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Old 2011-01-29, 00:18   #43
davar55's Avatar
May 2004
New York City

3·1,409 Posts

Originally Posted by Mini-Geek View Post
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?
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