2nd Hardy-Littlewood conjecture - Page 2

mart_r · 2020-10-23, 11:16

Quote:

Originally Posted by Dr Sardonicus

Once upon a time, long long ago, I posted a link to a paper discussing this, Prime Number Races.

It's like you can read my mind, I have a printout of that paper (actually one of the first math papers I have made a printout of) and wanted to look into it today to recap on logarithmic measures.

Also, https://www.semanticscholar.org/paper/The-hunt-for-Skewes'-number-Smith/a05c0eadc3eda9301f1b48e1622d3d62d581a04f

Dr Sardonicus · 2020-10-23, 12:37

Quote:

Originally Posted by R2357

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.

By 53#? I don't know where in the world you got that idea.

R2357 · 2020-10-23, 14:11

Quote:

Originally Posted by Dr Sardonicus

By 53#? I don't know where in the world you got that idea.

Simply because in any sequence 53#*n to 53#*n+1, all the numbers that are congruent to x*{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# will be divisible by a prime up to 53.

59²=3481, 3481>3159 and 59 is the next prime after 53.

That's why I think the conjecture likely!

mart_r · 2020-10-23, 16:12

Just because I can...

Take, for instance,
n = 1566280308578217520031412816790827048467516641360946961779273951
and set p = 163#, or in full,
p = 5766152219975951659023630035336134306565384015606066319856068810

then the numbers n+x are all coprime to p for the following 447 values of x:

Code:

And, viola, you have 447 potential prime number candidates in an interval of 3159.
The only thing you have to do now is to add p to n as many times until you find an example where all 447 numbers are prime. Simple as that

R2357 · 2020-10-23, 18:12

Co prime doesn't mean prime, and I still don't understand why the odds are to think that the conjecture is wrong, okay, it's incompatible with the first, but there is no particular reason to think it's false, why would we rather think that.

mart_r · 2020-10-23, 18:38

Then, I'm afraid, the odds that anyone can help you understand are about as good as the odds of finding an actual prime-447-tuplet.

Edit, FWIW:
If the first Hardy-Littlewood conjecture was wrong, this would have far-reaching consequences for prime number theory, probably of a magnitude as disproving the Riemann hypothesis.
The second conjecture is based merely on human intuition; conjectures of this kind have been proven wrong a myriad of times in the past.

Dr Sardonicus · 2020-10-23, 18:46

Quote:

Originally Posted by mart_r

Just because I can...

Take, for instance,
n = 1566280308578217520031412816790827048467516641360946961779273951
and set p = 163#, or in full,
p = 5766152219975951659023630035336134306565384015606066319856068810

then the numbers n+x are all coprime to p for the following 447 values of x:

Code:

0
2
6
8
12
<snip>
3158

And, viola, you have 447 potential prime number candidates in an interval of 3159.
The only thing you have to do now is to add p to n as many times until you find an example where all 447 numbers are prime. Simple as that

There is an "admissibility condition" to the effect that there can be no inevitable prime factor.

To check this, I told Pari-GP to compute the degree-447 polynomial

f = x*(x+2)*(x+6)*(x+8)*(x+12)*...*(x+3158)

and then to compute Mod(Mod(1,p)*f, x^p - x) for all primes p up to 447, to see if any were zero. None were. (Of course, for p > 447, the degree of f is less that p, so the polmod will be Mod(1,p)*f, which isn't 0.)

Looks like you're good to go.

Of course, as indicated by 447 tuples calculations, finding an example where the numbers are all prime might take a while.

R2357 · 2020-10-23, 19:20

Quote:

Originally Posted by R2357

Simply because in any sequence 53#*n to 53#*n+1, all the numbers that are congruent to x*{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# will be divisible by a prime up to 53.

59²=3481, 3481>3159 and 59 is the next prime after 53.

That's why I think the conjecture likely!

Okay, in my calculation, I overlooked the fact that numbers congruent to a composite up to 53# divisible by primes from 59 onwards, modulo 53# may be prime, mea culpa :|

But still, there remains two conditions :
- first the positions of the 3159 numbers must contain 447 numbers that are congruent to numbers not divisible by {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53#;
- secondly, the sequence chosen must not contain more numbers divisible by primes from 59 onwards than the difference (positive of course) between the numbers congruent to numbers not divisible by {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53} modulo 53# and 447.

Good luck successfully reaching the two above conditions :)

sweety439 · 2020-10-24, 14:27

Quote:

Originally Posted by R2357

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?

If Schinzel's hypothesis H is true, then the second Hardy-Littlewood conjecture is false, since Schinzel's hypothesis H covers both Bunyakovsky conjecture and Dickson's conjecture, Dickson's conjecture covers first Hardy–Littlewood conjecture, but the first & second Hardy–Littlewood conjecture cannot be both true since ....

primepi(3159) = 446, but there may be 3159 consecutive positive integers which contain 447 primes (unlike the case of 6 consecutive positive integers, any 6 consecutive positive integers >= {4,5,6,7,8,9} contain at most 2 primes, where prmiepi(6) = 3), this is a case of law of small numbers.

However, Schinzel's hypothesis H is widely believed to be true, like generalized Riemann hypothesis and abc conjecture.

R2357 · 2020-10-26, 17:07

Quote:

Originally Posted by mart_r

The second conjecture is based merely on human intuition;

I don't understand what you really mean by "human intuition", when we look at the primes from 1 to 3159, this makes more than 14% of the natural numbers, the non-multiples of one and two digit numbers make up barely 12% of them.

furthermore, it's not at all similar to the twin-prime conjecture, because here, we're looking for a bigger prime density than in the beginning.

And it's not just an intuition : each p# takes away the number of possible primes in pn-1# for every p#, not to mention all of the non-primes divisible by numbers bigger than p : in 17#, in the first sequence of 17# is already made out of more composites only by numbers from19, than primes, I let you imagine in for example, let's say 10³³*17# to 10³³+1*17# :)

CRGreathouse · 2020-10-26, 19:30

Quote:

Originally Posted by R2357

I don't understand what you really mean by "human intuition", when we look at the primes from 1 to 3159, this makes more than 14% of the natural numbers, the non-multiples of one and two digit numbers make up barely 12% of them.

There are 446 primes up to 3159, so 14.1%. There are 422 numbers coprime to 1, 2, ..., 99, or 13.4%. What does this show? 3159 isn't a counterexample to the second Hardy-Littlewood conjecture. But there are infinitely many other cases to check, and we should be cautious in generalizing from 'one case holds' to 'all cases hold': as is often the case in number theory, the first counterexample may be large.

2020-10-23, 18:38	#17
mart_r Dec 2008 you know...around... 29·31 Posts	Then, I'm afraid, the odds that anyone can help you understand are about as good as the odds of finding an actual prime-447-tuplet. Edit, FWIW: If the first Hardy-Littlewood conjecture was wrong, this would have far-reaching consequences for prime number theory, probably of a magnitude as disproving the Riemann hypothesis. The second conjecture is based merely on human intuition; conjectures of this kind have been proven wrong a myriad of times in the past. Last fiddled with by mart_r on 2020-10-23 at 19:13

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2020-10-23, 18:12	#16
R2357 "Ruben" Oct 2020 Nederland 46₈ Posts	Co prime doesn't mean prime, and I still don't understand why the odds are to think that the conjecture is wrong, okay, it's incompatible with the first, but there is no particular reason to think it's false, why would we rather think that.