4 ms·

According to German science magazine 2 is not prime

deleted 8y ago
[deleted]
heyjudy 8y ago
That's not a primary source and wrong. A prime is most simply: it has no factors other than itself and 1. "Not being even" is a corollary, and would be redundant if included in the definition of primality.
- adtac 8y ago
  >That's not a primary source Heh.
- schoen 8y ago
  Another way to see this is to observe that the Fundamental Theorem of Arithmetic would be wrong (or more difficult to state succinctly) if 2 were not considered prime or if 1 were considered prime. For example, if 2 were not prime, it would be impossible to represent the number 256 (or the number 123456) as a product of primes.
  - throwawaymath 8y ago
    This is an excellent rebuttal, nice.
    - lurchedsawyer 8y ago
      It's not, though. It's one group of mathematicians appropriating prime numbers and jack-booting them into their favourite theorem. There are valid reasons to treat 1 as a prime number, just as there are for defining 0^0 as 1 in some cases. Also for a fun pastime try asking an arithmetician to apply the fundamental theorem to 1 and watch them squirm.
      - schoen 8y ago
        Presumably the arithmetician will answer that its factorization is the empty set. That's what sympy says, for example. >>> sympy.factorint(6) {2: 1, 3: 1} >>> sympy.factorint(1) {} This definition is also sensible because it also preserves uniqueness in both directions. The empty set is the only prime factorization of 1, and 1 is the only natural number whose prime factorization is the empty set. (Wikipedia has a footnote that "Using the empty product rule one need not exclude the number 1 [from the fundamental theorem of arithmetic], and the theorem can be stated as: every positive integer has unique prime factorization.") The fundamental theorem doesn't have to state that the prime factorization is nonempty. It's true that some mathematicians have defined 1 as a prime number and there's nothing logically inconsistent about doing so, but it makes most theorems and formulas in number theory more complex and so this definition has fallen out of favor. Edit: I think the Wikipedia article on the empty product gives some quite nice examples of the benefits of a closely related concept. https://en.wikipedia.org/wiki/Empty_product https://en.wikipedia.org/wiki/Empty_product
        lurchedsawyer 8y ago
        See? Squirming. Somehow multiplying 0 factors is supposed to result in 1. I'm not saying it's incorrect, but you have to explain that edge case away. At the same time allowing 1 to be prime would remove that problem given that e.g. 6 = 2^1 * 3 ^1 and 6 = 1^81 * 2^1 * 3^1 are the same unique factorisation because, well, monoids. Edit: `s/fields/monoids`
        srtjstjsj 8y ago
        How many prime factors does the product of two numbers have? Not the sum of the number of prime factors of the individual numbers. Now you are the one squirming. Cleanly accounting for edge cases isn't squirming, it's logical thinking.
        impendia 8y ago
        > Presumably the arithmetician will answer that its factorization is the empty set. Arithmetician (Ph.D. in number theory) here. This answer is completely correct. As another reason why the factorization of 1 should be the empty set: suppose you have two positive integers m and n. Write S(m) and S(n) for their sets of prime divisors, counted with multiplicity. Then S(mn) is the union of S(m) and S(n). We need S(1) to be the empty set to make this rule consistent. Analogous to how log(1) is equal to 0.
  - srtjstjsj 8y ago
    Alternatively: "even" means "multiple of 2". Saying "no primes are even" is isomorphic to saying "no primes are multiples of 3 [or any other p you choose]". A claim "n can't be prime if none of the other primes are multiples of n", implies that cannot exist at all.

deleted 8y ago

[deleted]

That's not a primary source and wrong. A prime is most simply: it has no factors other than itself and 1. "Not being even" is a corollary, and would be redundant if included in the definition of primality.

adtac 8y ago
>That's not a primary source Heh.
schoen 8y ago
Another way to see this is to observe that the Fundamental Theorem of Arithmetic would be wrong (or more difficult to state succinctly) if 2 were not considered prime or if 1 were considered prime. For example, if 2 were not prime, it would be impossible to represent the number 256 (or the number 123456) as a product of primes.
- throwawaymath 8y ago
  This is an excellent rebuttal, nice.
  - lurchedsawyer 8y ago
    It's not, though. It's one group of mathematicians appropriating prime numbers and jack-booting them into their favourite theorem. There are valid reasons to treat 1 as a prime number, just as there are for defining 0^0 as 1 in some cases. Also for a fun pastime try asking an arithmetician to apply the fundamental theorem to 1 and watch them squirm.
    - schoen 8y ago
      Presumably the arithmetician will answer that its factorization is the empty set. That's what sympy says, for example. >>> sympy.factorint(6) {2: 1, 3: 1} >>> sympy.factorint(1) {} This definition is also sensible because it also preserves uniqueness in both directions. The empty set is the only prime factorization of 1, and 1 is the only natural number whose prime factorization is the empty set. (Wikipedia has a footnote that "Using the empty product rule one need not exclude the number 1 [from the fundamental theorem of arithmetic], and the theorem can be stated as: every positive integer has unique prime factorization.") The fundamental theorem doesn't have to state that the prime factorization is nonempty. It's true that some mathematicians have defined 1 as a prime number and there's nothing logically inconsistent about doing so, but it makes most theorems and formulas in number theory more complex and so this definition has fallen out of favor. Edit: I think the Wikipedia article on the empty product gives some quite nice examples of the benefits of a closely related concept. https://en.wikipedia.org/wiki/Empty_product https://en.wikipedia.org/wiki/Empty_product
      - lurchedsawyer 8y ago
        See? Squirming. Somehow multiplying 0 factors is supposed to result in 1. I'm not saying it's incorrect, but you have to explain that edge case away. At the same time allowing 1 to be prime would remove that problem given that e.g. 6 = 2^1 * 3 ^1 and 6 = 1^81 * 2^1 * 3^1 are the same unique factorisation because, well, monoids. Edit: `s/fields/monoids`
        srtjstjsj 8y ago
        How many prime factors does the product of two numbers have? Not the sum of the number of prime factors of the individual numbers. Now you are the one squirming. Cleanly accounting for edge cases isn't squirming, it's logical thinking.
      - impendia 8y ago
        > Presumably the arithmetician will answer that its factorization is the empty set. Arithmetician (Ph.D. in number theory) here. This answer is completely correct. As another reason why the factorization of 1 should be the empty set: suppose you have two positive integers m and n. Write S(m) and S(n) for their sets of prime divisors, counted with multiplicity. Then S(mn) is the union of S(m) and S(n). We need S(1) to be the empty set to make this rule consistent. Analogous to how log(1) is equal to 0.
- srtjstjsj 8y ago
  Alternatively: "even" means "multiple of 2". Saying "no primes are even" is isomorphic to saying "no primes are multiples of 3 [or any other p you choose]". A claim "n can't be prime if none of the other primes are multiples of n", implies that cannot exist at all.