Mathematical proof turned down by JAMS.

[Rudy Toody]

Perhaps, someone could tell me why it only took them a day to decide.

It is my first proof and it has a bunch of warts, buI I believe the logic is sound.

I have empirically verified that it finds all twin primes below 1 000 000.

Simple Twin Prime Sieve

Could it have been turned down because I have no formal education?

--Fred

 

[f95toli]

This is definitly not my area (I am a physicist) but -assuming that your proof is correct (btw, there is something wrong with your link)- have you verified that you simply haven't "re-invented" something that is already known?

Rember that people have been working on this for well over 2000 years, doing something original when it comes to primes is presumably EXTREMLY difficult.

 

[iCyborg]

Proving the twin prime conjecture would definitely not be reinventing the wheel.

I find the steps and notation hard to follow, perhaps you should do the first 1-2 steps as an example?
And some steps are confusing / don't make sense to me. Like Modification 6 - I can't read that superscript well (some deltas times something, and there seems to be mod 6), and what is 's' in '+s'? It's not mentioned anywhere. Have no idea what are you mapping here, and how do you get the set C from there. I didn't get past this...

 

[Rudy Toody]

I fixed the link (I had grabbed the wrong one).

I have been a professional programmer since 1963, so I understand logic. My logic is sound; it just doesn't look very pretty. Also, because the sieve maps 100% to Euler's Sieve, it should have the same provability. The sieve identifies all twin primes. I have almost completed another mapping that proves that double prime twins are infinite. Those twins are {{11,13},{17,19}} and {{101,103},{107,109}}, etc.

 

[Rudy Toody]

Originally posted by: iCyborg
Proving the twin prime conjecture would definitely not be reinventing the wheel.

I find the steps and notation hard to follow, perhaps you should do the first 1-2 steps as an example?
And some steps are confusing / don't make sense to me. Like Modification 6 - I can't read that superscript well (some deltas times something, and there seems to be mod 6), and what is 's' in '+s'? It's not mentioned anywhere. Have no idea what are you mapping here, and how do you get the set C from there. I didn't get past this...

s is the input to the mapping function that is applied to the set: it is the product from the previouse multiply.

That is a simple mapping: we take the product to be removed and create its partner. If the product is the lower part of the twin (5 == s mod 6), we add 2, otherwise we subtract 2. This means that the non-product number cancelled by the products in that set (PT), must be transferred to set NT. This keeps the mapping to Euler's Sieve intact.


Edit: you could download the .gif to an image viewer to zoom to a level that is readable.
Edit2: I will make up some Venn diagrams that show the relationships amoung all sets. I don't have any software to do that, so I will do some free-hand drawing.

 

[Rudy Toody]

At the end of the Create step the PT set will contain:

{1,(5,7},{11,13},{17,19},{23,25},{29,31},{35,37},{41,43}.......}

At the first Select we identify {5,7} as the first twin prime >3.

At the end of the first Remove step PT will contain:

{1,{11,13},{17,19},{29,31},{41,43}.......}

And the set NT will contain:

{7,23,37}.......}

For the next selection the lowest prime will be the 7, which we sieve.

Then select 11 from PT, 13 from NT,
Then select 17 from PT, 19 from NT,
Then select 23 from NT,
Then select 29 from PT, 31 from NT,

All selections from PT identify twin primes, because we have removed pairs-at-a-time from PT.

 

[Schmide]

The way it reads to me, there is no "provable moment" where you reduce your facts into a concise conclusion. The point where you declare the existence of infinite twin primes because of a, f and g is really not an application of inductive reasoning. Constructing a method for finding twin primes then highlighting some very broadly defined steps, does not lead to a conclusion that there are infinite twin primes. Otherwise, I do not see the point in your proof where your logic affirms the existence of the next logical twin prime; thus, deducing that the series will continue forever.

Edit reduction induction same thing.

 

[Cogman]

Your "proof" assumes that there are an infinite number of twin primes. It does not prove their existence. Euler's Sieve does not prove the existence of infinite primes, it only provides a method of finding primes. There is a big difference between the two. (The proofs of an infinite number of primes is quite interesting btw).

Empirical proofs mean squat when it comes to real proofs, and they are NEVER accepted. Just take a look at the 3x+1 conjecture. Tons of empirical data says that this is true, yet no mathematical proof comes close to proving it.

I would recommend reading other proofs before trying to solve this ancient problem. If it has stood for over 100 years, chances are it will take more then a 1 page paper to prove its truthfulness.

 

[CycloWizard]

Is the document you linked to what you actually submitted? If so, it got rejected because it's not an article, does not contain any references, nor is it formatted for the journal. What did the rejection letter say?

In my experience, people with little or no formal education don't get published because they don't put their work in the proper context by referencing prior work in the field. They then like to blame it on elitist academics who think they are better simply because of education. In reality, journals want to publish any good work that is sent to them, but they don't have the manpower to track down and check whether what you're claiming to have solved has already been solved. So post the rejection and let's see what this is really all about.

 

[Rudy Toody]

Originally posted by: CycloWizard
Is the document you linked to what you actually submitted? If so, it got rejected because it's not an article, does not contain any references, nor is it formatted for the journal. What did the rejection letter say?

In my experience, people with little or no formal education don't get published because they don't put their work in the proper context by referencing prior work in the field. They then like to blame it on elitist academics who think they are better simply because of education. In reality, journals want to publish any good work that is sent to them, but they don't have the manpower to track down and check whether what you're claiming to have solved has already been solved. So post the rejection and let's see what this is really all about.

The email just said it would not be published.
I did ask about the lack of credentials, and the editor said that was not it.
The paper had a couple of typos that I somehow missed and one of the proof statements referred to the incorrect previous statement, so I don't blame them for refusing to run with it.

Do I need to find the proof of the Sieve of Eratosthenes upon which Euler's Seive is based?

Where would I find the original Euler proof?

Where would I find the proof of the infinity of the primes? I have seen examples of it in many books and places on the internet: Same for the Euler proof.

Would I need to reference the proofs of the set theory for Union, Intersection, Append, and Complement to make this proof complete? If so, where do I find those. The books about proofs seem to use it without proving it.

I am sure I can show that the twin primes are infinite because of the sieving by pairs. I just need to clean it up a bit.

The main mistake I made was to attempt to create a one-page proof.

I am created some primitive Venn diagrams to show that at the end of each cycle that the union of my subsets is exactly equal to Euler's set.

I am also preparing an animated .gif to demonstrate the sieve.

Note: the proof of the double twin primes turns out to be more difficult than I had imagined, so I have put that on the back burner for now.

Edit: while I was drafting this post, I came up with an additional way to prove this without requiring multiple subsets and unions and intersections, etc.

Because I have noticed that a sieved set with the composites exed out like the Sieve of Eratosthes resembles Godel's Incompleteness Theorem, I can use a variation (i.e., to prove completeness) to prove this.

In case the method based on Godel doesn't pan out, I would still like answers for the preceding questions.

Thanks for your patience with me.

 

[Cogman]

Originally posted by: Rudy Toody

Originally posted by: CycloWizard
Is the document you linked to what you actually submitted? If so, it got rejected because it's not an article, does not contain any references, nor is it formatted for the journal. What did the rejection letter say?

In my experience, people with little or no formal education don't get published because they don't put their work in the proper context by referencing prior work in the field. They then like to blame it on elitist academics who think they are better simply because of education. In reality, journals want to publish any good work that is sent to them, but they don't have the manpower to track down and check whether what you're claiming to have solved has already been solved. So post the rejection and let's see what this is really all about.

The email just said it would not be published.
I did ask about the lack of credentials, and the editor said that was not it.
The paper had a couple of typos that I somehow missed and one of the proof statements referred to the incorrect previous statement, so I don't blame them for refusing to run with it.

Do I need to find the proof of the Sieve of Eratosthenes upon which Euler's Seive is based?

Where would I find the original Euler proof?

Where would I find the proof of the infinity of the primes? I have seen examples of it in many books and places on the internet: Same for the Euler proof.

Would I need to reference the proofs of the set theory for Union, Intersection, Append, and Complement to make this proof complete? If so, where do I find those. The books about proofs seem to use is with proving it.

I am sure I can show that the twin primes are infinite because of the sieving by pairs. I just need to clean it up a bit.

The main mistake I made was to attempt to create a one-page proof.

I am created some primitive Venn diagrams to show that at the end of each cycle that the union of my subsets is exactly equal to Euler's set.

I am also preparing an animated .gif to demonstrate the sieve.

Note: the proof of the double twin primes turns out to be more difficult than I had imagined, so I have put that on the back burner for now.

I don't know where you will find Eulers original proof. But here is a good paper demonstrating a few proofs of the infinite amount of primes.

http://www.maa.org/editorial/e...ly%20many%20primes.pdf

But again. Eulers Sieve and Eulers Proof are two separate beasts. A method for generating twin primes does not prove an infinite number of primes (Unless you are doing a proof by induction.. however for a sieve like this, induction will not work). Euler's proof is demonstrated in the proof above. Again, notice that Euler's Proof is not the same as Euler's Sieve.

 

[Rudy Toody]

Originally posted by: Cogman
I don't know where you will find Eulers original proof. But here is a good paper demonstrating a few proofs of the infinite amount of primes.

http://www.maa.org/editorial/e...ly%20many%20primes.pdf

But again. Eulers Sieve and Eulers Proof are two separate beasts. A method for generating twin primes does not prove an infinite number of primes (Unless you are doing a proof by induction.. however for a sieve like this, induction will not work). Euler's proof is demonstrated in the proof above. Again, notice that Euler's Proof is not the same as Euler's Sieve.

Here is a variation of Euler's Infinite Prime Product.

I found this fooling around with the pre-sieved set that is created from sieving the first two primes from it.

If you write the Euler Prime Product starting with the 3rd prime instead of the first, it converges to 2*Zeta[2] / 3. Which is exactly Pi^2 / 9, a square! Unfortunately, this does not extend to higher powers.

I am using the fact that there are an infinite number of primes to prove that there are an infinite number of twin primes. So, I guess I need to reference that proof.

 

[Born2bwire]

Could you link to the actual paper you submitted?

 

[iCyborg]

Originally posted by: Rudy Toody
I am using the fact that there are an infinite number of primes to prove that there are an infinite number of twin primes. So, I guess I need to reference that proof.

OK, I see what you're doing.

As Cogman said, your "proof" provides a way to generate all twin primes, but does not prove it will continue indefinitely. It's similar to how Sieve of Erathostenes generates all primes, but you need a separate proof that it will continue indefinitely.

The problem is that, while you're right that PT U NT = N, and because there are infinitely many primes you can continue with the process forever, you do not prove that PT cannot become empty (or equivalently NT=N at some point, so we're always picking stuff from NT), i.e. that C U (PT ^ P) is never equal to PT, so PT isn't wiped out. To prove this, and because the first two members of PT are always twin primes, you'd need to prove that there's always at least one more pair, which is what you're trying to prove in the first place.

It's a smart way to generate all twin primes, I'll give you that, but something like this may be already known.

 

[Schmide]

Originally posted by: iCyborg
As Schmide said then Cogman said, your "proof" provides a way to generate all twin primes, but does not prove it will continue indefinitely.

Fixed. Do you have me on ingnore?

 

[iCyborg]

No, I don't. It's just that Cogman compared it explicitly to the standard sieving algorithm, but you're right, you said it first, sorry, wasn't intentional. Fix approved

 

[Rudy Toody]

Originally posted by: iCyborg

Originally posted by: Rudy Toody
I am using the fact that there are an infinite number of primes to prove that there are an infinite number of twin primes. So, I guess I need to reference that proof.

OK, I see what you're doing.

As Cogman said, your "proof" provides a way to generate all twin primes, but does not prove it will continue indefinitely. It's similar to how Sieve of Erathostenes generates all primes, but you need a separate proof that it will continue indefinitely.

The problem is that, while you're right that PT U NT = N, and because there are infinitely many primes you can continue with the process forever, you do not prove that PT cannot become empty (or equivalently NT=N at some point, so we're always picking stuff from NT), i.e. that C U (PT ^ P) is never equal to PT, so PT isn't wiped out. To prove this, and because the first two members of PT are always twin primes, you'd need to prove that there's always at least one more pair, which is what you're trying to prove in the first place.

It's a smart way to generate all twin primes, I'll give you that, but something like this may be already known.

That is exactly what I'm trying to do! I think I have a way to prove that there is always a twin in the pipeline.

As for PT becoming empty---per Euler: when infinity is reached, PT should contain the "1" and NT should be back to null again. Not sure how I would prove this True?

I think I can prove that I always have two twins lined up when I begin processing the lower prime of the lower twin.

I have completely restructured the changes to Euler's Sieve to make it even easier to understand. I am in the process of coding two Mathematic programs to verify the changes. Then I will show them side-by-side as an exhibit. I have almost completed the annotated Venn diagrams as another exhibit. Then I will create an anotated example similar to the examples I posted earlier.

Thanks for you insight about targeting the second twin in the pipeline as the critical point. I will add that to the truth table to see if I can prove it.

I hope I can get a lot more done tonight, because tomorrow is a real life day and I won't get back to the problem until Friday.

 

[Schmide]

Originally posted by: Rudy Toody

As for PT becoming empty---per Euler: when infinity is reached, PT should contain the "1" and NT should be back to null again. Not sure how I would prove this True?

You'll be there in no time. :clock:

You can't reach infinity. The whole basis of the Euler proof was it became a harmonic series and thus went on forever. Your burden of proof is much harder than you believe it to be. How do you plan to show that as you go on forever, there always exists 2 primes right next to each other?

 

[Rudy Toody]

Originally posted by: Schmide

Originally posted by: Rudy Toody

As for PT becoming empty---per Euler: when infinity is reached, PT should contain the "1" and NT should be back to null again. Not sure how I would prove this True?

You'll be there in no time. :clock:

You can't reach infinity. The whole basis of the Euler proof was it became a harmonic series and thus went on forever. Your burden of proof is much harder than you believe it to be. How do you plan to show that as you go on forever, there always exists 2 primes right next to each other?

Like the tortoise, one step at a time.

I can show that when I am processing the lower twin prime that the next one is always prime. And the lower of the next twin. Not sure about the proving the higher of the next twin. That appears to be key to this proof.

Regarding the issue of this sieve having been done before, there is nothing in the literature. From my past experience as an inventor, I have learned to research prior art before doing anything. I didn't find any, so I submitted the paper. My mistake was to try to prove the conjecture in the same paper. I should have sent the paper in to show a new and novel way to generate twin primes.

 

[CycloWizard]

Originally posted by: Rudy Toody
The email just said it would not be published.

Can you paste the contents of the e-mail (minus whatever personally identifiable information)? That would allow me to point you in the right direction. As far as the other questions go, while I'm fascinated by number theory, I know virtually nothing about this sort of thing, so you're better off listening to these other guys. I do know something about getting things published, so I'll try to help you on that end if I can.

 

[Rudy Toody]

Originally posted by: CycloWizard

Originally posted by: Rudy Toody
The email just said it would not be published.

Can you paste the contents of the e-mail (minus whatever personally identifiable information)? That would allow me to point you in the right direction. As far as the other questions go, while I'm fascinated by number theory, I know virtually nothing about this sort of thing, so you're better off listening to these other guys. I do know something about getting things published, so I'll try to help you on that end if I can.

Dear Fred Daniel Kline,

I am sorry to inform you that JAMS will not publish your paper "Simple
Twin Prime Sieve".

SIncerely,

Karl Rubin
Editor, JAMS

I don't have a copy of the one I sent to them: It went through their submissions page.

 

[iCyborg]

Originally posted by: Rudy Toody
I can show that when I am processing the lower twin prime that the next one is always prime. And the lower of the next twin. Not sure about the proving the higher of the next twin. That appears to be key to this proof.

Not sure what you mean about lower/higher twin prime.
In each step you either pick from PT or from NT. If you pick from PT, its partner is prime too (this is your f) in the proof), and you've discovered one pair of twin primes. If you pick from NT, you're just removing primes. You only ever find a pair of twin primes if and only if you pick from PT.
Therefore, to prove the conjecture, you need to prove that PT will never become empty - not some limit in infinity, but for *each* K, after K steps, there's stuff in PT. Imagine that after K steps, PT is wiped clean. NT will become N and after this point you will always be picking from NT, you will never find any more twins, and your method will proceed just like the standard sieve algorithm:
step 2 - take the lowest from NT (PT U NT = NT since PT empty)
step 3 - multiply NT to get P
step 4 - remove P from NT
(step 5 - nothing since C will be empty, so nothing is inserted into NT).

You don't prove that this can't happen, and proving it is equivalent to proving the twin prime conjecture.

It's fun to play with these kind of problems, as long as you don't lose from sight that these problems are very hard, and that you will *very* likely not solve them. I've spent lots of time in HS/college on Goldbach and Collatz (3x+1) conjectures, it's fun if you don't get your hopes up.

 

[Rudy Toody]

Originally posted by: iCyborg
Not sure what you mean about lower/higher twin prime.

{5,7,11,13} in PT. I am processing the 5--the lower prime of the lower twin prime, etc...

I am working on an reductio ad absurdum argument for the 13-- the higher prime of the higher twin prime.

 

[Rudy Toody]

Euler's Sieve

thisIsASinglePrime[s_] :=
Block[{},
Print;
]
multiplyAndRemove[s_] :=
Block[{},
thisIsASinglePrime;
P = s*n;
n = Complement[n, P];
]
(* Main Loop *)
r = 100;
n = Table[i, {i, 1, r}];
While[1 != Length[n],
multiplyAndRemove[n[[2]]];
]

Simple Twin Prime Sieve

f[s_] := If[5 == Mod[s, 6], Return[s + 2], Return[s - 2]];
thisIsASinglePrime[s_] :=
Block[{},
Print;
]
thisIsLowerTwinPrime[s_] :=
Block[{},
Print[{s, s + 2}];
]
multiplyAndRemove[s_] :=
Block[{},
thisIsASinglePrime;
P = s*Union[PT, NT];
NT = Complement[NT, P];
c = Map[f, Intersection[PT, P]];
PT = Complement[PT, Union[c, P]];
NT = Union[NT, c];
]
(* Main Loop *)
r = 100;
PT = Table[i, {i, 1, r}];
PT = Complement[PT, 2*PT];
PT = Complement[PT, 3*PT];
NT = {Infinity};

While[1 < Length[PT] && 1 <= Length[NT],
If [PT[[2]] < NT[[1]],
thisIsLowerTwinPrime[PT[[2]]];
multiplyAndRemove[PT[[2]]];
];
multiplyAndRemove[NT[[1]]];
]

These are both working Mathematica 7 programs.

 

[Born2bwire]

Originally posted by: Rudy Toody

Originally posted by: CycloWizard

Originally posted by: Rudy Toody
The email just said it would not be published.

Can you paste the contents of the e-mail (minus whatever personally identifiable information)? That would allow me to point you in the right direction. As far as the other questions go, while I'm fascinated by number theory, I know virtually nothing about this sort of thing, so you're better off listening to these other guys. I do know something about getting things published, so I'll try to help you on that end if I can.

Dear Fred Daniel Kline,

I am sorry to inform you that JAMS will not publish your paper "Simple
Twin Prime Sieve".

SIncerely,

Karl Rubin
Editor, JAMS

I don't have a copy of the one I sent to them: It went through their submissions page.

The question is, is what you linked essentially what you sent them? Because if it is it would not past a preliminary inspection regardless of the content. You should spend some time reading the papers in the journal(s) you wish to submit (and other journals on the same subject) to be more aware of the kinds of papers they publish and the standards. In addition, they usually have a standards pamphlet that you should read and follow but most of that is purely layout guidelines. What you posted lacks a cited review of previous work and why your work is notable and different, citations for the assumptions that you made, probably does not follow their layout guidelines, and a lot of other things.