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.

[Batalov]

Originally Posted by Trejack

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? ...

There are some known twins that are also SG, for example, this one (found back in 2000).
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.

[Batalov]

...and here's a back-of-the-napkin 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<10¹⁰. 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 bit-factors. One can expect ~1/70 sieve survivors to be prime, ~1/70² sieve survivors to be twins, ~1/70³ sieve survivors to be triples, and finally ~1/70⁴ sieve survivors to be BiTwin quads. In the k region of size 10¹⁰, 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 18-core spot instance at Amazon's EC2). It's relatively easy.

I am sieving the quadruplet series in NewPgen {k*3^7569-4, -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.

[Batalov]

Here's a small training-wheels quad:
501399201855*2^1666-1
501399201855*2^1666+1
501399201855*2^1667-1
501399201855*2^1667+1

[Batalov]

Originally Posted by Trejack

I am sieving the quadruplet series in NewPgen {k*3^7569-4, -2, +2, +4} and ...

Do you have the appropriate tools for that? I am asking because I have indeed searched for similar quads, but my modified NewPGen binary was not made available to the masses. Have you figured out how to make one (it is not too hard but not very straightforward either, even if you start from solving the initial hurdle of compiling newpgen from source - you need a "sort of a" time machine for that: you need to go back a few gcc major versions and rent a 32-bit linux node)? If you are sieving by running multiple passes, you are doing it wrong - you need to sieve for a quad in one pass. Or else you will be drowning in disk I/O.

[henryzz]

Originally Posted by Batalov

Do you have the appropriate tools for that? I am asking because I have indeed searched for similar quads, but my modified NewPGen binary was not made available to the masses. Have you figured out how to make one (it is not too hard but not very straightforward either, even if you start from solving the initial hurdle of compiling newpgen from source - you need a "sort of a" time machine for that: you need to go back a few gcc major versions and rent a 32-bit linux node)? If you are sieving by running multiple passes, you are doing it wrong - you need to sieve for a quad in one pass. Or else you will be drowning in disk I/O.

polysieve would be an option for starting the sieve. It might be faster to then continue sieving the 4 sequences separately with newpgen.

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.

[robert44444uk]

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^n-1
a*2^n+1
2^n+a
2^n-a
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

[henryzz]

It seems to me that the source is only available for octoproth 5 not 6. Does anyone have a copy?

[robert44444uk]

Maybe here

https://sites.google.com/site/robert...attredirects=0