## Re: [PrimeNumbers] Re: Unknown Mathematician Proves Elusive Property of Prime Numbers

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• ... but it s Wired, so the title has to appeal to the geeks and casual readers and play with the notion that even in the field of Mathematics, you too could
Message 1 of 18 , May 27 3:03 PM
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Le 2013-05-27 23:13, djbroadhurst a écrit :
> "Mohsen" <mafshin89@...> wrote:
>
>> Unknown Mathematician Proves Elusive Property of Prime Numbers
>> http://www.wired.com/wiredscience/2013/05/twin-primes/
>
> I think that "unknown", in this context, means "not known
> to the first pundit that was asked" (in this case, perhaps,
> Andrew Granville, who often gets asked to respond, on the hoof).

but it's Wired, so the title has to appeal to the geeks and casual
and play with the notion that even in the field of Mathematics,
you too could become a star.

Maths look more and more like a social science to me these days...
• I must say that some of the popular exposition of this discovery of Zhang has been to allow the impression that there is a limit to the size of prime gaps, a
Message 2 of 18 , May 28 3:54 PM
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I must say that some of the popular exposition of this discovery of Zhang has been to allow the impression that there is a limit to the size of prime gaps, a proposition that puzzled me for a while, but which would have never occured to a Mathematician worth of the name due to its implicit dismissal of established theory.

All that Zhang seems to have assured us, and this is the comforting news, is that no how far along the number line we go, that at some stage we will find a prime gap of less than about 70 million.

In fact, it seems to be to imply that for some value less than the "Zhang Number" which is about 70 million, there is an infinite number of prime gaps, which fact gives some hope of proving the twin primes is infinite. But, the obvious gets no Brownie points, which is not to say that Zhang should not get the credit he deserves.

>
> > "Mohsen" <mafshin89@> wrote:
> >
> >> Unknown Mathematician Proves Elusive Property of Prime Numbers
> >> http://www.wired.com/wiredscience/2013/05/twin-primes/
> >
> > I think that "unknown", in this context, means "not known
> > to the first pundit that was asked" (in this case, perhaps,
> > Andrew Granville, who often gets asked to respond, on the hoof).
>
> but it's Wired, so the title has to appeal to the geeks and casual
> and play with the notion that even in the field of Mathematics,
> you too could become a star.
>
> Maths look more and more like a social science to me these days...
>
• ... There is no limit to the maximum gap, consider n!+2, n!+3, ... n!+n, these are all composite for large n, but it is trivial to show that up to a point x
Message 3 of 18 , May 29 7:00 AM
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>I must say that some of the popular exposition of this discovery of Zhang has been to allow the impression that there is a limit to the size of prime
>gaps, a proposition that puzzled me for a while, but which would have never occured to a Mathematician worth of the name due to its implicit dismissal of established theory.

There is no limit to the maximum gap, consider n!+2, n!+3, ... n!+n, these are all composite for large n, but it is trivial to show that up to a point x there is a maximal gap.

> All that Zhang seems to have assured us, and this is the comforting news, is that no how far along the number line we go, that at some stage we will find a prime gap of less than about 70 million.

That is correct.

> In fact, it seems to be to imply that for some value less than the "Zhang Number" which is about 70 million, there is an infinite number of
> prime gaps, which fact gives some hope of proving the twin primes is infinite. But, the obvious gets no Brownie points, which is
> not to say that Zhang should not get the credit he deserves.

To have proved the first fixed limit is an amazing result and will probably prepare the way for a long sequence of results improving the result.
• One of the points I was interested in, and perhaps you have averted to it but I missed it, is the corollary that there is at least one prime gap equal to or
Message 4 of 18 , May 29 9:16 PM
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One of the points I was interested in, and perhaps you have averted to it but I missed it, is the corollary that there is at least one prime gap equal to or less than "Zhang's Number" that has an infinity of occurences.

This probably has been proven in another way. If so, could someone please inform us where we can look it up, or better still give some detail on it.

For some reason, the number two is the one everyone is looking forward to proving, if only because is the smallest candidate left.

--- In primenumbers@yahoogroups.com, Chris Caldwell <caldwell@...> wrote:
>
> >I must say that some of the popular exposition of this discovery of Zhang has been to allow the impression that there is a limit to the size of prime
> >gaps, a proposition that puzzled me for a while, but which would have never occured to a Mathematician worth of the name due to its implicit dismissal of established theory.
>
> There is no limit to the maximum gap, consider n!+2, n!+3, ... n!+n, these are all composite for large n, but it is trivial to show that up to a point x there is a maximal gap.
>
> > All that Zhang seems to have assured us, and this is the comforting news, is that no how far along the number line we go, that at some stage we will find a prime gap of less than about 70 million.
>
> That is correct.
>
> > In fact, it seems to be to imply that for some value less than the "Zhang Number" which is about 70 million, there is an infinite number of
> > prime gaps, which fact gives some hope of proving the twin primes is infinite. But, the obvious gets no Brownie points, which is
> > not to say that Zhang should not get the credit he deserves.
>
> To have proved the first fixed limit is an amazing result and will probably prepare the way for a long sequence of results improving the result.
>
• ... least one prime gap equal to or less than Zhang s Number that has an ... this is indeed true, since there is a finite number of different possible gaps
Message 5 of 18 , May 30 12:30 AM
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On Thu, May 30, 2013 at 12:16 AM, John <mistermac39@...> wrote:

> **
> One of the points I was interested in, and perhaps you have averted to it
> but I missed it, is the corollary that there is at
>
least one prime gap equal to or less than "Zhang's Number" that has an
> infinity of occurences.
>

this is indeed true, since there is a finite number of different possible
gaps less than 71 million, and Zhang's theorem asserts that there are
infinitely many gaps of such size, so at least one of these gaps must occur
infinitely often.

>
> This probably has been proven in another way. If so, could someone please
> inform us where we can look it up, or better still give some detail on it.
>
> No, if that was proven, (that there was one gap less than Zhang's number
known to occur infinitely often), then this would have "superseeded"
Zhang's theorem.

> For some reason, the number two is the one everyone is looking forward to
> proving, if only because is the smallest candidate left.
>
why "left" ? no candidate at all has been eliminated so far...
But of course the number 2 is the ultimate challenge, it is special in
several ways, which partially may be, but aren't necessarily directly, a
consequence of the fact that its the smallest possible gap.
(For example, to name a trivial one, the pair of twin primes are
consecutive odd numbers, which is not the case for any other gap).

Maximilian

[Non-text portions of this message have been removed]
• ... Just thinking about it, falsity of the TPC would be deeply disturbing. Just imagine the concept of being given a prime, and then being able to instantly
Message 6 of 18 , May 30 6:11 AM
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--- On Thu, 5/30/13, Maximilian Hasler <maximilian.hasler@...> wrote:
> But of course the number 2 is the ultimate challenge, it is special in
> several ways, which partially may be, but aren't necessarily directly, a
> consequence of the fact that its the smallest possible gap.

Just thinking about it, falsity of the TPC would be deeply disturbing.
Just imagine the concept of being given a prime, and then being able to instantly determine the primality a different number without knowing
any of its factors (in particular, knowing that it's composite). That's even spookier than magnets.

Does anyone seriously doubt the TPC's truth?

Phil
--
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[stolen with permission from Daniel B. Cristofani]
• I think that this Zhang s theorem could led us very near to demonstrate the Polignac s conjecture. If there are infinitely many consecutive pairs of primes
Message 7 of 18 , May 30 6:22 AM
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I think that this Zhang's theorem could led us very near to demonstrate the
Polignac's conjecture.

If there are infinitely many consecutive pairs of primes with some gap < 70
millions, why not all the other number bigger than 70 millions? After
all, all bigger gaps have the same or more probabilities - say, 1 - to
exist because the gaps between the primes tend to grow and the size of the
sample we can pick numbers from is infinite.

If only some number(s) below 70 million comply... Why this excepcionality?

And there are only a finite number of gaps < 70 millions. Which is an
infinitesimal part of all possble gaps (infinite).

There shouldn't exist excepcional gaps bellow 70 millions either.

If there are infinitely many gaps > 70 millions, the rest of gaps below 70
millions ought to exist too. I think.

On Thu, May 30, 2013 at 9:30 AM, Maximilian Hasler <
maximilian.hasler@...> wrote:

> **
>
>
> On Thu, May 30, 2013 at 12:16 AM, John <mistermac39@...> wrote:
>
> > **
>
> > One of the points I was interested in, and perhaps you have averted to it
> > but I missed it, is the corollary that there is at
> >
> least one prime gap equal to or less than "Zhang's Number" that has an
> > infinity of occurences.
> >
>
> this is indeed true, since there is a finite number of different possible
> gaps less than 71 million, and Zhang's theorem asserts that there are
> infinitely many gaps of such size, so at least one of these gaps must occur
> infinitely often.
>
>
> >
> > This probably has been proven in another way. If so, could someone please
> > inform us where we can look it up, or better still give some detail on
> it.
> >
> > No, if that was proven, (that there was one gap less than Zhang's number
> known to occur infinitely often), then this would have "superseeded"
> Zhang's theorem.
>
>
> > For some reason, the number two is the one everyone is looking forward to
> > proving, if only because is the smallest candidate left.
> >
> why "left" ? no candidate at all has been eliminated so far...
> But of course the number 2 is the ultimate challenge, it is special in
> several ways, which partially may be, but aren't necessarily directly, a
> consequence of the fact that its the smallest possible gap.
> (For example, to name a trivial one, the pair of twin primes are
> consecutive odd numbers, which is not the case for any other gap).
>
> Maximilian
>
> [Non-text portions of this message have been removed]
>
>
>

[Non-text portions of this message have been removed]
• Hello, ... I don t and as mentioned by Jose, the de Polignac conjecture should be addressed too. I have serious reasons to think that TPC and dPC require the
Message 8 of 18 , May 30 9:05 AM
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Hello,

Le 2013-05-30 15:11, Phil Carmody a écrit :
> --- On Thu, 5/30/13, Maximilian Hasler <maximilian.hasler@...>
> wrote:
>> But of course the number 2 is the ultimate challenge, it is special
>> in
>> several ways, which partially may be, but aren't necessarily
>> directly, a
>> consequence of the fact that its the smallest possible gap.
>
> Just thinking about it, falsity of the TPC would be deeply
> disturbing.
> Just imagine the concept of being given a prime, and then being able
> to instantly determine the primality a different number without
> knowing
> any of its factors (in particular, knowing that it's composite).
> That's even spookier than magnets.
>
> Does anyone seriously doubt the TPC's truth?

I don't and as mentioned by Jose, the de Polignac conjecture
should be addressed too. I have serious reasons to think that
TPC and dPC require the same demonstration and will be proved
at the same time. It's easier than you might think but it is
still a lot of work.

> Phil
Yann
• I agree, If we take two consecutive primes separated by a gap n, and prove that there is an infinite quantity of consecutive pairs separated by such a gap (as
Message 9 of 18 , May 30 11:24 AM
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I agree,

If we take two consecutive primes separated by a gap n, and prove that
there is an infinite quantity of consecutive pairs separated by such a gap
(as Dr. Zhang has proven for a gap below 70 million), it would be true too
for every other n because every imaginable n is equally infinitesimal with
respect infinite.

On Thu, May 30, 2013 at 6:05 PM, <whygee@...> wrote:

> **
>
>
> Hello,
>
> Le 2013-05-30 15:11, Phil Carmody a �crit :
>
> > --- On Thu, 5/30/13, Maximilian Hasler <maximilian.hasler@...>
> > wrote:
> >> But of course the number 2 is the ultimate challenge, it is special
> >> in
> >> several ways, which partially may be, but aren't necessarily
> >> directly, a
> >> consequence of the fact that its the smallest possible gap.
> >
> > Just thinking about it, falsity of the TPC would be deeply
> > disturbing.
> > Just imagine the concept of being given a prime, and then being able
> > to instantly determine the primality a different number without
> > knowing
> > any of its factors (in particular, knowing that it's composite).
> > That's even spookier than magnets.
> >
> > Does anyone seriously doubt the TPC's truth?
>
> I don't and as mentioned by Jose, the de Polignac conjecture
> should be addressed too. I have serious reasons to think that
> TPC and dPC require the same demonstration and will be proved
> at the same time. It's easier than you might think but it is
> still a lot of work.
>
> > Phil
> Yann
>
>
>

[Non-text portions of this message have been removed]
• ... --I suspect Zhang s proof could easily be modified to show that there are an infinite set of integers K 0, such that, for each K in the set, there are an
Message 10 of 18 , May 30 2:43 PM
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> If there are infinitely many consecutive pairs of primes with some gap < 70
> millions, why not all the other number bigger than 70 millions?

--I suspect Zhang's proof could easily be modified to show that there are an infinite set of
integers K>0, such that, for each K in the set, there are an infinite set of primes P
with P+K simultaneously prime.

Zhang's present proof creates a set S of integers within [2, 70000000]
and proves that are an infinity of N such that N+S contains at least two primes.

Instead create a finite set T of integers with all gaps between set members >70000000
then prove there are an infinity of N such that N+T contains at least two primes...

then continue on, each set having min-gaps > the previous set's max-element.

This might be a good project for anybody trying to understand the Zhang proof.
• A line of thought that occurs to me to advance the cuase for the truth of TPC is this. The number of primes of the form 6n-1 and 6n-1 tends to be equal in the
Message 11 of 18 , May 30 6:55 PM
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A line of thought that occurs to me to advance the cuase for the truth of TPC is this.

The number of primes of the form 6n-1 and 6n-1 tends to be equal in the long run.

It the TPC is not true, then beyond that n which constitutes the largest higher twin 6n+1, then every prime of the form 6n-1 demands that the next gap will be at least 6. This creates a bias towards primes 6n+7, 6n+13, 6n+19,...... for all n greater than the maximum n referred to, and away from 6n+1 which is no longer permissible.

Thus the form 6n+1 would seem to have to compensate somehow by a process unknown, or covered by the word infinitesmally, which as in the delta y and delta x in differential calculus is conveniently disregarded with no harm to the calculus itself, so we are assured as the dy/dx is a limit to the ratio of delta y and delta x.

How does the form 6n+1 "catch up" if the TPC be not true? That is my question.

I hope the question is not too spooky.

John

> --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@> wrote:
> >
> > --- On Thu, 5/30/13, Maximilian Hasler <maximilian.hasler@> wrote:
> > > But of course the number 2 is the ultimate challenge, it is special in
> > > several ways, which partially may be, but aren't necessarily directly, a
> > > consequence of the fact that its the smallest possible gap.
> >
> > Just thinking about it, falsity of the TPC would be deeply disturbing.
> > Just imagine the concept of being given a prime, and then being able to instantly determine the primality a different number without knowing
> > any of its factors (in particular, knowing that it's composite). That's even spookier than magnets.
> >
> > Does anyone seriously doubt the TPC's truth?
> >
> > Phil
> > --
> > () ASCII ribbon campaign () Hopeless ribbon campaign
> > /\ against HTML mail /\ against gratuitous bloodshed
> >
> > [stolen with permission from Daniel B. Cristofani]
> >
>
• ... Well, if you can assume something spooky enough to fundamentally unbalance twins, there s no reason to expect it to not rebalance other gap sizes to
Message 12 of 18 , May 30 10:22 PM
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--- On Fri, 5/31/13, John <mistermac39@...> wrote:
> How does the form 6n+1 "catch up" if the TPC be not true?
> That is my question.

Well, if you can assume something spooky enough to fundamentally unbalance twins, there's no reason to expect it to not rebalance other gap sizes to compensate.

Phil
--
() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]
• ... Indeed. Very insightful! I did take a peek at the paper, and whilst I am in awe at the directness of the first paragraph of the abstract, I did notice that
Message 13 of 18 , May 30 10:36 PM
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--- On Fri, 5/31/13, WarrenS <warren.wds@...> wrote:
> > If there are infinitely many consecutive pairs of primes with some gap < 70
> > millions, why not all the other number  bigger than 70 millions?
>
> --I suspect Zhang's proof could easily be modified to show
> that there are an infinite set of
> integers K>0, such that, for each K in the set, there are
> an infinite set of primes P
> with P+K simultaneously prime.
>
> Zhang's present proof creates a set S of integers within [2, 70000000]
> and proves that are an infinity of N such that N+S contains
> at least two primes.
>
> Instead create a finite set T of integers with all gaps
> between set members >70000000
> then prove there are an infinity of N such that N+T contains
> at least two primes...
>
> then continue on, each set having min-gaps > the previous
> set's max-element.
>
> This might be a good project for anybody trying to
> understand the Zhang proof.

Indeed. Very insightful!

I did take a peek at the paper, and whilst I am in awe at the directness of the first paragraph of the abstract, I did notice that the body of the paper seemed to contain an outragious number magic numbers, and it wasn't clear where they came from. And no, I'm not concerned about 1,2,3,pi, and phi - it's 19, 292 and 293, 100, 88.4, and 48 and 56 - presumably all those are parametrised, and would need to be worked out again - and they might not lead to the same conclusions, of course.

And 32 sigmas on a single page - woh, that must be a record!

Phil
--
() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]
• ... 32 sigmas ? Wow ! we in the computer field struggle to reach 6 sigmas ;-) (though the LHC guys have between 5 and 7 sigmas on the Higgs boson) ... yg
Message 14 of 18 , May 30 10:45 PM
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Le 2013-05-31 07:36, Phil Carmody a écrit :
> And 32 sigmas on a single page - woh, that must be a record!

32 sigmas ?
Wow ! we in the computer field struggle to reach 6 sigmas ;-)
(though the LHC guys have between 5 and 7 sigmas on the Higgs boson)

> Phil
yg (still sieving...)
• Good point you make. Probably the 4-gaps get a filip also. That is if they have not already ceased to be around to balance the 6n+1 forms and the 6n-1. In all
Message 15 of 18 , May 31 1:54 AM
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Good point you make. Probably the 4-gaps get a filip also. That is if they have not already ceased to be around to balance the 6n+1 forms and the 6n-1.

In all this it must be remembered, at least by me, that rhere are still plenty of greater than "Zhang Number" gaps to help out yhe balancing act.

The fabled prime race may get a bit more attention as a result of all this, but it seems that at the smaller numbers, the lead does not change very often from what I have read.

John

--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- On Fri, 5/31/13, John <mistermac39@...> wrote:
> > How does the form 6n+1 "catch up" if the TPC be not true?
> > That is my question.
>
> Well, if you can assume something spooky enough to fundamentally unbalance twins, there's no reason to expect it to not rebalance other gap sizes to compensate.
>
> Phil
> --
> () ASCII ribbon campaign () Hopeless ribbon campaign
> /\ against HTML mail /\ against gratuitous bloodshed
>
> [stolen with permission from Daniel B. Cristofani]
>
• ... Let s raise the stakes. Does anyone seriously doubt the truth of the Bateman-Horn conjecture?
Message 16 of 18 , May 31 4:08 AM
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