5 ms·

One is not a prime number. If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. This m

by drdec 5y ago

One is not a prime number. If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. This makes the concept of prime numbers much more useful when one is excluded.

Igelau 5y ago
> If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. I don't dispute that, by definition, 1 is not prime, but I don't see how this statement would follow if we considered it prime. Edit: it seems more like it would be that every factorization would implicitly have 1^n tacked onto it, and while that isn't exactly useful, it doesn't break the game.
- drdec 5y ago
  That breaks the uniqueness of the prime factorization. Mathematicians love uniqueness, almost as much as existence.
  - Igelau 5y ago
    Could you please demonstrate an example of such a break? I think if we really look at it, we might see that it's really just convention and semantics. I don't want to seem like I'm cherry picking by providing my own example.
    - drdec 5y ago
      The unique prime factorization of 15 is 3⋅5. Let's call a number an Igelau prime if it is the number 1 or a prime number. Then an Igelau prime factorization of 15 is 3⋅5. Another Igelau prime factorization of 15 is 1⋅3⋅5. Another Igelau prime factorization of 15 is 1²⋅3⋅5. And so on. There are infinitely many Igelau prime factorizations of 15, thus there is no uniqueness. Edit: Clarifying what Igelau primes are.
thealig 5y ago
and there will be no prime numbers above 1 if 1 is considered a prime (based on existing definitions) Though if you think about it they can just modify the definition in that case
- drdec 5y ago
  The definition I usually use (speaking loosely for the moment) is that prime numbers are natural numbers greater than one whose divisors are only one and itself. Speaking more formally, a natural number p is a prime if and only if: a) p > 1 b) for any natural number n satisfying 1 < n < p, p mod n ≠ 0
JJMcJ 5y ago
Basically it's excluded because many theorems become cumbersome to state. Endlessly saying "primes other than 1" gets kind of tedious.

by drdec 5y ago

Igelau 5y ago

> If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. I don't dispute that, by definition, 1 is not prime, but I don't see how this statement would follow if we considered it prime. Edit: it seems more like it would be that every factorization would implicitly have 1^n tacked onto it, and while that isn't exactly useful, it doesn't break the game.

drdec 5y ago
That breaks the uniqueness of the prime factorization. Mathematicians love uniqueness, almost as much as existence.
- Igelau 5y ago
  Could you please demonstrate an example of such a break? I think if we really look at it, we might see that it's really just convention and semantics. I don't want to seem like I'm cherry picking by providing my own example.
  - drdec 5y ago
    The unique prime factorization of 15 is 3⋅5. Let's call a number an Igelau prime if it is the number 1 or a prime number. Then an Igelau prime factorization of 15 is 3⋅5. Another Igelau prime factorization of 15 is 1⋅3⋅5. Another Igelau prime factorization of 15 is 1²⋅3⋅5. And so on. There are infinitely many Igelau prime factorizations of 15, thus there is no uniqueness. Edit: Clarifying what Igelau primes are.

thealig 5y ago

and there will be no prime numbers above 1 if 1 is considered a prime (based on existing definitions) Though if you think about it they can just modify the definition in that case

drdec 5y ago
The definition I usually use (speaking loosely for the moment) is that prime numbers are natural numbers greater than one whose divisors are only one and itself. Speaking more formally, a natural number p is a prime if and only if: a) p > 1 b) for any natural number n satisfying 1 < n < p, p mod n ≠ 0

JJMcJ 5y ago

Basically it's excluded because many theorems become cumbersome to state. Endlessly saying "primes other than 1" gets kind of tedious.