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 jzakiya 2019-11-23 00:02

On the Infinity of Twin Prime and other K-tuples

[B]Abstract:[/B] The paper uses the structure and math of Prime Generators to show there are an infinity of twin primes, proving the Twin Primes Conjecture, as well as establishing the infinity of other k-tuples of primes.

In 1849 Polignac conjectured that for any consecutive pairs of primes there are an infinity of them that differ by any even value n. The TPC is the specific case for prime pairs that differ by 2.

Using the properties of Prime Generators I show we can directly generate all the primes and then directly examine and count the gaps between them to establish there are an infinitely increasing frequency of all gaps n. Data and graphs are provided to empirically show this.

I also provide a deterministic expression to exactly determine the number of residue gaps of size 2 and 4, which are equal and odd for all prime generators Pn, and a simple deterministic expression to estimate their numbers for all Pn within their interval p to p*p, which are all primes.

[URL]https://www.scribd.com/document/436414788/On-the-Infinity-of-Twin-Primes-and-other-K-tuples[/URL]

 Viliam Furik 2019-11-23 17:28

Is this what I think it is? (The proof of twin prime conjecture?)

 Batalov 2019-11-23 17:36

[QUOTE=Viliam Furik;531328]Is this what I think it is? (The proof of twin prime conjecture?)[/QUOTE]
No, it's not.

[COLOR="DarkRed"]MOD NOTE: thread is moved to Misc.Math subforum[/COLOR]

 Viliam Furik 2019-11-23 18:20

It definitely looks like one...

 jzakiya 2019-12-13 23:47

I've posted an updated version of my paper, which now includes an Appendix.

[I]On The Infinity of Twin Primes and other K-tuples[/I]

 sweety439 2019-12-13 23:52

This is a special case of [URL="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H"]Schinzel's hypothesis H[/URL].

 jzakiya 2020-06-14 17:10

As an aide to help focus people's understanding of the paper I've created this Quiz.
The answers to the questions come directly from the content in the paper.
If you can answer these (fairly easy) questions then you have a good grasp of the paper.
People can email me any questions, and their answers, or post them here.

The following questions pertain to the Prime Generator [B]P41[/B], unless otherwise stated.
1) What is its modulus value?
2) How many residues does it have?
3) What is the value of its first residue?
4) What are the forms of the last two (2) residues?
5) What are the values of the last 2 residues?
6) What is the value of the 3rd from last residue?
7) What are the total number of gaps for its prime generator sequence (PGS)?
8) How many gaps of size 2 does it have?
9) How many gaps of size 4 does it have?
10) What gap size occurs with the greatest frequency?
11) What is the maximum possible gap size it can contain?
12) What is the max number of possible different gaps sizes it can contain?
13) How many primes exist within its range r0 - (r0)^2? (Use PG properties to determine)
14) What's the last prime within the range?
15) What are the approximate number of Twin|Cousin pairs in the range?
16) How many primes make up the modulus?
17) What is the modular complement to residue 103?
18) What is the gap size of the pivot element of the PGS?
19) What PG|PGS would you predict the first gap size of 1,000 would appear within?
20) How many residues does P50 have? (Hint: ETF or brute force it.)

Extra Credit
For P41, what is the ratio value: (total number of residues) / (max number of different gaps)?
What does this ratio tell you about the distribution of gaps within the PGS? (Explain reasoning)

Extra, Extra, Credit
The odd primes start with the (3, 4) k-tuple {3:5:7} which is an instance of 2 consecutive twin primes.
Is it the only one, or are there more instances of 2 consecutive twin primes? (Explain reasoning)

 jzakiya 2020-06-25 22:56

[B]Update[/B]

In the paper I give the expression to compute the gap coefficient values a1 = a2 for all prime generators.

On Tu/Wed this week (June 23/24 2020) I determined the iterative expressions to directly compute
the gap coefficients values a3, a4, and a5 (for gap sizes 6, 8, and 10) for all primes generators.

This now provides an empirical basis to estimate the occurrences of these additional gap sizes within
the range p to p^2 for all prime generators, as shown for gap size 2 and 4 for a1 = a2.

The construction of these gap coefficient expressions show again the gap values are solely a function of
the modulus primes for any generator, and are always increasing functions. Thus, it is mathematically
impossible for a gap coefficient to disappear (become '0') once it comes into existence. It can only
increase thereafter for all larger prime generators, ensuring their infinite prime pairs.

I will eventually update the paper to include this additional information, probably by extending the appendix.
I also want to see how many other gap coefficients I can find expressions for first.
I would encourage/challenge others to see if they can find expressions for other coefficients too.

Below are the computational forms for the gap coefficients a1 to a5.
Each coefficient value computed for the current generator is a function of coefficients from the previous one.

The (ai') coefficients are from the previous generator, i.e. (Pn'_ai).
pn is the prime value of the current generator to compute the coefficients for. Thus p5 = 5; p7 = 7, etc.

[B]for Pn: a1 = (a1')(pn - 2)

for Pn: a2 = (a2')(pn - 2)

for Pn: a3 = (a3')(pn - 3) + 2(a2')

for Pn: a4 = (a4')(pn - 4) + (a3')

for Pn: a5 = (a5')(pn - 5) + 2(a4') + (a3')[/B]

Examples: Refer to Fig. 3 in paper for all values.

We know from the paper: a1 = a2 = (pn - 2)#, starting with p3 = 3

Ex for a3: P7_a3 = (P5_a3)(p7 - 3) + 2(P5_a2) = (2)(7-3) + 2(3) = 8 + 6 = 14

Ex for a4: P13_a4 = (P11_a4)(p13 - 4) + (P11_a3) = (28)(13-4) + (142) = 28*9 + 142 = 394

Ex for a5: P17_a5 = (P13_a5)(p17 - 5) + 2(P13_a4) + (P13_a3) = (438)(17-5) + 2(394) + (1690) = 7734

I would encourage people to confirm these expressions generate all the values in Fig. 3, and also to
then generate the values for a few larger generators not shown.

** I also discovered a misprinted value for P23_a3, it should be 280,[B]323,[/B]050 not ...232... **

It's also been over 2 weeks since I posted the Quiz problems and I find it interesting no one is even curious
enough to try to answer any of the problems and send them to me, or post in the forum.

 jzakiya 2020-07-08 15:27

[B]Update: Wed 2020/7/8[/B]

I've now also determined the computational form for
gap coefficients a6 and a7, which are included in
the list below. I also computer generated the PGS
gaps for P37, which is also included (which took
19 hrs 42 mins, with a multi-threaded algorithm).

When I update my paper I will include all this new
information in it, along with other new findings.

Notation:
Pn is the Prime Generator you want the coefficient (an) for.
So P11_a5 is a5 for P11.
The (an') are the coefficients for the prior Pn.

[B]for Pn: a1 = (a1')(pn - 2)

for Pn: a2 = (a2')(pn - 2)

for Pn: a3 = (a3')(pn - 3) + (a2')(1) + (a1')(1)

for Pn: a4 = (a4')(pn - 4) + (a3')(1)

for Pn: a5 = (a5')(pn - 5) + (a4')(2) + (a3')(1)

for Pn: a6 = (a6')(pn - 5) + (a5')(6) - (a4')(2)

for Pn: a7 = (a7')(pn - 7) + (a6')(3) - (a5')(3) + (a4')(4)[/B]

P37
a1 = 217,929,355,875
a2 = 217,929,355,875
a3 = 293,920,842,950
a4 = 91,589,444,450
a5 = 108,861,586,050
a6 = 83,462,164,156
a7 = 34,861,119,734
a8 = 16,996,070,868
a9 = 21,218,333,416
a10 = 4,814,320,320
a11 = 5,454,179,550
a12 = 4,073,954,144
a13 = 918,069,454
a14 = 857,901,000
a15 = 535,673,924
a16 = 58,664,256
a17 = 69,404,898
a18 = 46,346,428
a19 = 7,381,190
a20 = 10,176,048
a21 = 4,153,336
a22 = 526,596
a23 = 291,342
a24 = 239,760
a25 = 91,392
a26 = 8,912
a27 = 25,320
a28 = 2,952
a28 = 1,654
a30 = 452
a31 = 26
a32 = 48
a33 = 24

 jzakiya 2020-08-17 23:40

I've released the 2nd revision to my paper, with corrections, a substantial rewrite of the proofs sections, and the addition of the new findings of gap coefficients and a section on their numerical derivations. In all, it's now 4 pages longer, though it should read easier.