mersenneforum.org  

Go Back   mersenneforum.org > Prime Search Projects > Twin Prime Search

Reply
 
Thread Tools
Old 2016-04-01, 22:23   #1
pythonic
 
Apr 2016

116 Posts
Default Largest known prime p such that 6p-1 and 6p+1 are prime?

What is the largest known prime p such that 6p-1 and 6p+1 are twin primes? Some primes of this form are listed in the OEIS at http://oeis.org/A060212.
pythonic is offline   Reply With Quote
Old 2016-04-02, 01:30   #2
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22·2,281 Posts
Default

This is easy to make with a modified NewPgen.
There is a sieve mode for triple sets, but you need to change the multiplier to 6 in the code (then use mode).
It is likely that you can easily find a p with at least 10,000 or 20,000 digits.

The proof of course will not be a problem, because p will be a helper for 6p-1 and 6p+1 regardless of their form.
Batalov is offline   Reply With Quote
Old 2016-04-02, 05:40   #3
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22×2,281 Posts
Default

Here is a small one for you, for starters:
9975332*997#/35+1 = p (421 digits)
59851992*997#/35+5 = 6p-1
59851992*997#/35+7 = 6p+1
_____________________________

Then, twice larger...
14411087*2003#/35+1 (852 digits)
twin pair = 86466522*2003#/35+5, 86466522*2003#/35+7
_____________________________

And over 1000+ digits:
547561666*3001#/210+1 (1284 digits)
twin pair = 547561666*3001#/35+5, 547561666*3001#/35+7
_____________________________

And over 2000 digits:
p = 251637551*2^6666-1
twin pair = 3*251637551*2^6667-5, 3*251637551*2^6667-7

Last fiddled with by Batalov on 2016-04-03 at 02:36
Batalov is offline   Reply With Quote
Old 2016-04-05, 18:58   #4
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22·2,281 Posts
Default

2280196563*2^9982+1 = p (3015 digits)
and twin pair =
3*2280196563*2^9983+5 = 6p-1
3*2280196563*2^9983+7 = 6p+1
Batalov is offline   Reply With Quote
Old 2016-04-06, 18:10   #5
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

10010101002 Posts
Default

There is a table at the University of Tennessee Martin webpage with largest twin primes known -

https://primes.utm.edu/top20/page.php?id=1

The largest is
3756801695685 · 2666669 - 1 with 200700 digits.

Regards
Matt
MattcAnderson is offline   Reply With Quote
Old 2016-04-06, 18:22   #6
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

216448 Posts
Default

For all of the values in that table, is x = {{p_{TWIN} + 1} \over 6} prime? (So that the twins would be 6x-1 and 6x+1, you know?)

Hint: no.
Batalov is offline   Reply With Quote
Old 2016-04-06, 19:27   #7
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by Batalov View Post
For all of the values in that table, is x = {{p_{TWIN} + 1} \over 6} prime? (So that the twins would be 6x-1 and 6x+1, you know?)

Hint: no.
that's one way to look at it (aka are those twin primes known, not can the twin primes be proven to exist) the other way to look at is are any of the largest known primes (including those primes) not of the form such that 6p+1 or 6p-1 have to be composite.
science_man_88 is offline   Reply With Quote
Old 2016-04-08, 17:48   #8
PawnProver44
 
PawnProver44's Avatar
 
"NOT A TROLL"
Mar 2016
California

3058 Posts
Post

I believe there are infinitely many primes of the form 6q+1, 6q-1, for prime q. Don't know how to prove this.

Likewise, there should be infinitely many primes q such that 15*q-4, 15*q-2, 15*q+2, 15*q+4, are all prime. Still don't know how to prove this.

In fact, This is probable the case of minimum sets, where n consecuative integers have the n smallest factors.

For your problem this is equivalent that there are infinitely many primes q such that:

2q-1, q, and (2q+1)/3 are prime.

(2q-1)/3, q and 2q+1 are prime.

6q-1 and 6q+1 are prime.

Anyone please find an example for this.

Last fiddled with by PawnProver44 on 2016-04-08 at 17:48
PawnProver44 is offline   Reply With Quote
Old 2016-04-08, 18:12   #9
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

20C016 Posts
Default

Quote:
Originally Posted by PawnProver44 View Post
Likewise, there should be infinitely many primes q such that 15*q-4, 15*q-2, 15*q+2, 15*q+4, are all prime. Still don't know how to prove this.
well a possible start would be to show all the cases possible

q=2,3,6x-1,6x+1 and what each is equivalent to.


case q=2:

all parts are even so the result would be even so q=2 fails to meet the requirements

q=3:

produces 49 for the last one so q=3 is out.

q=6x-1:

produces : 90x-19, 90x-17,90x-13,90x-11

q=6x+1

produces: 90x+11,90x+13,90x+17,90x+19

now you need to show that for any x values that these are all prime create primes 6x+1 or 6x-1 or both infinitely often. x must already be of a certain form for 6x-1 or 6x+1 to be composite so prove infinitely often that these forms are not met ?

Last fiddled with by science_man_88 on 2016-04-08 at 18:15
science_man_88 is offline   Reply With Quote
Old 2016-04-08, 20:26   #10
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

10111000111112 Posts
Default

Quote:
Originally Posted by PawnProver44 View Post
I believe there are infinitely many primes of the form 6q+1, 6q-1, for prime q. Don't know how to prove this.
You shouldn't be able to -- this is a hard problem. Lots of progress has been made toward it in recent years
http://arxiv.org/abs/math/0606088
http://arxiv.org/abs/1009.3998
http://arxiv.org/abs/1409.1327
http://arxiv.org/abs/1511.04468
http://arxiv.org/abs/1603.07817
but we're still far from proving Dickson's conjecture.
CRGreathouse is offline   Reply With Quote
Old 2016-04-08, 20:52   #11
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

22×2,281 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
... x must already be of a certain form for 6x-1 or 6x+1 to be composite so prove infinitely often that these forms are not met ?
In addition to what Charles wrote, infinitude of composite members of any sequence has nothing to do with the infinitude of prime members.

sm88, you need to start thinking at least about the basic properties of infinities: that is, (even for complementary sets, e.g. a set "A" and a set of all others "non-A"):
- a sum of a finite set and and infinite set is an infinite set;
- a sum of an infinite set and and infinite set is an infinite set, so
- knowing that one subset is infinite (e.g. "non-A") gives you no information about its complement's infinitude.
Batalov is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
probable largest prime. sudaprime Miscellaneous Math 11 2018-02-05 08:10
NEW MERSENNE PRIME! LARGEST PRIME NUMBER DISCOVERED! dabaichi News 561 2013-03-29 16:55
Largest known prime Unregistered Information & Answers 24 2008-12-13 08:13
Largest 64 bit prime? amcfarlane Math 6 2004-12-26 23:15
need Pentium 4s for 5th largest prime search (largest proth) wfgarnett3 Lounge 7 2002-11-25 06:34

All times are UTC. The time now is 08:50.

Fri Sep 18 08:50:15 UTC 2020 up 8 days, 6:01, 0 users, load averages: 1.97, 1.88, 1.80

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.