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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: More complex numbers than reals?
Date: Thu, 11 Jul 2024 12:39:04 -0700
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On 7/11/2024 5:44 AM, Moebius wrote:
> Am 11.07.2024 um 07:33 schrieb Chris M. Thomasson:
> 
>> Is there a countable number of infinite[ly many] primes, just like 
>> there is a countable number of infinite[ly many] naturals? 

Thanks for the corrections in my wording. :^)

> Yes.
> 
> Hint: "In mathematics, a set is countable if either it is finite or it 
> can be made in one to one correspondence with the set of natural numbers."

Are the gaps in prime numbers "random" wrt their various length's?

(2, 3) has no gap wrt the naturals, however, (3, 5) does wrt (3, 4, 5). 
So a gap list where zero means no gap. { ... } denotes the gap:
_____________
(2, 3)  = 0 = (2, { }, 3) // no gap
(3, 5)  = 1 = (3, { 4 }, 5)
(5, 7)  = 1 = (5, { 6 }, 7)
(7, 11) = 3 = (7, { 8, 9, 10 }, 11)
....
_____________




> https://en.wikipedia.org/wiki/Countable_set

We can index the primes:

[0] = 2
[1] = 3
[2] = 5
[3] = 7
....