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-   -   2nd Hardy-Littlewood conjecture (https://www.mersenneforum.org/showthread.php?t=26118)

 R2357 2020-10-22 14:07

2nd Hardy-Littlewood 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 Hardy-Littlewood conjecture, which was expained to me as being incompatible with the first Hardy-Littlewood 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 Hardy-Littlewood 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?

 CRGreathouse 2020-10-22 14:32

[QUOTE=R2357;560711]The second Hardy-Littlewood 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.[/QUOTE]

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.

 Dr Sardonicus 2020-10-22 15:34

[url=http://www.opertech.com/primes/k-tuples.html]K-TUPLE Permissible Patterns[/url][quote]This conjecture fails if the k-tuple conjecture is true with a value of y = 3159.
An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
while π(3159) = 446.

Exhaustive searching has verified the Hardy-Littlewood conjecture is true for intervals up to 2529.
Exhaustive searching has identified all maximum density k-tuple patterns in intervals of 3 to 2331.[/quote]

 bsquared 2020-10-22 15:54

[QUOTE=Dr Sardonicus;560722][url=http://www.opertech.com/primes/k-tuples.html]K-TUPLE Permissible Patterns[/url][/QUOTE]

Maybe I'm not understanding something... how can the search be exhaustive if x is unbounded?

 axn 2020-10-22 16:14

[QUOTE=bsquared;560724]Maybe I'm not understanding something... how can the search be exhaustive if x is unbounded?[/QUOTE]

They're searching for permissible patterns, not actual occurrences of any pattern.

 bsquared 2020-10-22 16:16

[QUOTE=axn;560728]They're searching for permissible patterns, not actual occurrences of any pattern.[/QUOTE]

Ah, ok thanks.

 R2357 2020-10-22 16:35

[QUOTE=CRGreathouse;560713]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.[/QUOTE]

But the region where they are searching is over 10^174, so, if we want to find within a sequence of 3159, 447 prime numbers, that means that the interval, this high, must contain even less composite numbers than in the first 3159, this seems really unlikely.

 CRGreathouse 2020-10-22 17:06

[QUOTE=R2357;560731]But the region where they are searching is over 10^174, so, if we want to find within a sequence of 3159, 447 prime numbers, that means that the interval, this high, must contain even less composite numbers than in the first 3159, this seems really unlikely.[/QUOTE]

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.

 mart_r 2020-10-22 19:22

[QUOTE=R2357;560711]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 Hardy-Littlewood 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.
[/QUOTE]

That's merely the fallacy by looking at small numbers only. You probably have seen the calculations that the first example is expected in the vicinity of 10[SUP]1198[/SUP]. With current methods, there's really no hope of ever actually finding a 447-tuplet (or any larger tuplet where there are more primes in an interval than at the beginning of the number line). But every empirical study and all data available so far is in favor of Hardy-Littlewoods first conjecture, so one can assume that these super-dense patterns do exist. And, for any fixed p, one can calculate patterns of these 447-tuplets without a prime factor < p. (In fact, I have done so a few years ago, with p > 1327.)

For comparison: Five years ago, the first large 21-tuplet was found. We don't yet know about a 24-tuplet in the range of p+[0...100], but I hope that I live long enough to witness the discovery of a 24-tuplet.

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[SUP]316[/SUP], and in fact there are infinitely many such instances where Li(x) < pi(x).

 Dr Sardonicus 2020-10-23 00:26

[QUOTE=mart_r;560749]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[SUP]316[/SUP], and in fact there are infinitely many such instances where Li(x) < pi(x).[/QUOTE]
Once upon a time, long long ago, I posted a link to a paper discussing this, [url=https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf]Prime Number Races[/url].

 R2357 2020-10-23 08:11

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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