k.Mp +/- 1
 

Hi all,

[url]https://primes.utm.edu/primes/page.php?id=98189[/url]

What are primes of this form called?

Are there any collaborative efforts at discovering primes oh this form?

Thanks in advance.

ETA Just realized that the thread title and the example don't match.
I am interested to find out about both formats.
Thanks again.

In an old 15k Search [url='https://www.mersenneforum.org/showthread.php?t=3197']thread[/url] they're called 'General Mersenne primes" (as in Top5000 "Generalized Woodall/Cullen/Fermat") but not officially accepted.

Thank you Kar_bon for the reference.

There doesn't seem to be any large scale collaborative effort for the type-s.

What I find surprising is that there seem to be no prime of the form
k.Mp +/- 1

in the top 5k primes. the closest match seems to be :
[url]https://primes.utm.edu/primes/page.php?id=118696[/url]

It is surprising because at lower 100k dd they would take minuets to prove prime using N-1 (not sure about N+1). I also think they should not be too difficult to find.

I have no 100k dd+ ones to back that up, but plenty of lower ones.:smile:

The Largest Prime of the form that I have found so far with very little effort.

*** 17507*2*(2^216091-1 )+1 ***
CPS-600-A by Rashid Naimi
** 65055 dd **
a = 17

Proven prime using PFGW as well as my own Pari-GP code.

Seems to be a new prime::smile:

Some more primes of the form that had not been reported to factorDb prior to today:

1713*2*(2^9941-1)+1
291*6*(2^9689-1 )+1
873*2*(2^9689-1)+1
4689*6*(2^4423-1 )+1
335*6*(2^4253-1 )+1

and yet some more with very few digits which had not been reported to FactorDb prior to today:

2016*6*(2^3217-1 )+1
504*6*(2^1279-1 )+1
114*6*(2^607-1 )+1
56*6*(2^89-1 )+1

Factordb finished N-1 test:
[url]http://factordb.com/index.php?query=%282%5E216091-1%29*35014%2B1[/url]
:smile:

Look at the sub forum [URL="https://www.mersenneforum.org/forumdisplay.php?f=99"]Operazione Doppi Mersennes[/URL] which look for factors of double mersenne numbers. They are not interested in _all_ primes of the form k.Mp+1, but only those that can divide M(Mp) (hence 1 or 7 (mod 8))

Well, that leaves more quickly proven primes for the rest of us. Compare proving a 65k dd prime in less a minute, to months (or is it years) required for a general prime of the same size. The whole point of this thread is to wonder why no top 5k primes of the type exist when they are much less time consuming than many of them that do.:smile:
Thank you for the pointer though.

[QUOTE=a1call;511193]The whole point of this thread is to wonder why no top 5k primes of the type exist when they are much less time consuming than many of them that do.:smile:[/QUOTE]
The form k*2^n+/-1 is even easier, hence the bulk of Top 5000 consists of these numbers.

Quicker to prove, perhaps. Easier to find, might be debatable. That still doesn't explain why there is not a single prime of the form in the top 5k.

[QUOTE=a1call;511196]Quicker to prove, perhaps. Easier to find, might be debatable. That still doesn't explain why there is not a single prime of the form in the top 5k.[/QUOTE]

k*(2^p-1)+1 = k*2^p - (k-1). Top 5000 will list it in its normalized form.
[CODE]
579 205833*2^2976222-411665 895938 L4667 2017
580 18976*2^2976221-18975 895937 p373 2014
581 2^2976221-1 895932 G2 1997 Mersenne 36[/CODE]

Also [URL="https://primes.utm.edu/primes/page.php?id=115395"]31723 *2^1398273-507567[/URL]

Please look at the forum that linked above to see what has been done so far.

EDIT:- Hmmm...
[URL="https://primes.utm.edu/primes/page.php?id=116879"]280680 · (2^1257787 - 1) + 1[/URL]
[URL="https://primes.utm.edu/primes/page.php?id=118071"]1089904 · (2^1257787 - 1) + 1[/URL]

Not sure if the 1st one fits the bill but the last two certainly do.
You are a much better searcher than I am. My search did not turn up those.
Thank you for the reply.

Then again none of them are top 5k, are they?

[QUOTE=a1call;511201]Then again none of them are top 5k, are they?[/QUOTE]

You mean #579 & #580 are not top 5000?

Here's another one: 46821 · 2^3021380 - 374567 @ 562

I was taking about the linked items.
It's early morning here.
Thank you for the alternative listing format. I missed those earlier.

I am dabbling in finding Mersenne-Prime-based-Twin-Primes. They are generally eligible for N+/-1 deterministic tests since all the relevant prime factors would be known.
Just getting started so nothing astronomically large yet.:smile:
* Everything found using Pari=GP and PFGW
* Nothing bothered to be tested beyond being PRP's.

1.6.M3+/-1
10.6.M3+/-1
11.6.M3+/-1
21.6.M3+/-1
25.6.M3+/-1
26.6.M3+/-1
31.6.M3+/-1
34.6.M3+/-1
41.6.M3+/-1
46.6.M3+/-1
54.6.M3+/-1
55.6.M3+/-1

7.6.M5+/-1
8.6.M5+/-1
15.6.M5+/-1
22.6.M5+/-1
25.6.M5+/-1
27.6.M5+/-1
53.6.M5+/-1
60.6.M5+/-1
63.6.M5+/-1
77.6.M5+/-1

24.6.M7+/-1
30.6.M7+/-1
46.6.M7+/-1
49.6.M7+/-1
65.6.M7+/-1
101.6.M7+/-1
115.6.M7+/-1
119.6.M7+/-1
129.6.M7+/-1
149.6.M7+/-1
166.6.M7+/-1
175.6.M7+/-1
179.6.M7+/-1
231.6.M7+/-1
280.6.M7+/-1
305.6.M7+/-1
311.6.M7+/-1

367.6.M13+/-1
437.6.M13+/-1
493.6.M13+/-1
542.6.M13+/-1
592.6.M13+/-1
595.6.M13+/-1
623.6.M13+/-1
663.6.M13+/-1
672.6.M13+/-1
690.6.M13+/-1
697.6.M13+/-1

715.6.M17+/-1
833.6.M17+/-1
883.6.M17+/-1
893.6.M17+/-1
967.6.M17+/-1
970.6.M17+/-1
987.6.M17+/-1
1018.6.M17+/-1
1075.6.M17+/-1

24.6.M19+/-1
56.6.M19+/-1
101.6.M19+/-1
119.6.M19+/-1
201.6.M19+/-1
214.6.M19+/-1
311.6.M19+/-1

39.6.M31+/-1
131.6.M31+/-1
206.6.M31+/-1
290.6.M31+/-1

602.6.M61+/-1
740.6.M61+/-1

112.6.M89+/-1
265.6.M89+/-1

371.6.M107+/-1

184.6.M127+/-1

5772.6.M521+/-1

44649.6.M607+/-1
60840.6.M607+/-1
97769.6.M607+/-1
108911.6.M607+/-1
110320.6.M607+/-1
115411.6.M607+/-1
124801.6.M607+/-1

61939.6.M1279+/-1

Please feel free to add to the collection.:smile:

[B]151505.6.M2203 +/- 1 [/B]are minimal Twin-Primes for [B]M2203 [/B]with 670 dd each.:smile:

ETA:
[url]http://factordb.com/index.php?query=151505*6*%282%5E2203-1%29%2B1[/url]
[url]http://factordb.com/index.php?query=151505*6*%282%5E2203-1%29-1[/url]

[url]http://factordb.com/index.php?query=61939*6*%282%5E1279-1%29%2B1[/url]
[url]http://factordb.com/index.php?query=61939*6*%282%5E1279-1%29-1[/url]

64125.6.M2281 +/- 1 are minimal Twin-Primes for M2281 with 693 dd each.:smile:

ETA:
[url]http://factordb.com/index.php?query=64125*6*%282%5E2281-1%29%2B1[/url]
[url]http://factordb.com/index.php?query=64125*6*%282%5E2281-1%29-1[/url]

[B]533040.6.M3217 +/- 1[/B] are a pair of minimal Twin-Primes for [B]M3217[/B] with 975 dd each.:smile:
Found using Pari-GP and PFGW.

ETA:
[url]http://factordb.com/index.php?query=533040*6*%282%5E3217-1%29%2B1[/url]
[url]http://factordb.com/index.php?query=533040*6*%282%5E3217-1%29-1[/url]

[QUOTE=a1call;532318][B]533040.6.M3217 +/- 1[/B] are a pair of minimal Twin-Primes for [B]M3217[/B] with 975 dd each.:smile:
Found using Pari-GP and PFGW.
[/QUOTE]

How do you use Pari/GP in this venture?

Well, I do have a detailed algorithm which may or may not speed thins up:smile:, but in short Pari-code does the sieving and if the candidates don't have small factors it writes the input file for the PFGW and launches it using the system command. I even have counter outputs for multi-threading, but at this stage the PFGW is too fast to use it. It takes longer to read and write to harddisk than finish the primality test,:smile:

[CODE]273249*6*(2^4423-1)-1 is 3-PRP! (0.0079s+0.0031s)
273249*6*(2^4423-1)+1 is 3-PRP! (0.0079s+0.0029s)
[/CODE]

Found in 10 minutes with [C]./pfgw64 -N -f mersenne_twin [/C] where mersenne_twin contains:

[CODE]ABC2 $a*6*(2^4423-1)-1 & $a*6*(2^4423-1)+1
a: from 1 to 100000000
[/CODE]

[QUOTE=paulunderwood;532327][CODE]273249*6*(2^4423-1)-1 is 3-PRP! (0.0079s+0.0031s)
273249*6*(2^4423-1)+1 is 3-PRP! (0.0079s+0.0029s)
[/CODE]

Found in 10 minutes with [C]./pfgw64 -N -f mersenne_twin [/C] where mersenne_twin contains:

[CODE]ABC2 $a*6*(2^4423-1)-1 & $a*6*(2^4423-1)+1
a: from 1 to 100000000
[/CODE][/QUOTE]

:shock:
I am still at 196998 since last night and that's for M4253.:picard:

There are ranges where sieving for small factors on PARI would be faster than PFGW, I am pretty sure.

[QUOTE=a1call;532331]:shock:
I am still at 196998 since last night and that's for M4253.:picard:

There are ranges where sieving for small factors on PARI would be faster than PFGW, I am pretty sure.[/QUOTE]

1. Just write a simple sieve using libgmp. ...or use Polysieve? it will be way faster.

2. Any number less than 10,000 digits won't be worth the paper it is written on. Why not just start with M44497 (...and that's only for a warm up) ?

[QUOTE=Batalov;532333]1. Just write a simple sieve using libgmp. ...or use Polysieve? it will be way faster.

2. Any number less than 10,000 digits won't be worth the paper it is written on. Why not just start with M44497 (...and that's only for a warm up) ?[/QUOTE]

I am aware that any twins less than 50k dd won't make it to top 10 listing, but there are 3 points. I remember I read an article somewhere that Tibetans believed that if they finish counting to some number the world would come to an end. I can't find that article but this is interesting:

[QUOTE]
if you ask a Tibetan to do finger-counting, they won't bend down one finger at a time, rather they will use their thumb to count the phalanges of the finger (the three bones that make up every digit).
[/QUOTE]
[url]https://theculturetrip.com/asia/china/articles/10-things-you-didnt-know-about-the-tibetan-language/[/url]

* There's is a value to a complete set even if it includes single digit primes.
* I am doing this as a hobby
* There is already indications towards the facts that there is a bias for small k values which is an indication for infinitude of twin primes.

Plus I have 3 cores working for the 80k Mersenne for days now and on last check they only found a single non-twin prime so far.

Thank you for the sieving tips. I will look into them.

[B]407635.6.M4253 +/- 1 [/B]are a pair of minimal Twin-Primes for [B]M4253 [/B]with 1287 dd each.

Found Using PFGW and PaulUnderwood.:smile:


ETA:
Looks like someone fed these to factorDB more than a year ago:

[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29%2B1+[/url]
[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29-1+[/url]

[QUOTE=a1call;532336][B]407635.6.M4253 +/- 1 [/B]are a pair of minimal Twin-Primes for [B]M4253 [/B]with 1287 dd each.

Found Using PFGW and PaulUnderwood.:smile:


ETA:
Looks like someone fed these to factorDB more than a year ago:

[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29%2B1+[/url]
[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29-1+[/url][/QUOTE]

[QUOTE]Primality proving
Proven by certificate
Size 202,645 bytes
Status Verified
Uploaded by Edwin Hall
Date July 4, 2012, 4:39 am[/QUOTE]

A BLS (Brillhart-Lehmer-Selfridge) proof is quicker than ECPP. This can be done by putting the [U]prime factors[/U] in a helper file:

[code]cat > mersenne_twin_4423.helper
2
3
3
3
97
113
2^4423-1
[/code]

[code]./pfgw64 -N -tp -q"273249*6*(2^4423-1)-1" -h"mersenne_twin_4423.helper"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 273249*6*(2^4423-1)-1 [N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file mersenne_twin_4423.helper
Running N+1 test using discriminant 3, base 3+sqrt(3)
273249*6*(2^4423-1)-1 is prime! (0.1872s+0.0158s)
[/code]

[code]
./pfgw64 -N -t -q"273249*6*(2^4423-1)+1" -h"mersenne_twin_4423.helper"
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 273249*6*(2^4423-1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file mersenne_twin_4423.helper
Running N-1 test using base 2
273249*6*(2^4423-1)+1 is prime! (0.0355s+0.0001s)
[/code]

[QUOTE=paulunderwood;532337][QUOTE]

Primality proving
Proven by certificate
Size 202,645 bytes
Status Verified
Uploaded by Edwin Hall
Date July 4, 2012, 4:39 am


[/QUOTE][/QUOTE]

I have no idea where that quote is from (except that it seems to be from the "The Prime Database"), but it certainly does not seem to be from any source that is indexed by Google.

[QUOTE=a1call;532344]I have no idea where that quote is from (except that it seems to be from the "The Prime Database"), but it certainly does not seem to be from any source that is indexed by Google.[/QUOTE]

It comes from factorDB for that number.

FTR:

For [B]M9689 [/B]I rewrote my Pari-GP code so that it does sieving in Pari-GP but without multitthreading (writing out a counter where multiple instances can read a common counter ), and it is much faster to sieve in Pari and feeding it to PFGW, than running ABC2 in PFGW alone.

FTR:
I still don't know how Paul got the July 2012 information.
in this link:
[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29%2B1+[/url]

If i click on More information I get:
[QUOTE]
Others:
Create time
Before November 4, 2018, 12:20 am
[/QUOTE]

[QUOTE=a1call;532407]
I still don't know how Paul got the July 2012 information.
in this link:
[url]http://factordb.com/index.php?query=407635*6*%282%5E4253-1%29%2B1+[/url]

If i click on More information I get:[/QUOTE]

It seems to have vanished.

[QUOTE=paulunderwood;532409]It seems to have vanished.[/QUOTE]

Because, an N-1 test has been run since your post, which will remove any traces of the ECPP test.

Well it wasn't done by me, since when I opened the link it was already labeled P.
If the purge was done in November 2018, then it was not done since Paul's post yesterday.

No twin primes for M9689. for k <= 607638

I will rewrite my code to reduce the read/writes for the kCounter by some multiple to get back at multi-threading when I get a chance.:smile:

Well, my rewrite is done so I can get back to my hobby.

[B]3901012.6.M9689 +/- 1[/B] are minimal Twin-Primes for [B]M9689 [/B]with 2925 dd each.:smile:
Found using [B]Pari-GP[/B] & [B]PFGW[/B].


ETA:
[url]http://factordb.com/index.php?query=3901012*6*%282%5E9689-1%29%2B1+[/url]
[url]http://factordb.com/index.php?query=3901012*6*%282%5E9689-1%29-1+[/url]

[B]718187.6.M9941 +/- 1[/B] are minimal Twin-Primes for [B]M9941[/B] with 3000 dd each.:smile:
Found using Pari-GP & PFGW.

ETA:
[url]http://factordb.com/index.php?query=718187*6*%282%5E9941-1%29%2B1+[/url]
[url]http://factordb.com/index.php?query=718187*6*%282%5E9941-1%29-1+[/url]

[B]3527063.6.M11213 +/- 1 [/B]are minimal Twin-Primes for [B]M11213[/B] with 3383 dd each.:smile:
Found using [B]Pari-GP[/B] & [B]PFGW[/B].

ETA:
[url]http://factordb.com/index.php?query=3527063*6*%282%5E11213-1%29%2B1+[/url]
[url]http://factordb.com/index.php?query=3527063*6*%282%5E11213-1%29-1+[/url]

[QUOTE=Batalov;532333]1. Just write a simple sieve using libgmp. ...or use Polysieve? it will be way faster.
[/QUOTE]
a1call, just in case you weren't listening, I will repeat. Use Polysieve. It makes this whole search entirely trivial and very fast.

Illustration:
Warming up: ... [URL="http://factordb.com/index.php?id=1100000001415037066"]120729852*(2^23209-1)-1[/URL] and [URL="http://factordb.com/index.php?id=1100000001415037017"]120729852*(2^23209-1)+1[/URL]
... 292133940*(2^23209-1)±1
... 771902880*(2^23209-1)±1
... 904094700*(2^23209-1)±1
This takes just a couple hours to find.

Next, k*(2^44497-1)+-1 ...

120M would definitely take me more than a day on 16 threads even with my fancy algo. Assuming you got that on 1 core that would be astonishing.
Thank you I will definitely research polySieve with a higher priority now.
Thanks again.
ETA: BTW if anyone would like to pursue any of the remaining Mersenne prime minimal twins, please be my guest.
It works probably help avoid wasting resources if anyone doing so announces it here 1st.
I am still at about k=10M on the next Mersenne prime on the queue.A solution for 60k dd primes of any type would be a world top 10.

I found a solution for M19937.
I will post here when I get a chance.

[B]9522163.6.M19937 +/- 1[/B] are minimal Twin-Primes for[B] M19937[/B] with 6010 dd each.:smile:
Found using [B]Pari-GP[/B] & [B]PFGW[/B].


[B]10073020.6.M19937 +/- 1[/B] are secondary minimal Twin-Primes for[B] M19937[/B] with 6010 dd each.:smile:
Found using [B]Pari-GP[/B] & [B]PFGW[/B].


ETA:
[URL="http://factordb.com/index.php?query=9522163*6*%282%5E19937-1%29%2B1+"]9522163*6*(2^19937-1)+1[/URL]
[URL="http://factordb.com/index.php?query=9522163*6*%282%5E19937-1%29-1+"]9522163*6*(2^19937-1)-1[/URL]
[URL="http://factordb.com/index.php?query=10073020*6*%282%5E19937-1%29%2B1+"]10073020*6*(2^19937-1)+1[/URL]
[URL="http://factordb.com/index.php?query=10073020*6*%282%5E19937-1%29-1+"]10073020*6*(2^19937-1)-1[/URL]

I will next run [B]M21701[/B] with the setup that I have as-is, until I get a chance to improve on it.
I also have a few cores running for [B]M216091[/B] as an effort in wishful thinking.
:smile:

Twin based on M44497 is relatively quite hard to find. The first one is ~3x higher than ME of its location.

Anyway, there, 2024053740*(2^44497-1)±1 are prime

Good stuff Serge,:smile:
Thanks for saving me the hassle. That would have definitely taken me more than a month to find.
Kudos to you sir.
But I think as difficult as it was, it is still a much smaller k than one would have expected for a 10k dd+ (Or is it 13k dd+) twin prime pair if the distribution was unbiased.
I will leave it to Mr. Greathouse to do the probability calculation. It must be about square of the probability getting a single prime of the size.

The sequence [B][URL="https://oeis.org/A126612"]A126612[/URL] [/B]was approved in 2007.
It seems to be missing the [B]M607 [/B]entry, [B]44649[/B].


The additional entries are:

[B]64125, 533040, 407635, 273249 (found by Paul Underwood), 3901012, 718187, 3527063, 9522163, [M21701 entry], 20121642 (found by Batalov), 337342290 (found by Batalov).[/B]


I do not have editing privileges.
what is the process of completing the list?

Thanks in advance.:smile:

For a price of $50 on aws I found this pair, too:
2332724220*(2^86243-1)±1 are prime

[QUOTE=a1call;533147]I do not have editing privileges.
what is the process of completing the list?

Thanks in advance.:smile:[/QUOTE]

If Batalov can vouch for the validity of these new entries he could add them to OEIS.

Ok I will fill in the k value for 21701, and will do a light doublecheck and will submit next week

I am following this thread but have not had the time to properly post to it.
The minimal k.6 for 21701 is
23128315.6.M21701 +/-1
Kudos to Batalov for the 25k dd+ solution.

To post a more complete post for [B]M21701[/B]:
[B]23128315.6.M21701 +/- 1[/B] are minimal Twin-Primes for [B]M21701[/B] with 6541 dd each.
Found using Pari-GP & PFGW.


ETA:

[URL="http://factordb.com/index.php?query=23128315*6*%282%5E21701-1%29%2B1+"]23128315*6*(2^21701-1)+1 [/URL]
[URL="http://factordb.com/index.php?query=23128315*6*%282%5E21701-1%29-1+"]23128315*6*(2^21701-1)-1[/URL]

[QUOTE=GP2;533301]If Batalov can vouch for the validity of these new entries he could add them to OEIS.[/QUOTE]

Thank you [B]GP2 [/B]for letting me know that.
I have a question for you:

i tried to create a Windows server instance on Google-Compute with 96 cores, but I keep getting the message that there is a quota limit of 24 cores for that region. When I check the availability for the region it states that up to 96 cores should be available with the skylake.
How do I setup a Windows instance with 96 cores?

Thank you in advance for your insight.:smile:

[QUOTE=Batalov;533200]For a price of $50 on aws I found this pair, too:
2332724220*(2^86243-1)±1 are prime[/QUOTE]

Again, kudos to [B]Serge [/B]for finding the [B]25972 dd[/B] pair of Twin-Primes.:smile:

I will setup a run for the next Mersenne-Prime in the queue: [B]M110503[/B].
I do not expect a timely resolution. BTW if anyone wants to find Twins for [B] M216091[/B], please be my guest. there are no Twin-Primes for it below 1241000.6.M216091.:smile:

[QUOTE=a1call;533147]The sequence [B][URL="https://oeis.org/A126612"]A126612[/URL] [/B]was approved in 2007.
It seems to be missing the [B]M607 [/B]entry, [B]44649[/B].
[/QUOTE]
I lightly double-checked and posted the correct sequence and some comments. ("various researchers extended sequence"). The sequence is now correct (in the sense: all primes are checked, minimality double-checked for a(1-23); it only takes a couple hours if done optimally).

With regards to individual contributions: The editors' position is that everyone can post only their results, so my comments about others' contributions were edited out. Those who have anything to add are encouraged to [URL="https://oeis.org/wiki/Special:RequestAccount"]register[/URL] and post their own edits.