5 ms·

> If I say all odd numbers are prime, 1, 3, 5, and 7 may count in my favor, but at 9 the game is over. This is besides the point but those are wrong straight f

> If I say all odd numbers are prime, 1, 3, 5, and 7 may count in my favor, but at 9 the game is over. This is besides the point but those are wrong straight from the start, whichever way you cut it 1 is definitely not prime.

a1369209993 3y ago
Technically that depends on how "prime" is defined. The proper definition (roughly "has a prime factorization of cardinality exactly one") doesn't include one (and for good reason), but the version typically(?) taught in school ("has no factors other than one and itself") obviously does. A LLM could legitimately assume you mean "prime"-as-defined-by-mathematically-illiterate-math-teachers, rather than actually prime, and it would be in good company, since many bright school children have assumed the same.
- contravariant 3y ago
  Of course it depends on the definition but none of the ones still in use consider it a prime. It's kind of hard to nail down precisely why, the best I've been able to come up with is that Z/(p) is an integral domain for all primes p (and this is roughly equivalent to the definition of a prime) but not for p=1. Furthermore if you then plough on and redefine the Z/(1) i.e. zero ring to be an integral domain it becomes a pain to construct its field of fractions. Arguably 0 is a better prime, as it is a prime element.
  - a1369209993 3y ago
    > none of the ones still in use consider it a prime. "has no factors other than one and itself", AFAIK, is still used by school teachers (and others) introducing the concept of prime numbers. I agree that that definition is wrong, but that wasn't the point I was making.
roenxi 3y ago
People cut it so that 1 is prime: https://en.wikipedia.org/wiki/Prime_number#Primality_of_one https://en.wikipedia.org/wiki/Prime_number#Primality_of_one However, there is also controversy over whether 1 is a number. I tend to agree with you overall.
- Tainnor 3y ago
  > However, there is also controversy over whether 1 is a number. What?
  - roenxi 3y ago
    If you want more details about what someone said and they posted a link, my suggestion is to click on the link.
    - Tainnor 3y ago
      "The ancient Greeks thought" does not reflect what has happened in the two and a half millenia since. There's no debate over whether or not 1 is a number in modern mathematics.
kens 3y ago
There's a good reason why 1 is not considered prime: it would break the property that numbers can be uniquely factorized into primes (i.e. the fundamental theorem of arithmetic). 1 is not composite either; it is called a "unit". Interestingly, 1 has been considered prime in the past (and 2 has also been excluded): https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.p...

a1369209993 3y ago

Technically that depends on how "prime" is defined. The proper definition (roughly "has a prime factorization of cardinality exactly one") doesn't include one (and for good reason), but the version typically(?) taught in school ("has no factors other than one and itself") obviously does. A LLM could legitimately assume you mean "prime"-as-defined-by-mathematically-illiterate-math-teachers, rather than actually prime, and it would be in good company, since many bright school children have assumed the same.

contravariant 3y ago
Of course it depends on the definition but none of the ones still in use consider it a prime. It's kind of hard to nail down precisely why, the best I've been able to come up with is that Z/(p) is an integral domain for all primes p (and this is roughly equivalent to the definition of a prime) but not for p=1. Furthermore if you then plough on and redefine the Z/(1) i.e. zero ring to be an integral domain it becomes a pain to construct its field of fractions. Arguably 0 is a better prime, as it is a prime element.
- a1369209993 3y ago
  > none of the ones still in use consider it a prime. "has no factors other than one and itself", AFAIK, is still used by school teachers (and others) introducing the concept of prime numbers. I agree that that definition is wrong, but that wasn't the point I was making.

roenxi 3y ago

People cut it so that 1 is prime: https://en.wikipedia.org/wiki/Prime_number#Primality_of_one https://en.wikipedia.org/wiki/Prime_number#Primality_of_one However, there is also controversy over whether 1 is a number. I tend to agree with you overall.

Tainnor 3y ago
> However, there is also controversy over whether 1 is a number. What?
- roenxi 3y ago
  If you want more details about what someone said and they posted a link, my suggestion is to click on the link.
  - Tainnor 3y ago
    "The ancient Greeks thought" does not reflect what has happened in the two and a half millenia since. There's no debate over whether or not 1 is a number in modern mathematics.

kens 3y ago

There's a good reason why 1 is not considered prime: it would break the property that numbers can be uniquely factorized into primes (i.e. the fundamental theorem of arithmetic). 1 is not composite either; it is called a "unit". Interestingly, 1 has been considered prime in the past (and 2 has also been excluded): https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.p...