Re: twin prime conjecture

Hi Professor Haugland,

Thank you very much for your comment. I agree with you about the
mistake that is commonly made regarding the density of primes (just do
a Google search on the twin primes conjecture) but I don't believe that
your analysis applies to my argument.

If you take a look at Lemma 5 in my proof, for example, you will see
that the argument hinges on gcd(p,6)=1 and the argument is specific to
integers of the form 6(i+c) + 3 + h where h=2 or 4.

If you apply my argument to your example, you will see that my claim is
upheld.

I make the claim that you can apply my floor function only when you are
analyzing twin pairs (that is, numbers where x is congruent to 2 modulo
3 and y is congruent to 1 modulo 3 and y = x + 2).

Let's look at the twin pairs in the range 114 to 126 ("twin pair" is my
definition, I apologize for the nonstandard nature of it but I couldn't
think of a better term. Perhaps, "partial twin prime" since they are
relatively prime to 6 or "potential twin prime" would be better; I
rejected these other terms because I thought that they were too wordy)

The closest integer of the form is 111 so if we set i = 18, we get
6*18+3=111. I am also assume that the upper bound is of the form 6i+3
so I will use 123 which is 6*20+3.

So my analysis is regard to the number of twin pairs that are
relatively prime to 2,3,5,7,11 is equal to the total number of twin
pairs in the range multiplied by [(5-2)/5][(7-2)/7][(11-2)/11]

The total number of twin pairs (from Lemma 4) is: (123-111)/6 = 2.

So my formula for the minimum number of twin pairs relatively prime to
2,3,5,7,11 is:

floor(2*[3/5][5/7][9/11]) = 0

I apologize if I misunderstood your argument. Please let me know if
you have questions about any of my points. You might want to look at
my exchange with John Roberts-Jones for more details on the
justification for my formula:
http://groups.google.com/group/sci.math/browse_frm/thread/650388de35e91582/12cf790b87ab6022?lnk=raot#12cf790b87ab6022

Thank you very much for your feedback! If you have any suggestion for
how I can make it clearer about my assumptions regarding the prediction
of density of twin primes and I will be glad to update the proof. :-)

I suspect that many number theorists will assume that I am making a
probability argument when in fact I am not. There is no probability in
this argument. 1 argument is based on congruences modulo 6 (that is
where gcd(p,6)=1 comes in, see Lemma 5) and 1 argument is based on a
complete residue system (that is where the
floor(c*[p1-2/p1][p2-2/p2]*...*[pn-2/pn]) comes in (see Lemma 8).

Cheers,

-Larry


jankrihau@xxxxxxxxxxx wrote:
larry.freeman@xxxxxxxxx wrote:
Hi Math Experts,

My name is Larry Freeman and I am a math amateur. I have a draft of a
proposed proof of the twin primes conjecture which can be found here:
http://proposedproofs.blogspot.com/2006/10/proposed-proof-twin-prime-conjecture.html

I would greatly appreciate it if someone who is an expert in number
theory could take a look at it and explain to me where I went wrong.
This solution came too easily to me for it to be correct.

You are quite right about that. :-) The problem is the step from the
fact that the density of "pairs of odd integers relatively prime to 6
that are not divisible by any of the primes p1 through pn" is

(*) [1 - 2/p1][1 - 2/p2]*...*[1-2/pn]

(which appears to be correct) to the claim that the number of such
pairs among c consecutive pairs is floor(c*[1 - 2/p1][1 -
2/p2]*...*[1-2/pn]). It's just not that simple. I think you may have
been deceived by the use of a very crude lower bound for (*). It is
possible to give an asymptotically correct esimate of (*), and your
claim would then imply that the twin primes (or primes, with a modified
argument) are extremely evenly distributed. But that's not the case.

How about estimating the number of integers between 114 and 126 that
are coprime to 2, 3, 5, 7 and 11? Note that floor(13 *
(1-1/2)*(1-1/3)*(1-1/5)*(1-1/7)*(1-1/11)) = 2. But an integer less than
13^2 that is coprime to 2, 3, 5, 7 and 11 must be 1 or a prime, and
there are no primes in the interval.

---
J K Haugland
http://home.no.net/zamunda

.

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