Twin Prime Sieve - A Unique and Efficient Method for Generating Twin Primes

[Rudy Toody]

http://forums.anandtech.com/messageview.aspx?catid=50&threadid=2331345&enterthread=y"

I discovered this sieve a couple of weeks ago and could not find anything similar in the liturature. I made a very ugly proof and sent it off to JAMS, where it was rejected immediately.

Since then, we have been discussing this sieve.

One poster suggested that I come over here to get more insights.

If one of you could check out that thread and post here about the viability of the sieve, it would be of great help.

Thanks.

 

[CRGreathouse]

Finding the first 8169 twin primes (those below a million) isn't hard. Mathematicians have searched 1,000,000,000,000 times further: see for example Tables of values of pi(x) and of pi2(x).

Critiques on http://pics.bbzzdd.com/users/Rudy_Toody/SimpleTwinPrimeSieve.jpg :
Your justification for (a) is wrong. The step is correct, but this is a celebrated theorem of Euclid, not a definition.
(b) is also not true by definition.
I don't see any justification for (h) or (g).

I can't read Modification 6 in the jpg, so I can't say too much more.

 

[Rudy Toody]

Eratosthenes requires that you cycle through the sieve until the next uncrossed number is found. That is the prime.

Euler's Sieve requires handling each number only once. It is based on Eratosthenes.

My sieve is based on Euler's Sieve in that it handle numbers only twice, because I am sieving twin primes and using the non-twins to finish the sieving.

 

[robert Ihnot]

I could not understand b either. Under 2, you say Select the prime,(the lowest number >1 from N). But, the lowest number need not be prime. I assume you thought of these sets as changing from operation to operation, but that is not clear.

Remove (N intersection P) from N, finally...repeat step 2. Well, now we have a different N, like say, N'.
Furthermore, the writing in Modificaton 6, under 4, is so faint, I don't know what it says.

Usually these kind of things involve a lot of ambiguity in their "proofs."

 

[Rudy Toody]

If you ignore the bad proof and read further down the thread you will find the explanation of what I meant to say.

I have a newer sieve that provides more information to be used for a proof. I have only shown the Mathematica program that does the sieving. It is fairly simple, too, It has been changed in one small regard since it was posted. However, the change has not changed the logic of the program. It has only changed how the program relates to the upcoming proof.

This was my first attempt at a proof, and I am self taught, and so it was ugly and incorrect. The next one will be much better.

 

[CRGreathouse]

When you say "dual twin primes" I assume you mean prime quadruplets (p, p+2, p+6, p+8). In that case your list to 100,000 is correct, apart from missing (in your format) "{{5,7},{11,13}}" which I imagine is intentional.

These have been counted up to 10¹⁶; see Enumeration of the prime quadruplets to 1e16.

I didn't see the newer version of the proof you mentioned in the other thread, just new Mathematica code. I'm not too good with Mathematica -- Pari/GP is better at number theory, so that's what I use -- so I'll admit your code didn't look easy to figure out.