On the Infinity of Twin Prime and other K-tuples

[retina]

Originally Posted by jzakiya

If you bother to read the paper I explain why this is the case, give you the function to exactly determine the number of their residues frequency, and how to estimate them for any region.

Again, if you're not willing to study the paper you won't understand the reasons why the gap structure of primes exist in the totally deterministic structure I show you that they do.

Please do a little reading before commenting.

I was commenting on your assertion that testing up to <some_limit> is enough to confirm or refute. It isn't. It would merely provide evidence to support or not support. It can't confirm or refute anything.

[jzakiya]

If you read my statement before, I already said that what you just stated would be what some people would say about counting the (2,n) tuples up to some finite number.

If you aren't going to actually read, and understand, anything I write why comment on it?

[retina]

Originally Posted by jzakiya

If you aren't going to actually read, and understand, anything I write why comment on it?

I only commented on the part I quoted. And that part I quoted was either attempting to mislead, or shows a basic misunderstanding you have. Either way it was wrong to say it IMO.

[mart_r]

Originally Posted by jzakiya

Update: Wed 2020/7/8

I've now also determined the computational form for
gap coefficients a6 and a7, which are included in
the list below. I also computer generated the PGS
gaps for P37, which is also included (which took
19 hrs 42 mins, with a multi-threaded algorithm).
(...)

for Pn: a1 = (a1')(pn - 2)
for Pn: a2 = (a2')(pn - 2)
for Pn: a3 = (a3')(pn - 3) + (a2')(1) + (a1')(1)
for Pn: a4 = (a4')(pn - 4) + (a3')(1)
for Pn: a5 = (a5')(pn - 5) + (a4')(2) + (a3')(1)
for Pn: a6 = (a6')(pn - 5) + (a5')(6) - (a4')(2)
for Pn: a7 = (a7')(pn - 7) + (a6')(3) - (a5')(3) + (a4')(4)

P37
a1 = 217,929,355,875
(...)

I did just about the same calculations two years ago until I concocted a Pari program that lets you find the number of gaps up to gap size = 212 in "prime generators", as I understand it, the ring of integers coprime to p#.

If you want to do some number crunching, I've attached the program (Pari code together with three data files for the code). Note that the algorithm doesn't work when gap > 2*p.

(The data for "a_z.txt" was taken from S. Herzog, I think it was http://zigherzog.net/primes/mathconstants/trk.html, a link which is now dead. Similar data can be found under http://pauillac.inria.fr/~harley/brent.html)

Alas, though, it won't actually prove Polignac's conjecture. Sorry I can't give any details, as I'm not deep enough into the matter. But I'm sure if it was proven, you'd read it in the newspapers (yes, I'm a child of the 20th century...).

[CRGreathouse]

Originally Posted by jzakiya

If you bother to read the paper I explain why this is the case, give you the function to exactly determine the number of their residues frequency, and how to estimate them for any region.

Again, if you're not willing to study the paper you won't understand the reasons why the gap structure of primes exist in the totally deterministic structure I show you that they do.

Please do a little reading before commenting.

There isn't any actual mathematical content in the paper. I would enjoy a discussion of the paper but there simply isn't anything there to discuss. You don't seem to have understood the basic premise of what a proof entails if you think that you've presented one here.

I don't say this to be unkind, but merely to explain why no one will be able to give you what you would like. Normally I'd glance at a proof to see if I could find a tell-tale mistake, but here there isn't even an attempt to prove the proposition at hand.

[bsquared]

Originally Posted by jzakiya

This is an application of Prime Generator Theory (PGT) I began developing in 2008, and applied to creating the fastest/most efficient prime sieves, as independently verified in the link below, and a Rubygem.

https://github.com/hathix/prime-algorithms

https://github.com/jzakiya/primes-utils

Your SSoZ is not new; sieving over residue classes has been done for a long time. An early example is the (IMO lovely named) black-key sieve in 1993. (It is considered bad form, by the way, to name a technique/algorithm/proof after yourself.) Yafu also sieves by residue class; admittedly by arranging the bits slightly differently than you do (if I understand your algorithm correctly), but performing > 6x faster in most cases. GPU k-generators for GIMPS trial factoring also sieve this way.

[edit]
I am intrigued by the thought of packing the residue-class bits contiguously, and then sieving at fixed bit-positions for each class. Yafu makes independent arrays for each class and sieves each in turn. There are pros and cons to each approach.

Independent array pros:
* trivial to parallelize across classes
* fewer segments to represent a given number-line range, per class

Cons:
* variable-bit memory access per sieve step (so that one must compute a bit-offset in the current byte for each step). I'm actually not sure this is a bottleneck though, because the computation is so much faster than the memory access.

Bit-packed pros:
* fixed-bit sieving (perhaps an improvement, see cons above)
* a given segment remains cached throughout class sieving (this is probably a bigger win)
* primes can be generated in-order

Cons:
* more segments to represent a given number-line range
* parallelization would probably be best done across sub-segments of the range

I might have to play around with some code to see how things compare

[jzakiya]

Originally Posted by bsquared

Your SSoZ is not new; sieving over residue classes has been done for a long time. An early example is the (IMO lovely named) black-key sieve in 1993. (It is considered bad form, by the way, to name a technique/algorithm/proof after yourself.) Yafu also sieves by residue class; admittedly by arranging the bits slightly differently than you do (if I understand your algorithm correctly), but performing > 6x faster in most cases. GPU k-generators for GIMPS trial factoring also sieve this way.

Had never heard or seen anything about the black-key sieve. It obviously hasn't been adopted much. But again, I provide a comprehensive mathematical framework which explains how to use any even number as the modulus for a prime generator, i.e. a generalized method to explain how it works, and why, with coded examples.

Also its interesting that apparently the Sieve of Eratosthenes, Sieve of Atkin, and Sieve of Sundaram are totally good form, but the Sieve of Zakiya somehow is not. Why? Who should my sieves and techniques be named after? So only certain types of people have work named after them?

https://en.wikipedia.org/wiki/Generation_of_primes

Thankfully, thousands of people have downloaded my Rubygems, read my papers, and are using my work.

[Uncwilly]

Originally Posted by jzakiya

Also its interesting that apparently the Sieve of Eratosthenes, Sieve of Atkin, and Sieve of Sundaram are totally good form, but the Sieve of Zakiya somehow is not. Why? Who should my sieves and techniques be named after? So only certain types of people have work named after them?

To have others name something after you is an honour. To name something for yourself is pride.
Eratosthenes came up with the idea. It was a good one, people called it after him.

[kruoli]

Originally Posted by jzakiya

Thankfully, thousands of people have downloaded my Rubygems, read my papers, and are using my work.

It is nice that you can be comfortable with your work. Nonetheless, we try to base everything here on facts, not on your reach or your "customer ratings".

You could even have had a great breakthrough once and your next proposal could as well be rubbish. But vice versa is of course also possible. In all cases, we need facts, at best a mathematical proof. CRGreathouse is really trying to push you forward in a friendly manner!

[CRGreathouse]

Originally Posted by Uncwilly

To have others name something after you is an honour. To name something for yourself is pride.
Eratosthenes came up with the idea. It was a good one, people called it after him.

I have a project tracking self-naming and non-self-naming in the OEIS at
https://oeis.org/wiki/User:Charles_R...ouse_IV/Vanity

It needs vastly more work than I am able to provide, but it does give some kind of an idea of the norms and expectations in the OEIS, which are somewhat more relaxed than those of academic mathematics, which in turn is less cut-throat than many (most?) other academic disciplines.

Incidentally I would be happy to receive feedback (by private message or email) on this or any of my other projects.

[jzakiya]

Originally Posted by kruoli

It is nice that you can be comfortable with your work. Nonetheless, we try to base everything here on facts, not on your reach or your "customer ratings".

Well, since you aren't going to address any of the "facts" presented in the paper there's little else left for me to say to you. For others who are interested/curious I can answer their questions to help them understand.