4 ms·

> primes are very common Isn't it thought that they get more and more rare the further you go away from zero in the positive direction? By the way I just real

> primes are very common Isn't it thought that they get more and more rare the further you go away from zero in the positive direction? By the way I just realized why no negative numbers are prime and that's because -1 is a factor in all of them. Quite obvious but I just never thought about it before.

tzahola 9y ago
That's not exactly true. E.g. -1 is also a "factor" of 3: -3 * -1 = 3 In number theory we define primes as a subset of the natural numbers, which excludes negative integers. But when you move on to abstract algebra, the definition of a prime element in a commutative ring is an element p, such that p|ab implies p|a or p|b. Using this definition, we can conclude that the prime elements in the ring of integers are both the positive and negative primes.
mkl 9y ago
-1 is not a prime, so that doesn't work. The actual reason is a little more boring - negative numbers are not prime by the definition of a prime number: "a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers" (Wikipedia).
- posterboy 9y ago
  There's another definition, that doesn't have to exclude the one explicitly. I think it's a natural number not divisible by any natural number smaller than itself.
  - tzahola 9y ago
    *other than one
    - posterboy 9y ago
      damn what was it though ... can't remember where I read it.
deleted 9y ago
[deleted]
tialaramex 9y ago
Yes, the primes become more sparse, but because there are countably infinitely many of them, and the natural numbers (of which they are a subset) are merely countably infinite, this counts as "common". Our intuitions about this stuff are not very helpful in mathematics. For example, _most_ numbers are normal in any base (if you wrote them out as a decimal fraction they'd have an evenly distributed amount of all the different digits, forever) and we can prove this is true of the set of real numbers - but even though we know it's true, we don't have any definite examples of such numbers, the closest are we think Pi might be normal but we can't prove it, and we know Chaitin's constants are normal for sure, but by definition there's no way to write one of those down...

tzahola 9y ago

That's not exactly true. E.g. -1 is also a "factor" of 3: -3 * -1 = 3 In number theory we define primes as a subset of the natural numbers, which excludes negative integers. But when you move on to abstract algebra, the definition of a prime element in a commutative ring is an element p, such that p|ab implies p|a or p|b. Using this definition, we can conclude that the prime elements in the ring of integers are both the positive and negative primes.

mkl 9y ago

-1 is not a prime, so that doesn't work. The actual reason is a little more boring - negative numbers are not prime by the definition of a prime number: "a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers" (Wikipedia).

posterboy 9y ago
There's another definition, that doesn't have to exclude the one explicitly. I think it's a natural number not divisible by any natural number smaller than itself.
- tzahola 9y ago
  *other than one
  - posterboy 9y ago
    damn what was it though ... can't remember where I read it.

deleted 9y ago

[deleted]

tialaramex 9y ago

Yes, the primes become more sparse, but because there are countably infinitely many of them, and the natural numbers (of which they are a subset) are merely countably infinite, this counts as "common". Our intuitions about this stuff are not very helpful in mathematics. For example, _most_ numbers are normal in any base (if you wrote them out as a decimal fraction they'd have an evenly distributed amount of all the different digits, forever) and we can prove this is true of the set of real numbers - but even though we know it's true, we don't have any definite examples of such numbers, the closest are we think Pi might be normal but we can't prove it, and we know Chaitin's constants are normal for sure, but by definition there's no way to write one of those down...