20110110, 19:39  #1 
Jan 2011
24_{10} Posts 
Twin Primes
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 
20110111, 08:58  #2 
Jan 2011
2^{3}·3 Posts 
I've put up a compressed version of this proof here:
http://barkerhugh.blogspot.com/2011/...dversion.html 
20110111, 16:52  #3  
"Forget I exist"
Jul 2009
Dartmouth NS
2·3·23·61 Posts 
Quote:


20110111, 17:06  #4  
"Bob Silverman"
Nov 2003
North of Boston
2·3^{3}·139 Posts 
Quote:


20110111, 17:19  #5 
Jan 2011
11000_{2} Posts 

20110111, 17:20  #6 
Jan 2011
2^{3}·3 Posts 
Incidentally, it might be worth looking at the compressed version as it leaves out some unnecessary detail:
http://barkerhugh.blogspot.com/2011/...dversion.html 
20110111, 17:59  #7  
"Bob Silverman"
Nov 2003
North of Boston
2·3^{3}·139 Posts 
Quote:
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". 

20110111, 19:16  #8  
Jun 2003
5×1,087 Posts 
Quote:


20110111, 19:30  #9  
Jan 2011
2^{3}·3 Posts 
Quote:
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... Last fiddled with by Hugh on 20110111 at 19:37 

20110111, 19:34  #10  
Jan 2011
2^{3}×3 Posts 
Quote:
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." Last fiddled with by Hugh on 20110111 at 19:37 

20110111, 19:44  #11  
Jan 2011
30_{8} Posts 
Quote:
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?? 

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