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At that time we can determine if 1 is prime.
If it’s found that Eratosthenes’ sieve is the only prime generating function then we have our answer.
If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course
Isn't "every positive integer can be written as a product of primes greater than 1 in one and only one way" incorrect? A prime number is a only product of itself * 1, isn't it?
I think we’ll need to wait for an answer to if there is a prime number generating function. At that time we can determine if 1 is prime. If it’s found that Eratosthenes’ sieve is the only prime generating function then we have our answer.
If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course
Isn't "every positive integer can be written as a product of primes greater than 1 in one and only one way" incorrect? A prime number is a only product of itself * 1, isn't it?
If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course
i remember something from math class about "1" and "prime" being special cases of "units" and "irreducible" (?) that made me think these kinds of definitions are much more complicated than we want them to be regardless.
0^0 = 1? Yes, it’s simpler that way.
0! = 1? Yes, it’s simpler that way.
0/0 = ∞? No, it’s undefined.
0.9999… = 1? Yes, it’s just two ways of expressing the same number.
1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.
I think we’ll need to wait for an answer to if there is a prime number generating function. At that time we can determine if 1 is prime. If it’s found that Eratosthenes’ sieve is the only prime generating function then we have our answer.
We'd have our answer in what way?
Namely, if the sieve is the only generating function for all of the primes then 1 would need to be omitted as prime as removing its factors would remove every number, thus failing to generate the list of primes.
One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".
This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently explaining that we declare that the empty set is not a set, to avoid complicating theorems, proofs, and exposition by the endless repetition of "non-empty set."
(I agree that this is different, because there's no interesting "unique factorization theorem" for sets, but I can still imagine things developing this way. And, indeed, there are complications caused by allowing the empty set in a model of a structure, and someone determined to do so can make themselves pointlessly unpopular by asking "but have you considered the empty manifold?" and similar questions. See also https://mathoverflow.net/questions/45951/interesting-example....)