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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: Fri, 19 Jul 2024 22:21:52 -0700
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On 7/11/2024 1:26 PM, Moebius wrote:
> Am 11.07.2024 um 21:39 schrieb Chris M. Thomasson:
> 
>> Are the gaps [between] prime numbers "random" wrt their various length's?
> 
> In a certain sense (maybe) "yes", but there are "rules". :-P
> 
> For example, for each and ever natural number n > 1 there's a prime 
> between n and 2n. :-)
> 
> See: https://en.wikipedia.org/wiki/Bertrand%27s_postulate
> 
> Moreover: "The prime number theorem, proven in 1896, says that the 
> average length of the gap between a prime p and the next prime will 
> asymptotically approach ln(p), the natural logarithm of p, for 
> sufficiently large primes. The actual length of the gap might be much 
> more or less than this. However, one can deduce from the prime number 
> theorem an upper bound on the length of prime gaps[.]"
> 
> Source: https://en.wikipedia.org/wiki/Prime_gap
> 
>> (2, 3) has no gap wrt the naturals, however, (3, 5) does [...]. 
> 
> For all n e IN, n > 1, there is a prime between n and 2n.
> 
> Let's check it for some n:
> 
> n = 2: 2, *3*, 4
> n = 3: 3, 4, *5*, 6
> n = 4: 4, *5*, 6, *7*, 8
> n = 5: 5, 6, *7*, 8, 9, 10
> :
> 
> On the other hand, it's NOT known if for all natural number n there is a 
> prime between n^2 and (n+1)^2.
> 
> See: https://en.wikipedia.org/wiki/Legendre%27s_conjecture
> 
> Fascinating, isn't it?
> 
> _______________
> 
> Back to countably infinite sets. :-)
> 
>>> https://en.wikipedia.org/wiki/Countable_set
>>
>> We can index the primes:
>>
>> [0] = 2
>> [1] = 3
>> [2] = 5
>> [3] = 7
>> ...
> 
> Exactly.
> 
> This means (using math terminology) that there is a bijektion between IN 
> and P (the set of primes).
> 
> Hence the set of primes is countably infinite.
> 

thanks.