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2005-12-28, 15:49   #254
Jean Penné

May 2004
FRANCE

2·281 Posts
Mersennes can not be part of Sophie-Germain pairs

The only exception is the pair 3 = M(2), 7 = M(3)

For all odd exponents p > 3 :

1) 2*M(p) + 1 = 2^(p+1) - 2 + 1 = M(p+1) cannot be prime because p+1 is even. So, M(p) cannot be Sophie-Germain(P).

2) (M(p) - 1)/2 = (2^p - 2)/2 = 2^(p-1) - 1 = M(p-1) cannot be prime because p-1 is even. So, M(p) cannot be Sophie-Germain(2*P+1).

3) For p>2, M(p) + 2 = 2^p + 1 is always multiple of 3, so M(p) cannot be the smallest number of a Twin pair...

For M(p) - 2 = 2^p - 3, I cannot conclude, so perhaps an M(p) can be the biggest of a Twin pair ?

Quote:
 Originally Posted by Citrix Has any one looked at, if mersenne numbers are a Sophie Germain pair etc. I did some work some time back.(1 number with no factor.) Has anyone else done such work? Does anyone want to do the latest prime? eg. 2*p+1 and 2*p-1 where p is a mersenne primes M(p) Smallest factor for 2*M(p)-1. 2 prime 3 prime 5 prime 7 11*23 13 prime 17 11*23831 19 prime 31 9241*464773 61 37 89 1823 107 11 127 11 521 193 607 11 1279 98807 2203 47 2281 131 3217 11 4253 23 4423 313 9689 289935200473 9941 419303 11213 Trial factor to 2^32; 20@b1=11000;50@b1=50000;composite by Base 3-PRP. 19937 11 21701 22313687645759 23209 37 44497 11 86243 6553 110503 313 132049 2348788960811 216091 23 756839 757 859433 193 1257787 11 1398269 96997 2976221 1451 3021377 11 6972593 103869349 13466917 11 20996011 86183 24036583 59 25964951 76382783

2005-12-28, 16:19   #255
Prime95
P90 years forever!

Aug 2002
Yeehaw, FL

2×3×1,193 Posts

Quote:
 Originally Posted by Kosmaj I think you mean this. I'll be interested to help such a search.
That is what I meant. What is needed is someone to organize the search. That is, contact Brent and Zimmermann and make sure they aren't already working on it. Get their recommendations and experiences in using their sieve and test software. Get their expected CPU time required. Put up a web page to hand out software and ranges and collect results.

2005-12-28, 20:20   #256
Citrix

Jun 2003

32·52·7 Posts

What are 2*p-1 kind of primes called? I think there is a special name to them? Does anyone know? Any analysis on this form?

What about 2*p^2-1? where p is a mersenne prime?

@ Jean, Thanks for your explanation.

Citrix

Quote:
 Originally Posted by Jean Penné The only exception is the pair 3 = M(2), 7 = M(3) For all odd exponents p > 3 : 1) 2*M(p) + 1 = 2^(p+1) - 2 + 1 = M(p+1) cannot be prime because p+1 is even. So, M(p) cannot be Sophie-Germain(P). 2) (M(p) - 1)/2 = (2^p - 2)/2 = 2^(p-1) - 1 = M(p-1) cannot be prime because p-1 is even. So, M(p) cannot be Sophie-Germain(2*P+1). 3) For p>2, M(p) + 2 = 2^p + 1 is always multiple of 3, so M(p) cannot be the smallest number of a Twin pair... For M(p) - 2 = 2^p - 3, I cannot conclude, so perhaps an M(p) can be the biggest of a Twin pair ?

2005-12-29, 00:04   #257
Xyzzy

"Mike"
Aug 2002

23×3×52×13 Posts

Quote:
 Originally Posted by Prime95 What is needed is someone to organize the search. That is, contact Brent and Zimmermann and make sure they aren't already working on it. Get their recommendations and experiences in using their sieve and test software. Get their expected CPU time required. Put up a web page to hand out software and ranges and collect results.
We could just do it as a sub forum under "Other Projects". I know nothing about the math but I'd be willing to help with the admin side.

 2005-12-30, 03:14 #258 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts The full official press release is up: http://www.mersenne.org/30402457.htm
 2005-12-30, 16:31 #259 Jeff Gilchrist     Jun 2003 Ottawa, Canada 100100100012 Posts I'm now 94% complete with the verification and it should be complete in about 16 hours from now.
 2005-12-31, 09:01 #260 tywebb   Dec 2005 7 Posts alright. time's up. the 16 hours are over. so is it finished? is it doubly verified now?
 2005-12-31, 12:24 #261 Jeff Gilchrist     Jun 2003 Ottawa, Canada 7·167 Posts Its all good! I found this waiting this morning: [Sat Dec 31 02:29:42 2005] UID: /20554, M30402457 is prime! G29: 00000000,0 NEW MERSENNE PRIME DISCOVERED !!!! PLEASE, send email to woltman@alum.mit.edu AND gbv_at_oxixares.com with the file "s30402457_last" attached. (You can find it in your working directory) Terminated all the queued job in file: glucas.que. Second verification complete, just in time for end of 2005. Last fiddled with by Prime95 on 2005-12-31 at 16:07
 2005-12-31, 12:32 #262 tywebb   Dec 2005 7 Posts holy tethered cow indeed! it's official! congratulations!
 2005-12-31, 13:38 #263 Numbers     Jun 2005 Near Beetlegeuse 1100001002 Posts Absolutely marvellous. Break out the bubbly!
 2005-12-31, 14:33 #264 T.Rex     Feb 2004 France 2·457 Posts Congratulations, Jeff ! The full verification of M43 has been done entirely in 2005 ! So we have 365 days for finding M44 ! (I guess the GIMPS will find it before 2007 !) Tony

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