no twin left behind?

[Mini-Geek]

I just finished checking on the list of Proth primes for Riesel counterpart twins for the 2010-11-24 list. No new primes found.

[Mini-Geek]

16294579238595022365*2^7±1 are prime (22 digits)
I've finished checking the Riesel list of 2010-11-26. This was the only new twin prime.

[kar_bon]

Originally Posted by Mini-Geek

16294579238595022365*2^7±1 are prime (22 digits)
I've finished checking the Riesel list of 2010-11-26. This was the only new twin prime.

This twin was in the 'old' list, too, as I can see.
I've nothing changed on the 10^10 Summary page since 2010-10-18.

[Mini-Geek]

Originally Posted by kar_bon

This twin was in the 'old' list, too, as I can see.
I've nothing changed on the 10^10 Summary page since 2010-10-18.

Oh, yeah you're right. Not sure how that happened...perhaps I accidentally included some numbers for testing that shouldn't have been, so when I found that I thought it was new and didn't properly check if it existed.

[kar_bon]

Originally Posted by Mini-Geek

Oh, yeah you're right. Not sure how that happened...perhaps I accidentally included some numbers for testing that shouldn't have been, so when I found that I thought it was new and didn't properly check if it existed.

I've checked my saved history-files and this twin is included since 2008-10-01.

But thanks for the checking of the twin list.

[Mini-Geek]

Originally Posted by gd_barnes

Another interesting effort would be to check k*b^n-1 primes for k*b^n-3 primes to make a twin in that manner. Of course you could not prove anything that would make the top 20 but you probably could prove n<=10K in a reasonable amount of time and the chances are very small that you would find anything larger than that anyway.

I am checking for -3 (from Riesel primes) and +3 (from Proth primes) twins for the Riesel list of 2010-11-26 and the Proth list of 2010-11-24. I have found 4470 primes or PRPs so far, the largest being 5789*2^15513+3 (4674 digit PRP, and the Primo primality test is queued; of course 5789*2^15513+1 is a known prime, so this makes a twin PRP/prime pair). None so far are of a reportable size, either for being a top 10000 PRP or being a top 20 twin. I'm sure most of the small ones are already known, just because they're so small you wouldn't have to notice their somewhat special form to run across them, but there are still a good number that are hundreds to thousands of digits that were most likely not previously known. All primes are attached, in two files. The ones in pfgw-prime.log were trivially proven to be prime by PFGW, the ones in pfgw.log were found to be PRP. Of the 1773 PRPs there, I have verified the 1755 smallest and submitted the certificates to the FactorDB. (it actually took close to 30 of those and verified the rest itself; it doesn't bother taking outside certificates for primes under 300 digits, but it does enter it into its system for it to verify) The rest are still running in Primo.
The PRP testing is currently at n=475K. There are 275 candidates remaining. In the event (with a dismally low chance of under 3%) that one of those is prime, I'll have found a probable twin prime. AFAIK that would only be officially recognized as a non-twin PRP on the top PRPs list, as it would be too large to prove, but it would still be highly likely to be the largest pair of twin primes known.

[henryzz]

What is the record for a non +-1 twin prime? Is there a type non +-1 that is easy to prove and sieve?

[Mini-Geek]

Originally Posted by henryzz

What is the record for a non +-1 twin prime? Is there a type non +-1 that is easy to prove and sieve?

The top 20 twin primes list consists solely of k*b^n+-1 (all but one have b=2). I don't know of any non +-1 that are easy to prove, because AFAIK efficient primality proving algorithms are generally based on N+1 or N-1 being factored to 33.33% (or PFGW's 'combined' test that uses largely-but-not-quite-33% factored N+1 and N-1, and of course things like Primo/ECPP that can work on any, but I wouldn't call ECPP 'efficient' in this context), so for twin primes, choosing (smooth number)+-1 is a clear choice. 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.
Status:
PRPing: n=513365, 209 candidates remain, sieving more before continuing.
proving PRPs: n=4414, 11 candidates remain. (over 1 hour per test; I'm not sure I'll go all the way with this, 2^15513 is starting to look pretty large...might take a few days just for that test)

[henryzz]

If you give in before 5789*2^15513+1, +3 then I would do it. This one is worth testing because it is the largest we know of in its class.

[Mini-Geek]

Originally Posted by henryzz

If you give in before 5789*2^15513+1, +3 then I would do it. This one is worth testing because it is the largest we know of in its class.

Ok, I'll leave that one to you. I am also not proving 7027*2^13017-3 and 755*2^13474-3. You or anyone else can do those, or they could stay just PRPs for a while (until someone wants to prove them).
For clarity, here are the largest PRPs I found and who has them reserved for proving:

517*2^6098+3 and below Mini-Geek (done, certificates in DB)
7315*2^6423-3 Mini-Geek (done, certificate in DB)
1381*2^6512+3 Mini-Geek (done, certificate in DB)
7027*2^13017-3 [unreserved]
755*2^13474-3 [unreserved]
5789*2^15513+3 henryzz

The first 1767 (all before 517*2^6098+3) took about 19 hours. The last one I completed, 517*2^6098+3, took a little over 5 hours. I'd guess the next two I'm doing are 6-8 hours each and the largest three are several days to a few weeks each (unless you do multicore Primo or a distributed ECPP).
Also, a status on my PRPing:
I have split it into 3 parts and am running it on Prime95, (multiple cores and automatic P-1 makes it much better than PFGW when the numbers are this size ) the lowest is at n=819630, and there are 29 candidates left to PRP. The largest one alone will probably take a couple days, since it is so large: 2^13466917-3. If that turns out PRP, that would be incredible! Besides being the largest PRP, with 4053946 digits, it would (probably) be a twin with a Mersenne prime. Unfortunately that is extremely unlikely...

[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.