## k-tuples

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• Have the k-tuples web page back up http://www.opertech.com/primes/k-tuples.html Lost server link -- main page is back up and will reload individual pages this
Message 1 of 19 , May 7 8:13 AM
Have the k-tuples web page back up
http://www.opertech.com/primes/k-tuples.html

Lost server link -- main page is back up
and will reload individual pages this weekend.

Tom
• Concerning the Hardy-Littlewood conjecture, pi(x+y)-pi(x)
Message 2 of 19 , Jan 9, 2005
Concerning the Hardy-Littlewood conjecture,
pi(x+y)-pi(x)<=pi(y).

latest find
pi(x+3243)-pi(x) = 457 = pi(3243) [0]

also, large grouping of -1's around 3429

past finds
pi(x+4323)-pi(x) = 590 = pi(4323) [0]
pi(x+4327)-pi(x) = 591 = pi(4327) [0]

and true crossover at
pi(x+4333)-pi(x) = 592 > pi(4333) [+1]

Tom
• New lower bound for crossover of pi(x+y)-pi(x)
Message 3 of 19 , Feb 9, 2005
New lower bound for crossover of
pi(x+y)-pi(x)<=pi(y) violation

If y=3243 the interval pi(x+3243)-pi(x)
can contain 458 primes while pi(3243)=457

Old record was y=4333 with 592 primes.

Also, many new near misses.

Will generate the modulii and update web pages tonight.

Web page will be at
http://www.opertech.com/primes/k-tuples.html

Tom
• Couldn t contain myself. Web pages have been updated. http://www.opertech.com/primes/k-tuples.html Tom
Message 4 of 19 , Feb 9, 2005
Couldn't contain myself.
Web pages have been updated.

http://www.opertech.com/primes/k-tuples.html

Tom
• In a message dated 10/01/2005 03:43:02 GMT Standard Time, tom@opertech.com writes: Concerning the Hardy-Littlewood conjecture, pi(x+y)-pi(x)
Message 5 of 19 , Feb 14, 2005
In a message dated 10/01/2005 03:43:02 GMT Standard Time, tom@...
writes:

Concerning the Hardy-Littlewood conjecture,
pi(x+y)-pi(x)<=pi(y).

latest find
pi(x+3243)-pi(x) = 457 = pi(3243) [0]

also, large grouping of -1's around 3429

past finds
pi(x+4323)-pi(x) = 590 = pi(4323) [0]
pi(x+4327)-pi(x) = 591 = pi(4327) [0]

and true crossover at
pi(x+4333)-pi(x) = 592 > pi(4333) [+1]

Tom,

I notice that your splendid page
_http://www.opertech.com/primes/k-tuples.html_
(http://www.opertech.com/primes/k-tuples.html)
says that you have obtained _exhaustive_ results on w(k) for k <= 305.
That is impressive!

My question is:-
what is the algorithm you use? (if it's not secret:-)

In particular, is it recursive?

I have spent a while trying to program such a provably exhaustive
enumeration of all admissible tuples, but without success: my 1 Gb of memory is (like
Fermat's margin) too narrow.

-Mike Oakes

[Non-text portions of this message have been removed]
• The landscape has changed. http://www.opertech.com/primes/kpiwchart.html While testing the 458-tuple for other variations, the program kicked out a 460-tuple
Message 6 of 19 , Feb 20, 2005
The landscape has changed.
http://www.opertech.com/primes/kpiwchart.html

While testing the 458-tuple for other variations,
the program kicked out a 460-tuple of the same size.
Spent some time rechecking program, and tested the
460-tuples and they are valid.

so pi(x+3243)-pi(x) could equal 460 while pi(3243)=457

This was not only a crossover but a major crossover of +3.

Looked back through listings and checking has a new upper
bound of 3159 with 447 primes.

Tom
• Anybody up to checking some calculations?? Sure would be appreciated. See http://www.opertech.com/primes/residues.html in particular the value of C2 (think it
Message 7 of 19 , Mar 2, 2005
Anybody up to checking some calculations??
Sure would be appreciated.

See http://www.opertech.com/primes/residues.html
in particular the value of C2 (think it is a little large)

Tom
• On the k-tuples, have just finished an update of the permissible patterns site www.opertech.com/primes/k-tuples.html At the site the trophies (contradiction
Message 8 of 19 , Jan 16, 2006
On the k-tuples, have just finished an update of the permissible
patterns site www.opertech.com/primes/k-tuples.html
At the site the trophies (contradiction patterns/super dense
constellations) are listed up to packing 100 additional primes in an
interval. In fact 17 additional prime can be packed in an interval of
length of just 8509 integers.
The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
(s) and the growth.
It is estimated that all intervals of more than 5980 integers can
demonstrate a super-dense condition.

Enjoy
Thomas J Engelsma
• Dear Tom, Thank you very much for your crackajack work. About your ¡°An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
Message 9 of 19 , Jan 16, 2006
Dear Tom, Thank you very much for your crackajack work.
About your “An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
while p(3159) = 446.”,
can I realize:
An admissible 447-tuple has been created in an interval of 3159 integers, while p(3159) = 446?
It is not:
An admissible prime 447-tuple has been created in an interval of 3159 integers,while p(3159) = 446.
Where we call a k-tuple is admissible, if it does not cover all congruence classes modulo any
prime p, we call the k-tuple a prime k-tuple when all of its components are primes by Daniel M. Gordon and Gene Rodemich.
So that: if the original k-tuple conjecture is true, then Hardy-Littlewood conjecture
p(x+y) - p(x) <= p (y)
is fails with a value of y = 3159.
In my paper I try prove that admissible prime k-tuples are infinite rather then admissible k-tuples will infinitely often be simultaneously primes. The original k-tuple conjecture may not true, example the prime of the form n^2-1.
Could you read my paper please and we will discuss some interesting problem.
Fengsui Liu.

Tom <tom@...> 写道： On the k-tuples, have just finished an update of the permissible
patterns site www.opertech.com/primes/k-tuples.html
At the site the trophies (contradiction patterns/super dense
constellations) are listed up to packing 100 additional primes in an
interval. In fact 17 additional prime can be packed in an interval of
length of just 8509 integers.
The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
(s) and the growth.
It is estimated that all intervals of more than 5980 integers can
demonstrate a super-dense condition.

Enjoy
Thomas J Engelsma

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• ... From: Tom ... Wonderful work, Tom. One I wish I d been part of! ... I absolutely will! I of course blindly accept the possible
Message 10 of 19 , Jan 17, 2006
From: "Tom" <tom@...>
>
> On the k-tuples, have just finished an update of the permissible
> patterns site www.opertech.com/primes/k-tuples.html
> At the site the trophies (contradiction patterns/super dense
> constellations) are listed up to packing 100 additional primes in an
> interval. In fact 17 additional prime can be packed in an interval of
> length of just 8509 integers.

Wonderful work, Tom. One I wish I'd been part of!

> The chart at www.opertech.com/primes/kpiwchart.html shows the crossover
> (s) and the growth.
> It is estimated that all intervals of more than 5980 integers can
> demonstrate a super-dense condition.
>
> Enjoy

I absolutely will!

I of course blindly accept the "possible therefore happens" approach in such
'linear' matters (i.e. this does not in any way apply to pseudoprime existance
questions). It's just a shame that there will probably never be any possibilty
of the human race actually finding such a tuple. I don't know if QC can reduce
the problem to a triviality, but I don't have much faith in QC either!

Are you still pushing the green zone to the right, or has your program hit an
architectural brick wall now?

Phil

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• Are you still pushing the green zone to the right, or has your program hit an architectural brick wall now? Phil, On the chart, the green zone is from
Message 11 of 19 , Jan 17, 2006
Are you still pushing the green zone to the right, or has your
program hit an architectural brick wall now?

Phil,
On the chart, the green zone is from exhautive searching, and cannot
(will not) be improved. The assembler program I wrote had a ceiling
of 2047. When I move to a 64-bit machine I intend to run the
exhaustive search some more. My initial calculations say I should be
able to run up to 2250 before I get exponentially stopped.
The red zone is in constant change above 3900, some lucky finds