Mathematical proof turned down by JAMS.

[Estrella]

Originally posted by: 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!

Dude, it would be great if you introduced much more rigor into your work. You have not proven that this is an unprovable statement given the axioms of the number system you are working in.

 

[Estrella]

Originally posted by: 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.

My point is is that almost not infinite carries no meaning in a mathematical sense.

 

[DrPizza]

Originally posted by: inspire

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.

Just reiterating, because your post doesn't make clear that you understood mine - while Fermat initially thought he had a fantastically elegant proof, he later came to realize that he was wrong.

Also, I'm not sure Godel's Incompleteness Theorem applies to the twin prime proof above.

And, lastly, I love the simplicity of the Goldbach conjecture - every even integer, greater than 2, can be expressed as the sum of two prime numbers. I used that in my Geometry class today (discussing the concept of proof) along with .999... = 1 (because unlike ATOT, they're at least average students - thus were able to understand it.) But, with Goldbach's conjecture, I was finally able to get students to seize on the idea of a counter-example.

 

[Rudy Toody]

Originally posted by: Estrella

Originally posted by: 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!

Dude, it would be great if you introduced much more rigor into your work. You have not proven that this is an unprovable statement given the axioms of the number system you are working in.

I interpret Gödel's Incompleteness Theorem to mean, If you have a set of rules that cannot provide enough certainty to prove something you know is true, that you must add another rule to do so. Then using that rule you will still have some things you cannot prove, so you must add another rule.

We have some rules that describe sieving for primes. Using just those rules we can find all of them, but we have no idea if they are infinite.

Now, using those same rules, we notice the twins and wonder if they are infinite. So we add a rule to sieve twins. However, we still cannot show they are infinite. However, we notice some quadruplets and create a rule to sieve quadruplets (my sieve) and find all quadruplets that we encounter. We still do not know if they run out. If I find a rule to sieve octuplets, etc., I will still be unable to determine if anything is infinite.

So, I have mapped my set of rules to the axioms of the number system to show they are equivalant and thus incomplete.

Edit: I just discovered that this is called the sieve parity problem.

 

[iCyborg]

This has nothing to do with Godel's theorem, which roughly states that no matter how many *axioms* you add (without losing consistency), you will always have *some* true statement, which cannot be proved. You could always add a rule "there are infinitely many twin primes" and be done with that particular statement. The point is that there isn't a single specific statement that is always unprovable, but rather that no matter how you add rules (axioms really), there will be some such statement.

 

[Cogman]

Originally posted by: Estrella

Originally posted by: 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.

My point is is that almost not infinite carries no meaning in a mathematical sense.

I understand that, so tell me how to express something to the effect that the maximum size of n approachs infinity? or that the domain of n is all positive integers, and even with that, it that the gap cannot be of infinite size.

 

[imported_inspire]

Originally posted by: DrPizza
Just reiterating, because your post doesn't make clear that you understood mine

Yeah, I quoted you because I was more or less building on your story. But yeah - Fermat didn't have the proof, and even though the concept is simple when you think about it, the proof is quite complex.

 

[Estrella]

Originally posted by: Cogman

Originally posted by: Estrella

Originally posted by: 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.

My point is is that almost not infinite carries no meaning in a mathematical sense.

I understand that, so tell me how to express something to the effect that the maximum size of n approachs infinity? or that the domain of n is all positive integers, and even with that, it that the gap cannot be of infinite size.

So, if numbers come in sizes, then is it possible for me to get a number in tall, grande, or large?

The domain of n? n does not have a domain, n is a number. The function f(n)=n! has a domain. Define gap more precisely.

 

[Estrella]

Originally posted by: Rudy Toody

Originally posted by: Estrella

Originally posted by: 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!

Dude, it would be great if you introduced much more rigor into your work. You have not proven that this is an unprovable statement given the axioms of the number system you are working in.

I interpret Gödel's Incompleteness Theorem to mean, If you have a set of rules that cannot provide enough certainty to prove something you know is true, that you must add another rule to do so. Then using that rule you will still have some things you cannot prove, so you must add another rule.

We have some rules that describe sieving for primes. Using just those rules we can find all of them, but we have no idea if they are infinite.

Now, using those same rules, we notice the twins and wonder if they are infinite. So we add a rule to sieve twins. However, we still cannot show they are infinite. However, we notice some quadruplets and create a rule to sieve quadruplets (my sieve) and find all quadruplets that we encounter. We still do not know if they run out. If I find a rule to sieve octuplets, etc., I will still be unable to determine if anything is infinite.

So, I have mapped my set of rules to the axioms of the number system to show they are equivalant and thus incomplete.

Edit: I just discovered that this is called the sieve parity problem.

No, see iCyborg's post below where you originally said this.

Math is never based on interpretation.

Also, just because your proof is insufficient does not mean the statement you tried to prove is unprovable in that system(I am sure you are mistaking the axiomatic system for the course you lay in your sieve).

Also Also, I have never seen a mapping from one set of axioms to another.

You actually get kudos for recognizing that the problem you are running into is the sieve parity problem.

 

[Rudy Toody]

Originally posted by: Estrella

Originally posted by: Rudy Toody

Originally posted by: Estrella

Originally posted by: 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!

Dude, it would be great if you introduced much more rigor into your work. You have not proven that this is an unprovable statement given the axioms of the number system you are working in.

I interpret Gödel's Incompleteness Theorem to mean, If you have a set of rules that cannot provide enough certainty to prove something you know is true, that you must add another rule to do so. Then using that rule you will still have some things you cannot prove, so you must add another rule.

We have some rules that describe sieving for primes. Using just those rules we can find all of them, but we have no idea if they are infinite.

Now, using those same rules, we notice the twins and wonder if they are infinite. So we add a rule to sieve twins. However, we still cannot show they are infinite. However, we notice some quadruplets and create a rule to sieve quadruplets (my sieve) and find all quadruplets that we encounter. We still do not know if they run out. If I find a rule to sieve octuplets, etc., I will still be unable to determine if anything is infinite.

So, I have mapped my set of rules to the axioms of the number system to show they are equivalant and thus incomplete.

Edit: I just discovered that this is called the sieve parity problem.

No, see iCyborg's post below where you originally said this.

Math is never based on interpretation.

Also, just because your proof is insufficient does not mean the statement you tried to prove is unprovable in that system(I am sure you are mistaking the axiomatic system for the course you lay in your sieve).

Also Also, I have never seen a mapping from one set of axioms to another.

You actually get kudos for recognizing that the problem you are running into is the sieve parity problem.

I meant to say that the problem cannot be solved using my sieve.

Edit: I guess mapping was a bad choice of words. I substituted my rules for the axioms in the incompleteness theorem. My reading of the incompleteness theorem shows that it would cover any consistant system whether it's axioms, sieving rules, or how to beat blackjack!

 

[Cogman]

Originally posted by: Estrella

Originally posted by: Cogman

Originally posted by: Estrella

Originally posted by: 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.

My point is is that almost not infinite carries no meaning in a mathematical sense.

I understand that, so tell me how to express something to the effect that the maximum size of n approachs infinity? or that the domain of n is all positive integers, and even with that, it that the gap cannot be of infinite size.

So, if numbers come in sizes, then is it possible for me to get a number in tall, grande, or large?

The domain of n? n does not have a domain, n is a number. The function f(n)=n! has a domain. Define gap more precisely.

now you are just being nit-picky. We aren't writing the math journal here, you know perfectly well what I mean.

Obviously, I was speaking of the domain of f(n)=n!. The upper bound of that domain is equal to maximum size of consecutive non-prime integers that can be "easily" found.

Happy? Now again, what would you call the domain of f(n)=n!. since you side stepped it by nit-picking my non-math statements.

 

[mutz]

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.

that actually proovs that the gap between primes goes bigger as the number does.
u can see it also at the OP program output.

 

[DrPizza]

Originally posted by: mutz

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.

that actually proovs that the gap between primes goes bigger as the number does.
u can see it also at the OP program output.

No, it doesn't prove that. i.e. if there are still twin primes, the gap between those numbers is 2. All it demonstrates is that you can find a set of consecutive integers of size n, such that none of those integers is a prime number. You are guaranteed to have such a set at (n+1)!

That primes tend to fall further and further apart as numbers get larger is fairly evident; this just didn't prove it.

 

[mutz]

All it demonstrates is that you can find a set of consecutive integers of size n

yeah that is understood, though it does necessarily means that the number of non-prime integers in the set will get bigger as n does, atleast accordingly to n, so the gap between distinctive primes does,
(or is it that obvious)...,
whether this is/was already evident i'm unaware.
not talking twin primes.
sorry if there's any mistake.

Edit: yeah, it is quite evident as it'll inevitably widen as self multiplications (5*5,7*7,9*9,11*11 etc.) will add up.

i wonder whether there could be really found any order in they're distribution and whether the proof of they're infinity demands finding that order first.
(it probably does).

 

[Estrella]

Since, phone browsers suck at the normal window.

F(n)=n! Is defined only on the non-negative integers. So the domain of F would be that.

 

[DrPizza]

Originally posted by: mutz

All it demonstrates is that you can find a set of consecutive integers of size n

yeah that is understood, though it does necessarily means that the number of non-prime integers in the set will get bigger as n does, atleast accordingly to n, so the gap between distinctive primes does,
(or is it that obvious)...,
whether this is/was already evident i'm unaware.
not talking twin primes.
sorry if there's any mistake.

i wonder whether there could be realy found any order in they're distribution and whether the proof of they're infinity demands finding that order first.
it probably does..

There are already several decent proofs/approximations of their distribution. Off the top of my head, I can't recall any of the formulas though. I've got a pile of math books that touch on primes though. Maybe if my wife falls asleep early tonight, I'll read the chapters on primes from a few of them.

 

[mutz]

I've got a pile of math books that touch on primes though. Maybe if my wife falls asleep early tonight, I'll read the chapters on primes from a few of them.

yeah, please do,
i'll go have a search too.

 

[jersiq]

Originally posted by: mutz

I've got a pile of math books that touch on primes though. Maybe if my wife falls asleep early tonight, I'll read the chapters on primes from a few of them.

yeah, please do,
i'll go have a search too.

Don't know if your search took you here, but it was a topics class I couldn't take this semester on the Riemann-Zeta function.

Riemann Zeta Function

Look under the Euler Product Formula heading.

 

[mutz]

yeah i was just looking at it few hours ago,
i'll study it much deeper in the next days,
regards.

p.s -
starting at 3:
(2),(2),4,(2),4,(2),4,
6,(2),6,4,(2),4,6,6,(2),6,
4,(2),6,4,6,8,4,(2),4,(2),
4,14,4,6,(2),10,(2),6,6,4,
6,6,(2),10,(2),4,(2),12,12,4,
(2),4,6,(2),10,6,6,6,(2),6,
4,(2),10,14,4,(2),4,14,6,10,
(2),4,6,8,6,6,4,6,8,4,
8,10,(2),10,(2),6,4,6,8,4,
(2),4,12,8,4,8,4,6,12,(2),
18,6,10,6,6,(2),6,10,6,6,
(2),6,6,4,(2),12,10,(2),4,6,
6,(2),12,4,6,8,10,8,10,8,
6,6,4,8,6,4,8,4,14,10,
12,(2),10,(2),4,(2),10,14,4,(2),
4,14,4,(2),4,10,4,8,10,8,
4,6,6,14,4,6,6,8,6,4
8,4,6,(2).

3->1021.

overall 171 prime numbers, 18 twins which makes 36 counted as a part of a twin, 2 out of them are 3&5,5&7 at the first row.
it's probably better searching higher numbers when looking for some order though at smaller numbers one can better appreciate & accumulate the differentiality.
it's very hard finding any relevance between them though a statistical computed graph might show it's benefits here,
i'm not sure it can be decoded using bare eye and simple logic...
sorry if there's any mistake.

i must add here that any number that could not be divided by any prime is the next potential prime, (else it is divided by another (earlier) prime.)
so it seems prime numbers "eat" they're own later distribution (naturally) meaning, they're existence and survival is being risked by they're own distribution, meaning,
they actually "live" inside some kind of a loop, which is always growing and expanding in order to escape it's own self created extinction...
as they grow (afar from zero) they become harder to find,
but as numbers are infinite,
so eventually they are.
it is something alive..
sounds logic...

p.s - Reimann zeta function is quite complexed...,
i'm not sure i can corrently fully understand it, (or ready to), though there's a nice wiki page related to all unsolved mathematical problems at :
http://en.wikipedia.org/wiki/U...roblems_in_mathematics,
try u'r luck .

 

[Gannon]

wrong thread...

 

[crosscut]

If you want the only book on primes go here.....it is all you need!
http://www.jmccanneyscience.co...andTableofContents.HTM

 

[CP5670]

I hope that isn't your book. It has all the usual signs of quackery.

An exact formula for the prime density function has been known for over 100 years, called the Riemann-von Mangoldt formula, and a few terms of that gives a good approximation in many cases. You can take the inverse to get an expression for the nth prime number.

On a side note, Terry Tao has posted a nice non-technical paper on this stuff on his blog here, which you guys might find interesting.

 

[mutz]

thanks for the link CP, though,
it's still quite complicated...,
i'll leave it for the math guys here, or for later on,
(but it is nice to go through though).

good luck with the book crosscut .