this post was submitted on 21 May 2024
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Doesn't that miss out n=1?
1 being prime breaks a lot of the useful properties of primes, such as the uniqueness of prime factorization.
Isn't that function listing all the numbers? Not only the primes?
Oh, no that's just the primes. I was responding to a person joking about how we don't even know all the primes, so I used a technical yet unhelpful definition of "the set of all primes" to be technically correct,xas is the mathematics way. :)
I don't know if prime factorization is the correct English word for it but the operation I am referring to takes a (non zero) natural number and returns a multiset of primes that give you the original number when multiplied together. Example:
pf(12)={2,2,3}if we allowed 1 to be a prime then prime factorization cease to be a function aspf(12)={1,2,2,3}andpf(12)={1,1,1,1,2,2,3}become valid solutions.You are correct. The person you're replying to misread my set as a fancy way of saying "all natural numbers", not "all primes".
So you're both right, in that if 1 were a prime, the primes would not work right, and if 1 were not a natural number then those would not work right.
Using the totient function to define the set of primes is admittedly basically just using it for the fancy symbol I'll admit, and the better name for where we keep all the primes is the blackboard bold P. 😊