You’ve certainly convinced me that ‘1’ should not be included in the set of things that are used to uniquely factor numbers. However, how I can I know if this set is the set of “primes”?
I guess I was thinking that the essence of primes was about their irreducibility/atomic-ness. The number 5 would be considered prime because you can’t describe it multiplicatively in any way except by using the number 5. Using my preferred notion, the number 0 and the number −1 would also be “prime” (as Mr Hen guessed). Is there a different word for this concept?
See wikipedia on natural generalizations of prime numbers. In particular note that most of the definitions say “units” instead of “1″, like “Irreducible elements are ones which cannot be written as a product of two ring elements that are not units.” which rules out 0 for the integers, +, x and includes the possibility of multiple units (-1 and 1).
I don’t know offhand of any nice, commonly referenced property P(S,O) that is: A,x,y in a structure S with operation O: A is P just when if x O y = A then either x = A or y = A. Which I believe is the general property you’re thinking about?
A second comment...
You’ve certainly convinced me that ‘1’ should not be included in the set of things that are used to uniquely factor numbers. However, how I can I know if this set is the set of “primes”?
I guess I was thinking that the essence of primes was about their irreducibility/atomic-ness. The number 5 would be considered prime because you can’t describe it multiplicatively in any way except by using the number 5. Using my preferred notion, the number 0 and the number −1 would also be “prime” (as Mr Hen guessed). Is there a different word for this concept?
See wikipedia on natural generalizations of prime numbers. In particular note that most of the definitions say “units” instead of “1″, like “Irreducible elements are ones which cannot be written as a product of two ring elements that are not units.” which rules out 0 for the integers, +, x and includes the possibility of multiple units (-1 and 1).
I don’t know offhand of any nice, commonly referenced property P(S,O) that is: A,x,y in a structure S with operation O: A is P just when if x O y = A then either x = A or y = A. Which I believe is the general property you’re thinking about?
Edit: with O commutative I do believe
Thank you. And yes, that is the property.