On the Infinity of Twin Prime and other K-tuples - Page 2

jzakiya · 2020-08-19, 19:37

I didn't realize you could do that.

Thanks for the suggestion/advice.

It's attached here.

Gelly · 2020-08-19, 22:02

I would like to bring my own personal attention, and perhaps yours, to section 7 - Prime Generator Properties. In particular, you mention for multiple paragraphs that the properties of prime numbers (and, resultingly, these generators) do not need proofs in order to realize their properties.

This seems to entirely sidestep the point of a proof in the name of using empirical evidence to support your claims. There exist many conjectures that are based off of "obvious" properties that are heuristic, but there's no known reason for them.

If you are at all interested in being taken seriously, I would consider removing the diatribe against proofs of very simple things - especially the line that says "You have to draw pictures... You cannot imagine these properties into existence just from numerical analysis, you have to observe them first."

If you cannot prove with numerical analyses even the simple properties of your generators, then this is not a proof. Perhaps you could gain a lot of insight on your generators - especially for the rest of us - to prove all of these things with no doubt.

jzakiya · 2020-08-19, 23:25

Thank you for your feedback.

What I was trying to (maybe ineleganatly) say in that section, is that most of the properties of prime generators are easily, and best, observed visually. In fact, this is how I came to recognize these properties.

I didn't mean to convey they couldn't be established by analysis, et al, and in fact stated these properties are the set in place by the requirements of the residues and other mathematical operations and their requirements.

I was trying to establish more of the philosophical basis for why prime generators are what they are. The statement about having to draw the pictures was to emphasize that for me, the visual expression of their properties led to their empirical understanding, which allowed me to apply them to my class of prime sieves, etc.

My background is in engineering, not as a mathematician, so I think and talk like one. That is why I appreciate the feedback because I know there is an epistemological difference how we think, and emphasize what's important.

Actually, the best way to present all of this (which I'm considering doing if I get the time a resources to do so) is create videos and animations to present it, because I can then draw the pictures and explain them at the same time.

If it is possible to get past the form/language used in the paper, I believe the merits of the math, and demonstration of the empirical results it achieves, makes my case that there are an infinite number of prime pairs for any gap size, as Polignac conjectured.

I'm really interested in feedback stating if people find anything they see is mathematically incorrect or not plausible.

CRGreathouse · 2020-08-20, 02:53

Quote:

Originally Posted by Gelly

If you cannot prove with numerical analyses even the simple properties of your generators, then this is not a proof.

Quote:

Originally Posted by jzakiya

My background is in engineering, not as a mathematician, so I think and talk like one. That is why I appreciate the feedback because I know there is an epistemological difference how we think, and emphasize what's important.

I have a great appreciation for engineers (having both a grandfather and a father-in-law of that profession) and engineering. In terms of how to communicate results in a mathematical paper, results without proof come in several forms:

Conjectures. These are statements which are believed to be true, but for which no proof can be found at present. (There are some subtleties here: by proposing a conjecture, you're implicitly suggesting that you think the problem is at least somewhat hard and worthy of study.)
Claims. These are statements which the author claims to be true, but for which the author does not provide a proof. (Actually, sometimes a proof does occur at a later point, but that's a different matter.) Sometimes this is used for informal statements that could not have a formal proof; otherwise there is some reason (e.g., article page limits, or because proving the claim is an exercise in the text) for which the proof is omitted. The understanding in such cases is that the author would be able to provide more details if contacted.
Propositions. Statements provided without comment on their importance or truth/falsity. (The term is also used, confusingly, in a sense like that of theorem or lemma. Sorry; I didn't make this stuff up.)
Axioms. Statements which some mathematician (physicist, etc.) may choose to assume to be true and use as the basis for a mathematical system (like ZFC, NF, etc.)

I would characterize your statements as claims, and as such, they can't be used in any of your proofs (as they aren't themselves proven).

jzakiya · 2020-08-20, 19:12

To me a "proof" is a way/process to "establish with certainty" if something can be shown to convincingly be absolutely true.

My epistemological basis for knowing is 'more' empirical than abstract.

The paper is more about just a proof of prime gaps, but about a general theory to characterize the understanding of primes, which I've used repeatedly prior to writing the paper, to create the most efficient/fastest prime sieves, primality tests, et al.

The discussion seems to always focus on the language used to describe the results than on the results. Nobody has ever tried to state where any of the mathematical framework is wrong or implausible, and the results are invalid.

The ultimate focus should be on are the results correct.
To me, after I did my twinprimes sieve, it was understood there are an infinity of them (there can't be no last one). And from the paper, I discovered/show Sexy Primes are the most abundant (prime gaps of 6).

So there is allot to learn in the paper, even if you don't think it constitutes a "proof" that satisfies your requirements.

Batalov · 2020-08-20, 19:36

Quote:

Originally Posted by jzakiya

The discussion seems to always focus on the language used to describe the results than on the results.

Don't get confused with people's polite way, when in this polite way they are telling you that you results are rubbish.

Quote:

Originally Posted by jzakiya

Nobody has ever tried to state where any of the mathematical framework is wrong or implausible, and the results are invalid.

Because there is nothing to discuss there. In the words of Pauli, "This is not even wrong."
Meaning if there was something solid and wrong - you can begin discussing it.
But there is nothing to discuss when "the paper" is about "It is easy to get convinced that most of the time <bla>." It is rhetorical exercise, not a mathemetical. Very good for a used car salesman, not good for math.

LaurV · 2020-08-25, 07:43

Quote:

Originally Posted by CRGreathouse

Conjectures.
Claims.
Propositions.
Axioms.

You missed:

Open problems. These are more like conjectures, but for the first, we have some idea in which direction they lean, while for the last, we have no freaking idea.
Hypotheses. These are like axioms, they can be used to build a whole theory on, except that they are not necessary true*. Example, our joker's "if any odd number is a semiprime, then..." in the parallel thread. Note that some conjectures become hypotheses after they are disproved. Famous and thematic example, the Chinese hypothesis.
Guesses. These are like claims, but weaker, we have an idea where they lean, but not so much confidence to make a claim.

-------
* now, retina

will argue that the axiom is not necessary true either, but only something we assume it is true, to build our theory on it, for example the parallel's axiom, which, if we consider untrue, a whole new branch of geometry is born.

CRGreathouse · 2020-08-25, 14:26

Quote:

Originally Posted by Batalov

But there is nothing to discuss when "the paper" is about "It is easy to get convinced that most of the time <bla>." It is rhetorical exercise, not a mathemetical. Very good for a used car salesman, not good for math.

Allow me to expand on this a little, because of the cultural clash issues we may be having. I know that you think that you have provided evidence, even though it falls short of the evidence which is customary in mathematical circles. But the evidence already known to mathematicians is vastly stronger. To wit: Chen proved that there are infinitely many primes p such that p+2 is either prime or semiprime; Zhang proved that there are infinitely many primes p such that the next prime is at most p + 10^7; an assortment of mathematicians improved the constant from 10^7 to 246; under the generalized Elliott-Halberstam conjecture, it can be improved to 6 (this last result is what's called a conditional proof). On the computational side, Tomás Oliveira e Silva computed the number of twin primes up to 4*10^18 to be 3023463123235320 which is a close fit to analytic predictions.

jzakiya · 2020-08-26, 14:05

I wrote a paper explaining a general theory of using what I call Prime Generators, and applied the theory to establish that Polignac's Conjecture is true, there are infinitely many pairs of primes that differ by any gap n.

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

It seems the last group of posters to this thread are not interested in learning any of the knowledge the paper presents, but are more interested playing word games to distract people from the paper's content.

If the paper was so much "rubbish" then a genuine critique of it would methodically point out all its mathematical deficiencies. But of course that can't be done, because none exist.

As stated, PGT has been codified/applied since 2008 in software to produce real work. This paper applies it to explain a mathematical conjecture.

Like with any foundational knowledge of math, you can't just read it once, you have to study it until you understand it philosophically and empirically. Clearly, some people here are not interested in doing that, and seem merely interested in making personal attacks on me, for whatever set of reasons driving them (which I suspect will continue after reading this).

I would suggest this to people who are curious about the premises posed in the paper.

For example, the paper explains and shows how Sexy Primes (gaps of 6) are the most abundant, while Twins and Cousins are the next most abundant. Either that statement is true or false. This is so easy to confirm or refute. Generate a large list of primes, say up to 10^9, 10^10, etc, and just count the (2, n) tuples and see their distribution. Piece of cake! I just told you which gaps occur the most frequently, show my claims breakdown for your list of primes. I know some people will say that doesn't prove anything, but since it will show what I say it will, it will suggest I know what I'm talking about.

Also, I establish how to predict when gaps of any size start to appear in Prime Generator (PG) residues, which we know with certainty will become (2,n) tuples within the range r0 to r0^2 for some larger Pn. Thus, take some large gap size, say 50,000, and see if it occurs for the Pn I say it first will. This is purely a numerical exercise, doable with sufficient hardware resources. Prove the predictive features of PGT wrong!

Unfortunately, what is going on here has nothing to do with the pursuit of knowledge, or a genuine interest to understand the contents of my papers. Differences in interpretation of results is legitimate, dismissal of the empirically established basis of the mathematical framework isn't.

I dare anyone to point out one thing in the paper that is mathematically wrong. Cite one equation, calculation, derivation, etc, just one. And don't just wave your hands and make polemical statements, provide a rigorous mathematical explanation to establish your claim, since you state I haven't done so in my paper.

Maybe, just maybe, if people actually start to study, and then discuss, the math and empirical results of the paper, we can have a fruitful discussion of what the contents actually means. We'll see if that is possible in this math forum.

retina · 2020-08-26, 14:10

Quote:

Originally Posted by jzakiya

For example, the paper explains and shows how Sexy Primes (gaps of 6) are the most abundant, while Twins and Cousins are the next most abundant. Either that statement is true or false. This is so easy to confirm or refute. Generate a large list of primes, say up to 10^9, 10^10, etc, and just count the (2, n) tuples and see their distribution.

That would only confirm or refute up to the limit you computed. But that says nothing about the asymptotic behaviour. We don't have computers that can compute up to infinity.

jzakiya · 2020-08-26, 14:17

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.

2020-08-19, 22:02	#13
Gelly May 2020 47₁₀ Posts	I would like to bring my own personal attention, and perhaps yours, to section 7 - Prime Generator Properties. In particular, you mention for multiple paragraphs that the properties of prime numbers (and, resultingly, these generators) do not need proofs in order to realize their properties. This seems to entirely sidestep the point of a proof in the name of using empirical evidence to support your claims. There exist many conjectures that are based off of "obvious" properties that are heuristic, but there's no known reason for them. If you are at all interested in being taken seriously, I would consider removing the diatribe against proofs of very simple things - especially the line that says "You have to draw pictures... You cannot imagine these properties into existence just from numerical analysis, you have to observe them first." If you cannot prove with numerical analyses even the simple properties of your generators, then this is not a proof. Perhaps you could gain a lot of insight on your generators - especially for the rest of us - to prove all of these things with no doubt. Last fiddled with by Gelly on 2020-08-19 at 22:02

2020-08-19, 23:25	#14
jzakiya Jul 2014 2²×3×5 Posts	Thank you for your feedback. What I was trying to (maybe ineleganatly) say in that section, is that most of the properties of prime generators are easily, and best, observed visually. In fact, this is how I came to recognize these properties. I didn't mean to convey they couldn't be established by analysis, et al, and in fact stated these properties are the set in place by the requirements of the residues and other mathematical operations and their requirements. I was trying to establish more of the philosophical basis for why prime generators are what they are. The statement about having to draw the pictures was to emphasize that for me, the visual expression of their properties led to their empirical understanding, which allowed me to apply them to my class of prime sieves, etc. My background is in engineering, not as a mathematician, so I think and talk like one. That is why I appreciate the feedback because I know there is an epistemological difference how we think, and emphasize what's important. Actually, the best way to present all of this (which I'm considering doing if I get the time a resources to do so) is create videos and animations to present it, because I can then draw the pictures and explain them at the same time. If it is possible to get past the form/language used in the paper, I believe the merits of the math, and demonstration of the empirical results it achieves, makes my case that there are an infinite number of prime pairs for any gap size, as Polignac conjectured. I'm really interested in feedback stating if people find anything they see is mathematically incorrect or not plausible. Last fiddled with by jzakiya on 2020-08-19 at 23:29

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2020-08-26, 14:05	#20
jzakiya Jul 2014 2²·3·5 Posts	I wrote a paper explaining a general theory of using what I call Prime Generators, and applied the theory to establish that Polignac's Conjecture is true, there are infinitely many pairs of primes that differ by any gap n. 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 It seems the last group of posters to this thread are not interested in learning any of the knowledge the paper presents, but are more interested playing word games to distract people from the paper's content. If the paper was so much "rubbish" then a genuine critique of it would methodically point out all its mathematical deficiencies. But of course that can't be done, because none exist. As stated, PGT has been codified/applied since 2008 in software to produce real work. This paper applies it to explain a mathematical conjecture. Like with any foundational knowledge of math, you can't just read it once, you have to study it until you understand it philosophically and empirically. Clearly, some people here are not interested in doing that, and seem merely interested in making personal attacks on me, for whatever set of reasons driving them (which I suspect will continue after reading this). I would suggest this to people who are curious about the premises posed in the paper. For example, the paper explains and shows how Sexy Primes (gaps of 6) are the most abundant, while Twins and Cousins are the next most abundant. Either that statement is true or false. This is so easy to confirm or refute. Generate a large list of primes, say up to 10^9, 10^10, etc, and just count the (2, n) tuples and see their distribution. Piece of cake! I just told you which gaps occur the most frequently, show my claims breakdown for your list of primes. I know some people will say that doesn't prove anything, but since it will show what I say it will, it will suggest I know what I'm talking about. Also, I establish how to predict when gaps of any size start to appear in Prime Generator (PG) residues, which we know with certainty will become (2,n) tuples within the range r0 to r0^2 for some larger Pn. Thus, take some large gap size, say 50,000, and see if it occurs for the Pn I say it first will. This is purely a numerical exercise, doable with sufficient hardware resources. Prove the predictive features of PGT wrong! Unfortunately, what is going on here has nothing to do with the pursuit of knowledge, or a genuine interest to understand the contents of my papers. Differences in interpretation of results is legitimate, dismissal of the empirically established basis of the mathematical framework isn't. I dare anyone to point out one thing in the paper that is mathematically wrong. Cite one equation, calculation, derivation, etc, just one. And don't just wave your hands and make polemical statements, provide a rigorous mathematical explanation to establish your claim, since you state I haven't done so in my paper. Maybe, just maybe, if people actually start to study, and then discuss, the math and empirical results of the paper, we can have a fruitful discussion of what the contents actually means. We'll see if that is possible in this math forum.

2020-08-26, 14:17	#22
jzakiya Jul 2014 60₁₀ Posts	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.