mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2019-11-23, 00:02   #1
jzakiya
 
Jul 2014

22·3 Posts
Default On the Infinity of Twin Prime and other K-tuples

Abstract: 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.



https://www.academia.edu/41024027/On...other_K-tuples


https://www.scribd.com/document/4364...other-K-tuples

Last fiddled with by jzakiya on 2019-11-23 at 00:03
jzakiya is offline   Reply With Quote
Old 2019-11-23, 17:28   #2
Viliam Furik
 
Jul 2018
Martin, Slovakia

22×3×7 Posts
Default

Is this what I think it is? (The proof of twin prime conjecture?)
Viliam Furik is offline   Reply With Quote
Old 2019-11-23, 17:36   #3
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

27·71 Posts
Default

Quote:
Originally Posted by Viliam Furik View Post
Is this what I think it is? (The proof of twin prime conjecture?)
No, it's not.

MOD NOTE: thread is moved to Misc.Math subforum
Batalov is offline   Reply With Quote
Old 2019-11-23, 18:20   #4
Viliam Furik
 
Jul 2018
Martin, Slovakia

22·3·7 Posts
Default

It definitely looks like one...
Viliam Furik is offline   Reply With Quote
Old 2019-12-13, 23:47   #5
jzakiya
 
Jul 2014

22·3 Posts
Default

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

On The Infinity of Twin Primes and other K-tuples

https://www.academia.edu/41024027/On...eraction_paper
jzakiya is offline   Reply With Quote
Old 2019-12-13, 23:52   #6
sweety439
 
sweety439's Avatar
 
Nov 2016

87B16 Posts
Default

This is a special case of Schinzel's hypothesis H.

Last fiddled with by sweety439 on 2019-12-13 at 23:55
sweety439 is offline   Reply With Quote
Old 2020-06-14, 17:10   #7
jzakiya
 
Jul 2014

22·3 Posts
Default

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 P41, 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 is offline   Reply With Quote
Old 2020-06-25, 22:56   #8
jzakiya
 
Jul 2014

1210 Posts
Default

Update

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.


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')


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 280,323,050 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.

Last fiddled with by jzakiya on 2020-06-25 at 23:03
jzakiya is offline   Reply With Quote
Old 2020-07-08, 15:27   #9
jzakiya
 
Jul 2014

148 Posts
Default

Update: Wed 2020/7/8

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.

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)


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 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Twin Prime Constellations robert44444uk Prime Gap Searches 21 2019-06-09 12:51
How do you efficiently sieve for prime 3/4-tuples? Puzzle-Peter Software 156 2019-06-03 20:19
find very easy twin prime in the infamy twin primes hal1se Miscellaneous Math 13 2018-11-05 16:34
Highest Prime is also a twin prime... NOT hydeer Lone Mersenne Hunters 9 2018-04-03 22:54
Twin Prime Days, Prime Day Clusters cuBerBruce Puzzles 3 2014-12-01 18:15

All times are UTC. The time now is 15:11.

Sat Aug 15 15:11:42 UTC 2020 up 2 days, 11:47, 0 users, load averages: 1.65, 1.85, 1.88

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.