no twin left behind?

[Mini-Geek]

I've started testing for twins (PRP or provable) of all primes on the top 5000 list that do not have a base of 2 (since I can test those by downloading Karsten's lists). The timestamp for the list was "Thu Jan 20 04:51:06 CST 2011". I made a Python script to parse it out, which I'll post when I have results to post. Due to the vastly varying bases, GFNs, Phi's, and factorials/primorials, I don't see how I can really presieve this efficiently, so I'm just running it in PFGW with -f.

[gd_barnes]

An excellent and very interesting work Tim. Nice job!

[gd_barnes]

David,

Just to be clear: Do you plan to primality prove 5789*2^15513+3 ?

Tim,

I updated the status in post 32 to reflect what has now been done.


Gary

[Mini-Geek]

Originally Posted by Mini-Geek

I've started testing for twins (PRP or provable) of all primes on the top 5000 list that do not have a base of 2 (since I can test those by downloading Karsten's lists). The timestamp for the list was "Thu Jan 20 04:51:06 CST 2011". I made a Python script to parse it out, which I'll post when I have results to post. Due to the vastly varying bases, GFNs, Phi's, and factorials/primorials, I don't see how I can really presieve this efficiently, so I'm just running it in PFGW with -f.

Results and Python 3.x parsing script attached. I also searched for twins of Proth primes (k*2^n+1) with a k over 10000 since those aren't on Karsten's list yet, and presieved that with srsieve. No primes found. Not counting numbers divisible by 3, I factored or tested 566 numbers. The other ones were of the type that they should be included on one of Karsten's lists.

[henryzz]

Originally Posted by gd_barnes

David,

Just to be clear: Do you plan to primality prove 5789*2^15513+3 ?

Tim,

I updated the status in post 32 to reflect what has now been done.


Gary

I will test it. It will take a while. I will try to use multicores. We are in no hurry otherwise I wouldn't do it. It will eventually get finished. This number is small enough for me to eventually finish but large enough that I feel the need to use multicores.

[henryzz]

It will be about 100 hours of processing for phase 1. 4% done now. Begining to think multithreading is a waste of time for this small a number. I am running 2 threads and thread one has had 5/6 of the successes so far.

[Mini-Geek]

Originally Posted by Mini-Geek

Results and Python 3.x parsing script attached.

I've updated the script to handle extra-long/multi-line lines correctly. It can now read the entire current top 5000 list without any errors or placing numbers in the "confused" file (at least, from rank 1 to 5000 - not sure what'll happen if it sees all the other things at the top and bottom). In case anyone's interested, I'm attaching the updated version here.

[mart_r]

Originally Posted by Mini-Geek

I don't know if a record list exists for non +-1 twin prime, or (probably the same) a pair of twin primes/PRPs, where at least one is only known to be a PRP. Given the obscurity of that, I'd guess my 5789*2^15513+1, +3 pair have a good chance of being that.

Uhm... What about the twin

(20431926447260679*4001#*(205881*4001#+1)+210)*(205881*4001#−1)/35 +5, +7 (5132 digits)

?

[Mini-Geek]

Originally Posted by mart_r

Uhm... What about the twin

(20431926447260679*4001#*(205881*4001#+1)+210)*(205881*4001#−1)/35 +5, +7 (5132 digits)

?

I was not aware of these twin primes. Is the pair listed on a list anywhere, or did you just find the pair, or what? A google search can't find it, and it's not large enough to be on any list I can see. I can confirm with PFGW that they are primes. (the number between the primes is easily trial factored to at least 33.34%)
What about narrowing the definition to twins where at least one of the pair would (at time of discovery) best be proved by general methods like ECPP, (e.g. no N-1, N+1, or N-1/N+1 combined test is useful) whether they have been proven or are still PRP? Maybe it qualifies for that record! If not, I give up and admit it's not a terribly interesting twin, it just happens to be the largest I found in my search.

[davar55]

Originally Posted by Mini-Geek

1381*2^6512+3 and below Mini-Geek (done, certificates in DB)
7027*2^13017-3 [unreserved]
755*2^13474-3 [unreserved]
5789*2^15513+3 henryzz

Done PRPing all candidates and proving all candidates I said I would. All results I saved (started a little after n=20K) along with all primes in pfgw.log or pfgw-prime.log according to current proven status are attached.

2^13466917-3 would have taken a little over two days, but I put it on two cores for most of it, so it took closer to one day. Because it wasn't sieved very well, (only to 5 billion, or about 2^32) Prime95 chose P-1 bounds that gave it a 20% chance of finding a factor. Unfortunately it did not find a factor, even with such generous bounds, so I had to test it. Alas, the largest known twin Mersenne prime (i.e. 2^p-1 and (2^p-3 or 2^p+1) are prime) is just p=5: 29 and 31.
Just for fun, here are all known primes that are twin Mersenne or Fermat primes:

2^16+1, +3 (65537, 65539)
2^4+1, +3 (17, 19)
2^5-1, -3 (29, 31)
3, 5, and 7, by various formulas (3=2^1+1=2^2-1, 5=2^1+3=2^2+1=2^3-3, 7=2^2+3=2^3-1)

I'd guess that such twin pairs are finite and fully listed there, even if there are infinite Mersenne, Fermat, and twin primes. AFAICT from a quick googling, the last time someone looked for Mersenne Twin Primes was in 1999, when the highest p known to make 2^p-1 prime was 3021377.

So has anyone factored

As you can see, there are only two numbers (2^4253-3 and 2^11213-3) in the table that I have not been able to factor.

either M(4253)-2 or M(11213)-2 since then?

[Mini-Geek]

Originally Posted by davar55

So has anyone factored

either M(4253)-2 or M(11213)-2 since then?

Not as far as I know, but I don't know of any serious ECM attempts, just that old TF to 2^32. Here are the FactorDB entries for those:
http://factordb.com/index.php?query=2^4253-3
http://factordb.com/index.php?query=2^11213-3