Mathematical proof turned down by JAMS.

[iCyborg]

Originally posted by: DrPizza
Anyone else? I'm usually really good at following math textbooks. I can't make rhyme or reason out of his first sieve for double primes. I can't read step 6 either (I can barely make out "mod")

I couldnt' figure it out either until he explained it a bit, I also asked about step 6 too...

My understanding is that he removes all the multiples of 2 and 3. This will leave all odd numbers of the form 6K+-1 in PT, and you can group these naturally into pairs (6k-1,6k+1). After that, it's basically the standard sieving algorithm, but whenever you remove a nonprime, you also remove its "twin". You have to keep track of these removed twins since you have to do the sieving with them too, and that's what that NT set is for.

@Rudy
you're missing (3,5) and (5,7) in that list

 

[Rudy Toody]

Originally posted by: iCyborg
@Rudy
you're missing (3,5) and (5,7) in that list

In this sieve, {3,5} is not considered a twin. It is the anomalous twin.

The list is of dual twins only, so the {5,7} doesn't pair with another twin.

Example: {41,43} pairs with {47,49}, but the 49 causes those values to be moved down a level for further sieving.

If I printed the twins in this range, the list would have 1224 lines.

There are over 9K primes in this range, also.

Edit:

At each multiply-remove step:
In PD, we remove 2² numbers, and move 2²-1numbers down a level.
In PT, we remove 2¹ numbers, and move 2¹-1 numbers down a level.
In NT, we remove 1 number.

 

[f95toli]

Both me and Born2Wire are active at www.physicsforums.com which has a very active sub-forum for number theory.
As far as I can tell (again, not my field) some of the people active in that forum know a LOT about primes so they should be able to give you some good feedback if that is what you are looking for.

 

[Rudy Toody]

Originally posted by: f95toli
Both me and Born2Wire are active at www.physicsforums.com which has a very active sub-forum for number theory.
As far as I can tell (again, not my field) some of the people active in that forum know a LOT about primes so they should be able to give you some good feedback if that is what you are looking for.

Could you post a link on that forum, so they could look at this thread, if they are interested?
Edit: Nevermind! I did it myself.

It looks like a good forum. Thanks, f95toli.

 

[Rudy Toody]

See the modified sub-title of this thread: I will have this proof completed in a couple of days. Stay tuned!

 

[DrPizza]

I hate to burst your bubble, but I really don't think your sieve does, in fact, support the twin prime conjecture. It appears to merely be an algorithm for finding the twins - it doesn't seem to make any predictions about where they're at. Nor does it predict that there will always be a twin. It simply seems to start with all the pairs of odd integers in the form 6n-1,6n+1 that are less than some value (100,000? in this case), and removes the pairs that aren't both primes. I don't see where it guarantees that , say, from 1 billion to 2 billion there is guaranteed to be another twin prime.

For what it's worth, in the last 6 months, at least 3 of the books I've read have devoted numerous chapters to prime numbers. I am always amazed at how much study has gone into prime numbers. Thus, I feel that it's almost naive (I don't mean to sound insulting) to think that someone is going to stumble upon something so simple to prove that there are an infinite number of double primes. If there were a simple proof, I believe it would have been found. If you spend a couple hours researching primes and twin primes, I think you'll be astonished at the depth at which people have researched primes and how much is known.

 

[Estrella]

Originally posted by: DrPizza
I hate to burst your bubble, but I really don't think your sieve does, in fact, support the twin prime conjecture. It appears to merely be an algorithm for finding the twins - it doesn't seem to make any predictions about where they're at. Nor does it predict that there will always be a twin. It simply seems to start with all the pairs of odd integers in the form 6n-1,6n+1 that are less than some value (100,000? in this case), and removes the pairs that aren't both primes. I don't see where it guarantees that , say, from 1 billion to 2 billion there is guaranteed to be another twin prime.

For what it's worth, in the last 6 months, at least 3 of the books I've read have devoted numerous chapters to prime numbers. I am always amazed at how much study has gone into prime numbers. Thus, I feel that it's almost naive (I don't mean to sound insulting) to think that someone is going to stumble upon something so simple to prove that there are an infinite number of double primes. If there were a simple proof, I believe it would have been found. If you spend a couple hours researching primes and twin primes, I think you'll be astonished at the depth at which people have researched primes and how much is known.

I was trying not to do that.

 

[Rudy Toody]

Originally posted by: Estrella

Originally posted by: DrPizza
I hate to burst your bubble, but I really don't think your sieve does, in fact, support the twin prime conjecture. It appears to merely be an algorithm for finding the twins - it doesn't seem to make any predictions about where they're at. Nor does it predict that there will always be a twin. It simply seems to start with all the pairs of odd integers in the form 6n-1,6n+1 that are less than some value (100,000? in this case), and removes the pairs that aren't both primes. I don't see where it guarantees that , say, from 1 billion to 2 billion there is guaranteed to be another twin prime.

For what it's worth, in the last 6 months, at least 3 of the books I've read have devoted numerous chapters to prime numbers. I am always amazed at how much study has gone into prime numbers. Thus, I feel that it's almost naive (I don't mean to sound insulting) to think that someone is going to stumble upon something so simple to prove that there are an infinite number of double primes. If there were a simple proof, I believe it would have been found. If you spend a couple hours researching primes and twin primes, I think you'll be astonished at the depth at which people have researched primes and how much is known.

I was trying not to do that.

I think you will be surprised. I think everyone will be thinking, "How did we not see this for 160 years?"

 

[CP5670]

Yes, something like this is not going to have a simple solution in a few pages. There is an enormous body of existing work on this problem, and people have been studying it from different approaches for hundreds of years.

I'm not a number theorist but have studied a few parts of it related to analysis. If you want to really get into this stuff, this book is well known and a good place to start.

 

[DrPizza]

For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

 

[DrPizza]

And also for what it's worth, I would be absolutely thrilled to be surprised & discover that your proof was a valid proof of the twin prime conjecture. And, also, once you tackle the twin primes conjecture, I suggest looking at the millenium problems. Perhaps you will be the person who is motivated enough to not only learn the mathematics necessary, but also be able to come up with the insight to finally solve these problems. One Million dollar prize is associated with each of these problems. http://www.claymath.org/millennium/

 

[Cogman]

Originally posted by: DrPizza
For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

Whoah, that is crazy. You basically have a way to provably show a n-wide gap of primes

I don't know, that is just weird to me to think about, Obviously we know that there are an infinite number of primes, however, this shows that for any length up to almost infinity (Since I don't believe you can do infinity!) there is a gap that big between primes. That's really crazy to think about.

BTW, rudy, if you take one of the million dollar questions, I would suggest the P=NP problem. If it is true, then we could see some very cool things, if it isn't true, then at least people will stop wasting their time trying to prove it.

 

[CP5670]

Actually, the Riemann hypothesis is much more closely related to what the OP is interested in. A solution to that would answer a lot of open questions in number theory, especially about how the prime numbers are distributed.

The thing about this problem is that it seems to be a rather simple question in its original form. There are several standard methods for finding zeros of functions, but they all fail to work somewhere along the way in this case.

 

[Rudy Toody]

Speaking of Riemann, did you see my previous post where I describe a curiousity involving the cosine of the square-root of 2 Zeta[2]/3 equaling one-half?

 

[CP5670]

Well, that is pretty obvious. It's well known that zeta(2)=pi^2 / 6, which can be shown in several ways.

 

[Rudy Toody]

Originally posted by: CP5670
Well, that is pretty obvious. It's well known that zeta(2)=pi^2 / 6, which can be shown in several ways.

You are correct.

How about taking the Euler Prime Product from 5 insteaq of 2? This converges to Pi squared over 9.

 

[Estrella]

Making a program to find twin primes and/or primes up to a certain number is nothing novel or new. This problem could be given to kids in a first year computer science class. You couple this program with the theorem stating that there are infinite primes, which proves absolutely nothing. Taking an existing sieve and making it throw those which are not pizza flavored(the equivalent of not having a twin) is nothing of note. There are better programs that can be modified for pizza flavoring that are faster and can be found on Wikipedia.

/thread

P.S. This isn't to get you to quit trying, you just need to do WAY more reading. WAY cannot be emphasized enough here.

 

[CP5670]

P.S. This isn't to get you to quit trying, you just need to do WAY more reading. WAY cannot be emphasized enough here.

:thumbsup:

Non-mathematicians often don't realize just how vast and well developed the subject is. You could probably spend an entire career on a problem like this, and all the different aspects of it and approaches towards it.

 

[DrPizza]

Originally posted by: Cogman

Originally posted by: DrPizza
For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

Whoah, that is crazy. You basically have a way to provably show a n-wide gap of primes

I don't know, that is just weird to me to think about, Obviously we know that there are an infinite number of primes, however, this shows that for any length up to almost infinity (Since I don't believe you can do infinity!) there is a gap that big between primes. That's really crazy to think about.

BTW, rudy, if you take one of the million dollar questions, I would suggest the P=NP problem. If it is true, then we could see some very cool things, if it isn't true, then at least people will stop wasting their time trying to prove it.

I know. It really blew my mind how simple such a "proof" was that you can prove a gap of any size exists between prime numbers.

Also, since we're talking about proofs to problems that look "simple", Fermat's last theorem - the one that many people point out that he at one point wrote that he had a simple proof that wouldn't fit in the margin of the paper - he didn't have such a proof. There's speculation over which "simple" but incorrect proof he may have thought of, but later, after he wrote that, he worked on proving at least one special case of that theorem. i.e. it's akin to claiming you have a proof that all animals with hair are mammals, then 2 years later, struggling over a proof that all dogs are mammals.

 

[Cogman]

Originally posted by: DrPizza

Originally posted by: Cogman

Originally posted by: DrPizza
For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

Whoah, that is crazy. You basically have a way to provably show a n-wide gap of primes

I don't know, that is just weird to me to think about, Obviously we know that there are an infinite number of primes, however, this shows that for any length up to almost infinity (Since I don't believe you can do infinity!) there is a gap that big between primes. That's really crazy to think about.

BTW, rudy, if you take one of the million dollar questions, I would suggest the P=NP problem. If it is true, then we could see some very cool things, if it isn't true, then at least people will stop wasting their time trying to prove it.

I know. It really blew my mind how simple such a "proof" was that you can prove a gap of any size exists between prime numbers.

Also, since we're talking about proofs to problems that look "simple", Fermat's last theorem - the one that many people point out that he at one point wrote that he had a simple proof that wouldn't fit in the margin of the paper - he didn't have such a proof. There's speculation over which "simple" but incorrect proof he may have thought of, but later, after he wrote that, he worked on proving at least one special case of that theorem. i.e. it's akin to claiming you have a proof that all animals with hair are mammals, then 2 years later, struggling over a proof that all dogs are mammals.

I almost wonder if this is an example of how close prime numbers really are to not being an infinite set of numbers. Perhaps it is that closeness that makes things like the twin prime conjecture hard to prove (That, and the seemingly random distribution that they exhibit).

Speaking of primes and Fermat. Fermat's little theorem has to be one of the coolest ones out there. I had a CS project that used Fermat's little theorem to probabilistically prove whether a random number is prime (after about 15 checks, it is 99.99% accurate).

Something that came as a result of that is the discovery that something like 70% of all randomly generated numbers are divisible by 2, 3, and 5. So one of the first checks I did was those three numbers. That eliminates a large amount of random number from prime testing very quickly. After that, it was diminishing returns.

 

[CP5670]

I almost wonder if this is an example of how close prime numbers really are to not being an infinite set of numbers. Perhaps it is that closeness that makes things like the twin prime conjecture hard to prove (That, and the seemingly random distribution that they exhibit).

This is one of the main reasons the Riemann hypothesis is considered to be so significant. The number of primes between 1 and x is roughly log(x) (or more precisely, the integral of 1/log(x) ) when x is large, but their local behavior can deviate from that quite a bit and is seemingly pretty random. These deviations are closely related to the locations of the zeta function's zeros.

 

[Estrella]

Originally posted by: Cogman

Originally posted by: DrPizza

Originally posted by: Cogman

Originally posted by: DrPizza
For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

Whoah, that is crazy. You basically have a way to provably show a n-wide gap of primes

I don't know, that is just weird to me to think about, Obviously we know that there are an infinite number of primes, however, this shows that for any length up to almost infinity (Since I don't believe you can do infinity!) there is a gap that big between primes. That's really crazy to think about.

BTW, rudy, if you take one of the million dollar questions, I would suggest the P=NP problem. If it is true, then we could see some very cool things, if it isn't true, then at least people will stop wasting their time trying to prove it.

I know. It really blew my mind how simple such a "proof" was that you can prove a gap of any size exists between prime numbers.

Also, since we're talking about proofs to problems that look "simple", Fermat's last theorem - the one that many people point out that he at one point wrote that he had a simple proof that wouldn't fit in the margin of the paper - he didn't have such a proof. There's speculation over which "simple" but incorrect proof he may have thought of, but later, after he wrote that, he worked on proving at least one special case of that theorem. i.e. it's akin to claiming you have a proof that all animals with hair are mammals, then 2 years later, struggling over a proof that all dogs are mammals.

I almost wonder if this is an example of how close prime numbers really are to not being an infinite set of numbers. Perhaps it is that closeness that makes things like the twin prime conjecture hard to prove (That, and the seemingly random distribution that they exhibit).

Speaking of primes and Fermat. Fermat's little theorem has to be one of the coolest ones out there. I had a CS project that used Fermat's little theorem to probabilistically prove whether a random number is prime (after about 15 checks, it is 99.99% accurate).

Something that came as a result of that is the discovery that something like 70% of all randomly generated numbers are divisible by 2, 3, and 5. So one of the first checks I did was those three numbers. That eliminates a large amount of random number from prime testing very quickly. After that, it was diminishing returns.

What does it mean for a set to be close to being not infinite? Suppose I were to take a couple elements out. I would still have an infinite set.

 

[Cogman]

Let me put it this way. given the n-gap proof that Dr. Pizza gave, If you had an infinite gap, then that would mean that prime are not infinite.

With factorials, any integer can be inserted into the ! operation, HOWEVER, you cannot use infinity in the factorial operation (There is no infinity! (to be read infinity factorial)).

It is like saying that you can make a gap that is as big as any positive integer, how many positive integers are there? an infinite number of them.

 

[Rudy Toody]

You fellows are correct. I cannot prove the infinitude of twins or any thing else through the use of a sieve.

I thought that by nesting subsets (thus nesting sieves) I could create some dependencies to ensure that the sieves never run out. Well, I can.

However, the top set would be idependent and thus we have no idea if it could be exhausted. We fix that by creating an even higher set.

Then we don't know if that will be exhausted, so we create a higher one.

Thus we create an infinite set of sieving rules to gaurantee the lower sets will not run out, but we still don't know about the highest.

This is Gödel's Incompleteness Theorem at it's best!

Thanks for your input!

 

[imported_inspire]

Originally posted by: DrPizza

Originally posted by: Cogman

Originally posted by: DrPizza
For what it's worth, and in a positive way, in one of the books I read a couple weeks ago, it said (paraphrasing) - as soon as the teacher says that something can't be done, although intuitively it seems that the teacher might be wrong, it is the best student who sees the challenge and goes about trying to prove the teacher wrong. A HUGE thumbsup to the OP of this thread for trying. Not that this is something that can't be done, but I honestly doubt that a solution to the twin primes conjecture will be anything brief.

But, studying prime numbers is always fascinating. And, to me, it's always fascinating to learn something new, especially when it's something simple that I had never encountered before nor thought of myself - but something I can take with me to my high school classes to "entertain" students with. i.e. something that instantly makes sense - Starting at 1,000,001 factorial, there are guaranteed to be 1 million consecutive integers that are not prime. (not that that's the first time there are a million consecutive non-primes).
Quick explanation, and it becomes obvious:
1*2*3*...*1,000,001 + 2 is divisible by 2
1*2*3*...*1,000,001 + 3 is divisible by 3
1*2*3*4*...*1,000,001 + 4 is divisible by 4
etc.

Whoah, that is crazy. You basically have a way to provably show a n-wide gap of primes

I don't know, that is just weird to me to think about, Obviously we know that there are an infinite number of primes, however, this shows that for any length up to almost infinity (Since I don't believe you can do infinity!) there is a gap that big between primes. That's really crazy to think about.

BTW, rudy, if you take one of the million dollar questions, I would suggest the P=NP problem. If it is true, then we could see some very cool things, if it isn't true, then at least people will stop wasting their time trying to prove it.

I know. It really blew my mind how simple such a "proof" was that you can prove a gap of any size exists between prime numbers.

Also, since we're talking about proofs to problems that look "simple", Fermat's last theorem - the one that many people point out that he at one point wrote that he had a simple proof that wouldn't fit in the margin of the paper - he didn't have such a proof. There's speculation over which "simple" but incorrect proof he may have thought of, but later, after he wrote that, he worked on proving at least one special case of that theorem. i.e. it's akin to claiming you have a proof that all animals with hair are mammals, then 2 years later, struggling over a proof that all dogs are mammals.

It's good that you brought up Fermat's Last Theorem, too, because it fits quite well into the discussion here. The theorem, while very easily understood, and, to a large degree, evident, went unproven for centuries. Fermat thought he had a fantastically elegant proof, but it never came.

It wasn't until 1995 that Andrew Wiles published an acceptable proof of Fermat's Last Theorem that was over 100 pages long and used mathematics that was unknown to Fermat at the time.

It simply shows that even the simplest ocnjectures can become quite difficult to prove and that such problems are very rarely simply solved. Though there is the case of the Euclidean Parallel Postulate, no other exceptions come to mind.