addition?
2026-03-28 23:58:36.1774742316
Analogue of prime numbers in addition?
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The most useful generalizations of "prime" occur in places where you have both addition and multiplication - in what we call "rings." It might seem like you only use one operation in the definition, but surprisingly that's not what happens in general. For example, in advanced rings, there is a notion of "prime" which is not even an element of the ring. For example, in ring theory, $2$ and $-2$ are essentially the same integer prime. It gets even more complicated than that, though.
That said, you can certainly define "prime" or "primitive" with any associative binary operation with an identity. Take the case of the natural numbers (including zero) under addition. Then the "primitive" elements are the non-units (they have no additive inverse) that cannot be expressed as the sum of two non-units. The only natural number satisfying this is $1$, as Daniel Fischer suggests.