20201022, 14:07  #1 
"Ruben"
Oct 2020
Nederland
46_{8} Posts 
2nd HardyLittlewood conjecture
Hello,
I've looked at the way primes behave in sequences which have a primorial length and, as I mentionned a few weeks ago, I came across the 2nd HardyLittlewood conjecture, which was expained to me as being incompatible with the first HardyLittlewood conjecture. What I don't understand is why do we believe that it's the first one which is right, personnaly, I believe it's the second, here's why I think so : The second HardyLittlewood conjecture states that with x>1, pi(x)>=pi(x+y)pi(y). That seems to me as very probable! Already, the first sequence of 30 contains 10 primes, none of the others will contain more than 7. Why would this conjecture seem to be false? 
20201022, 14:32  #2 
Aug 2006
3×1,993 Posts 
That case is certain: you can't fit that many primes into a run of 30 except at the beginning. But there are higher cases where the first conjecture says there is a really tight configuration, so tight that you could fit in even more primes than at the beginning. These configurations are known  the only question is whether there are examples of them appearing 'in the wild', as it were. But I certainly expect that they do appear, in fact infinitely often. The trouble is that you'd expect them to be fairly sparse and thus hard to find, and so it's not surprising we haven't found any so far.

20201022, 15:34  #3  
Feb 2017
Nowhere
2×47×53 Posts 
KTUPLE Permissible Patterns
Quote:


20201022, 15:54  #4  
"Ben"
Feb 2007
2×1,789 Posts 
Quote:


20201022, 16:14  #5 
Jun 2003
19·271 Posts 

20201022, 16:16  #6 
"Ben"
Feb 2007
2×1,789 Posts 

20201022, 16:35  #7  
"Ruben"
Oct 2020
Nederland
26_{16} Posts 
Quote:


20201022, 17:06  #8 
Aug 2006
3×1,993 Posts 
It's like looking for twin primes: sure, if you look really high they're rare, but no one doubts that there are plenty of them, even though they're really close together.

20201022, 19:22  #9  
Dec 2008
you know...around...
677 Posts 
Quote:
For comparison: Five years ago, the first large 21tuplet was found. We don't yet know about a 24tuplet in the range of p+[0...100], but I hope that I live long enough to witness the discovery of a 24tuplet. But I mainly wanted to point out a similar case in prime number theory: if you look at the function Li(x)pi(x) (Li(x) being the logarithmic integral), even for the largest values of x for which the exact value of pi(x) is known, hardly anyone would expect that it ever produces negative numbers. Yet it is known that Li(x) < pi(x) for some values around x=1.4*10^{316}, and in fact there are infinitely many such instances where Li(x) < pi(x). Last fiddled with by mart_r on 20201022 at 19:29 Reason: "the exact value of" 

20201023, 00:26  #10  
Feb 2017
Nowhere
2×47×53 Posts 
Quote:


20201023, 08:11  #11 
"Ruben"
Oct 2020
Nederland
26_{16} Posts 
Sequence of 3159 numbers containing 447 primes
Anyway, if there indeed is such a sequence, then the first or the occurrence will have been reached by 32 589 158 477 190 044 730, thus way below the lower band of 10^174.

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