Rarity of twin primes
 

I would put my money on Mersenne-Primes not being as rare as the mainstreame-rs think, based on intuition only. As a side I know that the Twin-Primes are nowhere nearly as rare as they are generally believed to be.:smile:

[QUOTE=a1call;503076]...as rare as they are generally believed to be.[/QUOTE]
By whom? Twins are not rare at all.

[QUOTE=Batalov;503077]By whom? Twins are not rare at all.[/QUOTE]

I agree. But you should say that to All-Knowing, All-Wise and Infallible Wikipedia.
[QUOTE]
Twin primes become increasingly rare as one examines larger ranges,[/QUOTE]


[url]https://en.m.wikipedia.org/wiki/Twin_prime[/url]

They actually become more common the higher up you go.
But a formal proof might need to wait till I retire.:smile:

ETA To clarify obviously number of primes in a given range (of constant number of integers) does decrease the higher up you go but the percentage of the Twin-Primes increases among the primes the higher you go. So the subjective/vague statement in the Wikipedia article is fair to be considered false.

[QUOTE=a1call;503079]To clarify obviously number of primes in a given range (of constant number of integers) does decrease the higher up you go but the percentage of the Twin-Primes increases among the primes the higher you go.[/QUOTE]

Your statement is false. Twin primes will thin out as a fraction of all primes, the higher you go.

[QUOTE=axn;503083]Your statement is false. Twin primes will thin out as a fraction of all primes, the higher you go.[/QUOTE]
Do you have any direct references to that assertion?
Thank you in advance.
Please feel free to split this side track of necessary.

[QUOTE=a1call;503084]Do you have any direct references to that assertion?[/QUOTE]
How did you come up with your original assertion?!

Rather than saying something is wrong, try to read up and understand.
[QUOTE=a1call;503079][url]https://en.wikipedia.org/wiki/Twin_prime[/url][/QUOTE]

Yes professor. My bad.:rolleyes:

[QUOTE=a1call;503079]To clarify obviously number of primes in a given range (of constant number of integers) does decrease the higher up you go but the percentage of the Twin-Primes increases among the primes the higher you go. So the subjective/vague statement in the Wikipedia article is fair to be considered false.[/QUOTE]

Wikipedia is correct and you are incorrect, as proved by Brun about 100 years ago.

[QUOTE=CRGreathouse;503112]Wikipedia is correct and you are incorrect, as proved by Brun about 100 years ago.[/QUOTE]

[QUOTE]That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor. [/QUOTE]

[url]https://en.m.wikipedia.org/wiki/Brun%27s_theorem[/url]

I do not see Anything in lines with a decrease in prefer of twins there. But I did not read that article in details yet.
If I missed something would appreciate a correction or a more direct Reference.
Thank you for the reference?
ETA never mind I see that the upper bound of the twin primes increases more slowly than Primes.
Thank you very much for the correction.

I'd like to point out that this is not some isolated result used to prove a curiosity, but rather a theorem that founded a field of study (sieve theory) that is extremely vibrant today -- it was the driving force behind Zhang's theorem, as well as the Ford-Green-Konyagin-Maynard-Tao proof(s) of a longstanding Erdős conjecture on prime gaps.

On the other hand there is a case to be made for the fact that Bounded-Ranges are not sufficient to prove one way or the other if there is a percentage increase/decrease or else, unless the bounded ranges happen to stop overlapping at some point which I assume is not the case here (Corrections are appreciated).
To clarify suppose that we happen to determine that the upper bound for the number of integers that are divisible by 5 is 1/3 which is true since 1/5 < 1/3 and
that the upper bound for the number of integers that are divisible by 7 is 1/2 which is also true since 1/7 < 1/2.
This can not be taken as a proof that the ratio of the integers which are divisible by 7 will exceed more rapidly than those which are divisible by 5 the higher we go.:smile: