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? 
[QUOTE=R2357;560711]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.[/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. 
[url=http://www.opertech.com/primes/ktuples.html]KTUPLE Permissible Patterns[/url][quote]This conjecture fails if the ktuple conjecture is true with a value of y = 3159.
An admissible ktuple of 447 primes can be created in an interval of 3159 integers, while π(3159) = 446. Exhaustive searching has verified the HardyLittlewood conjecture is true for intervals up to 2529. Exhaustive searching has identified all maximum density ktuple patterns in intervals of 3 to 2331.[/quote] 
[QUOTE=Dr Sardonicus;560722][url=http://www.opertech.com/primes/ktuples.html]KTUPLE Permissible Patterns[/url][/QUOTE]
Maybe I'm not understanding something... how can the search be exhaustive if x is unbounded? 
[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. 
[QUOTE=axn;560728]They're searching for permissible patterns, not actual occurrences of any pattern.[/QUOTE]
Ah, ok thanks. 
[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. 
[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. 
[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 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. [/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 447tuplet (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 HardyLittlewoods first conjecture, so one can assume that these superdense patterns do exist. And, for any fixed p, one can calculate patterns of these 447tuplets 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 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[SUP]316[/SUP], and in fact there are infinitely many such instances where Li(x) < pi(x). 
[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]. 
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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