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 [URL]http://oeis.org/A060212[/URL].

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.

[SPOILER]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.[/SPOILER]

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

2280196563*2^9982+1 = p (3015 digits)
and twin pair =
3*2280196563*2^9983+5 = 6p-1
3*2280196563*2^9983+7 = 6p+1

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

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

The largest is
[B]3756801695685 · 2666669 - 1 with 200700 digits.[/B]

[B]Regards[/B]
[B]Matt[/B]

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

Hint: [SPOILER]no[/SPOILER].

[QUOTE=Batalov;430882]For all of the values in that table, is [TEX]x = {{p_{TWIN} + 1} \over 6}[/TEX] [B]prime[/B]? (So that the twins would be 6x-1 and 6x+1, you know?)

Hint: [SPOILER]no[/SPOILER].[/QUOTE]

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.

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.

[QUOTE=PawnProver44;431048]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.[/QUOTE]

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 ?

[QUOTE=PawnProver44;431048]I believe there are infinitely many primes of the form 6q+1, 6q-1, for prime q. Don't know how to prove this.[/QUOTE]

You shouldn't be able to -- this is a hard problem. Lots of progress has been made toward it in recent years
[url]http://arxiv.org/abs/math/0606088[/url]
[url]http://arxiv.org/abs/1009.3998[/url]
[url]http://arxiv.org/abs/1409.1327[/url]
[url]http://arxiv.org/abs/1511.04468[/url]
[url]http://arxiv.org/abs/1603.07817[/url]
but we're still far from proving [url=https://books.google.com/books?id=i8MKAAAAIAAJ&pg=PA155]Dickson's conjecture[/url].

[QUOTE=science_man_88;431052]... x must already be of a certain form for 6x-1 or 6x+1 to be [B]composite[/B] so prove [B]infinitely often[/B] that these forms are not met ?[/QUOTE]
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 [B]no[/B] information about its complement's infinitude.

[QUOTE=Batalov;431065]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 [B]no[/B] information about its complement's infinitude.[/QUOTE]

okay thanks.


, the local king of the idiots.

For kicks and giggles, I found a larger triple set:
[URL="http://factordb.com/index.php?query=23660119537*2^18001-1"]23660119537*2^18001-1[/URL] = p (5430 digits)
[URL="http://factordb.com/index.php?query=2^18002*70980358611-5"]3*23660119537*2^18002-5[/URL] = 6p-1, twin
[URL="http://factordb.com/index.php?query=2^18002*70980358611-7"]3*23660119537*2^18002-7[/URL] = 6p+1, twin