20160506, 00:53  #1 
Apr 2016
2·13 Posts 
Sophie Germain Twins
I saw the last post of that thread and found another equally hard problem: What is the largest known prime p such that p, p+2, and 2p+1 are all prime or p, 2p+1, and 2p+3 are all prime? These should be just as hard to find and no ECPP proof is required for some of them.
And It is also possible to have two sets of twins (when both sets are prime): p, p+2, 2p+1, 2p+3 and not all require an ECPP proof. 13049445569, 13049445571, 26098891139 is a small example following the first set. 14288181899, 28576363799, 28576363801 is a small example following the second set. I could not find a quick and easy example for the last set, sorry. Would someone please give a quick example for the last set. (The smallest is 29, 31, 59, 61). Thanks for the help on finding these. 
20160506, 01:25  #2  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,281 Posts 
Quote:
You can search for the known ones using advanced search form at UTM server (and select 'all verified primes'). You can easily find a larger example (and the quad example). Use NewPGen and in it there's BiTwin option. First sieve, then run the battery of tests until the complete set will pop up. 

20160506, 02:52  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
9124_{10} Posts 
...and here's a backofthenapkin estimate how much work it is to find a BiTwin quad of UTM recordable size (that's >1000 digits).
I sieved {k*2^3777+1, k*2^3778+1} sets with k<10^{10}. You only need to sieve to 1G, so that the removal rate is comparable to PFGW testing speed. I oversieved to ~8G, that's up to 33 bitfactors. One can expect ~1/70 sieve survivors to be prime, ~1/70^{2} sieve survivors to be twins, ~1/70^{3} sieve survivors to be triples, and finally ~1/70^{4} sieve survivors to be BiTwin quads. In the k region of size 10^{10}, I found 460 primes and 6 twins. So you only need to go for ~60000 such chunks. I.e. k < 6E14, and you can do that in 4E9 sized chunks (set BitmapThreshold = 4000000000 in NewPGen.ini file), and a bit of scripting (see NewPGen's command line options, then pfgw N k chunk$i ) I assure you that you can find such a quad before this week is over on a small computer (or perhaps by tomorrow if you rent a 18core spot instance at Amazon's EC2). It's relatively easy. 
20160506, 04:02  #4 
Apr 2016
2×13 Posts 
I am sieving the quadruplet series in NewPgen {k*3^75694, 2, +2, +4} and for the SG/twin quad, {k*2^4991+1, 1, k*2^4992+1, 1} expect k to be no more than 10^15 for both of them. In comparison to SG twin quads, this seems easier, (I do not belive using an ABC file is the best way to save time on any of this because It took me a day to sieve to k= 1,200,000 for the prime quadruplet series, and k = 945,000 today for the SG/twin quads.). Unless you know of a better way, that is. Thanks.

20160506, 05:22  #5 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10001110100100_{2} Posts 
Here's a small trainingwheels quad:
501399201855*2^16661 501399201855*2^1666+1 501399201855*2^16671 501399201855*2^1667+1 
20160507, 04:31  #6  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,281 Posts 
Quote:


20160507, 19:09  #7  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
7×19×43 Posts 
Quote:


20160510, 04:33  #8 
Apr 2016
2×13 Posts 
A sieve for {p, p+2, p+6, p+8} is most efficient, though I expect it first to sieve p, then sieve the remaining candidates for p+2, and so on, which I should have very few candidates left for about a range of 10^15. I have used multiple passes for Newpgen, so I guess I did that wrong.

20160621, 16:30  #9 
Jun 2003
Oxford, UK
753_{16} Posts 
Unless I am mistaken, this looks like a simpler variant of the octoproth search. That looked for 8 values, all prime. Highlights shows the SG constituents.
a*2^n1 a*2^n+1 2^n+a 2^na a*2^(n+1)1 a*2^(n+1)+1 2^(n+1)a 2*(n+1)+a Similar series going out for longer strings of SGs. so providing a set of 12 or 16 values all prime. Various software was put on this site to find these octoproths. They will be buried somewhere in the archives  here is a link to all of the threads: http://www.mersenneforum.org/forumdisplay.php?f=63 
20160621, 22:14  #10 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
7·19·43 Posts 
It seems to me that the source is only available for octoproth 5 not 6. Does anyone have a copy?

20160623, 15:10  #11 
Jun 2003
Oxford, UK
3×5^{4} Posts 

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