Twin Primes

[Hugh]

I've not posted here before but have been an interested reader. I wanted to run an attempted proof of the Twin Prime Conjecture (and perhaps Polignac's Conjecture in general) past the forum members - I know that most attempted proofs are flawed. But I'm optimistic enough about this attempt to try to get some feedback.

It's related to the infinite product fraction Euler inverted to show that the reciprocals of the primes diverge. There's a fairly brief summary at the start, then more detail beneath. If you do have time to take a look I'd be extremely grateful.

http://barkerhugh.blogspot.com/2011/...onjecture.html

[Hugh]

I've put up a compressed version of this proof here:

http://barkerhugh.blogspot.com/2011/...d-version.html

[science_man_88]

Originally Posted by Hugh

I've not posted here before but have been an interested reader. I wanted to run an attempted proof of the Twin Prime Conjecture (and perhaps Polignac's Conjecture in general) past the forum members - I know that most attempted proofs are flawed. But I'm optimistic enough about this attempt to try to get some feedback.

It's related to the infinite product fraction Euler inverted to show that the reciprocals of the primes diverge. There's a fairly brief summary at the start, then more detail beneath. If you do have time to take a look I'd be extremely grateful.

http://barkerhugh.blogspot.com/2011/...onjecture.html

I have google chrome developer tools  works on my end.

[R.D. Silverman]

Originally Posted by Hugh

I've not posted here before but have been an interested reader. I wanted to run an attempted proof of the Twin Prime Conjecture (and perhaps Polignac's Conjecture in general) past the forum members - I know that most attempted proofs are flawed. But I'm optimistic enough about this attempt to try to get some feedback.

It's related to the infinite product fraction Euler inverted to show that the reciprocals of the primes diverge. There's a fairly brief summary at the start, then more detail beneath. If you do have time to take a look I'd be extremely grateful.

http://barkerhugh.blogspot.com/2011/...onjecture.html

It is total nonsense.

[Hugh]

Originally Posted by R.D. Silverman

It is total nonsense.

Thanks for looking - but would you mind letting me know what it is about it you regard as nonsense? Happy to take criticism, but it would be nice to get some detail.

[Hugh]

Originally Posted by R.D. Silverman

It is total nonsense.

Incidentally, it might be worth looking at the compressed version as it leaves out some unnecessary detail:

http://barkerhugh.blogspot.com/2011/...d-version.html

[R.D. Silverman]

Originally Posted by Hugh

Thanks for looking - but would you mind letting me know what it is about it you regard as nonsense? Happy to take criticism, but it would be nice to get some detail.

It is just handwaving with bad notation, failure to conform to accepted
mathematical vocabulary, failure to properly define your own terminology,
failure to properly define variables, failure to accurately state internal
lemmas, etc. etc. Need I go on? It simply isn't a proof. It is nonsense.

The question is not "what is wrong", but rather "what is right?" There is
very little that is right.

This paper is "not even wrong".

[axn]

Originally Posted by the site

There must be a “last prime” that, when added to the pattern of composite symmetries, casts out the final remaining pair (or pairs) of twin prime candidates. (Meaning that no prime larger than this is needed to cast out further twin prime candidates).

This is where your proof breaks down. Why "must" there be such a "last prime"?

[Hugh]

Originally Posted by R.D. Silverman

It is just handwaving with bad notation, failure to conform to accepted
mathematical vocabulary, failure to properly define your own terminology,
failure to properly define variables, failure to accurately state internal
lemmas, etc. etc. Need I go on? It simply isn't a proof. It is nonsense.

The question is not "what is wrong", but rather "what is right?" There is
very little that is right.

This paper is "not even wrong".

OK, happy to accept that I'm not familiar enough with how people expect proofs to be laid out. I don't think there is anything there that is actually that unclear though, so I think all you're really saying here is "it made me grumpy that it wasn't laid out in the correct way so I dismissed it".

I can understand that, but I'm hoping for a more genuine engagement with the actual argument which I think does have some interest, although I am prepared to accept it may have flaws.

For instance: Do you disagree that the symmetries of twin prime pairs formed by multiples of sets of primes exist? Do you disagree that there must always be infinite gaps in the twin prime symmetry formed by a finite set of primes? Or do you disagree with the final step in the argument...

[Hugh]

Originally Posted by axn

This is where your proof breaks down. Why "must" there be such a "last prime"?

And many thanks to you for bothering to engage more closely with the argument in spite of my informal layout.

If there is a flaw this may well be where it lies. Certainly that step needs a bit more scrutiny. However, I'd argue that it may be correct for a few reasons.

First, how does one ever move from the situation in which there are an infinite number of gaps in the twin prime pair symmetry to a situation in which there are none if there is no such "last prime."

Second, one could argue this

"All pairs of numbers will end up in a twin prime pair symmetry (either as the initial pair or by being cast out by another prime). Since, as we go up through the sieve, no prime can ever remove all the subsequent pairs, there must always remain more pairs that will have to be removed by becoming the initial twin prime within a "composite symmetry."

[Hugh]

Originally Posted by axn

This is where your proof breaks down. Why "must" there be such a "last prime"?

Or one more reverse way of looking at it (a little experimental...)

If we start by assuming "there is an infinite set of primes, but a highest pair of twin primes".

Then we gradually remove primes from the twin prime pattern.

At a certain point then either:

We must be able to find "the largest set of primes for which there is a highest pair of twin primes." (Meaning that having removed all other primes from the sieve there will still be a highest pair of twin primes). In which case the next prime we remove from the sieve, which results in an infinite repeating pattern of "twin prime pair candidates" is "the last prime."

or

It is impossible to remove even one prime from the sieve without the twin prime candidate pairs becoming infinite. In which case that prime is "the last prime".

I don't see any other ways that the infinite symmetry left by a finite set can be reconciled with the initial assumption: that "there is an infinite set of primes, but a highest pair of twin primes".

Therefore I can't see how this statement could be true given the rest of my argument.

Does that help any??