20050421, 03:17  #1 
Aug 2004
Melbourne, Australia
98_{16} Posts 
The New Mersenne Conjecture
Does anyone have any current information regarding the testing status of the New Mersenne Conjecture? Or any other information regarding this, that I could include in my project?
Let p be any odd natural number. If two of the following conditions hold, then so does the third: p = 2^k+/1 or p = 4^k+/3 2^p1 is a prime (2^p+1)/3 is a prime. 
20050421, 06:18  #2 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 

20050421, 13:01  #3  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×1,877 Posts 
Quote:
I was present when John Selfridge first posed this conjecture. His remarks included the somewhat obvious statement that the conjecture was clearly true, that we know all the instances where it is true, and that it is impossible of proof. It is clear, even on *very* loose probabilistic grounds that 2^n1 and (2^n+1)/3 are simultaneously prime only finitely often. This conjecture is another instance of what Richard Guy calls his "Strong Law of Small Numbers". Even John says that the conjecture is a minor curious coincidence. 

20050422, 12:53  #4  
Feb 2004
France
3×311 Posts 
Quote:
Tony 

20050422, 17:58  #5  
∂^{2}ω=0
Sep 2002
República de California
5×2,351 Posts 
Quote:
Quote:
Don't get me wrong  I also know John and have great respect for him and his work, that's why something the NMC seems somewhat out of character for him (at least to me). But unlike Bob I wasn't there when he proposed it, so perhaps in the interim it's taken on more gravitas than it deserves or JS ever intended for it. 

20050423, 00:07  #6 
Aug 2004
Melbourne, Australia
10011000_{2} Posts 
My project is on Mersenne numbers, with a focus on Euler's work.

20050529, 17:28  #7 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Mere curious coincidence or not, I have been trying to prove a few probable primes from the NMC table for a while, particularly (2^42737+1)/3. I just found a new prime factor of 2,5342L, an algebraic factor of (2^42737+1)/31:
p48 = 352752377673314068757006632344836735032049556013 2,5342L has lot of prime factors, 8 of them (179 digits worth) found so far. The cofactor is down to 626 digits and still composite. Alex I should add: this discovery was very lucky, the factor was found with B1=3M (p40 parameters)! The group order is 2^2 3^3 167 193 911 1789 17327 69191 91867 2614223 2650897 81465869 with sigma=6006597770531109. Last fiddled with by akruppa on 20050529 at 17:32 
20051015, 18:15  #8 
Jun 2003
3×5×107 Posts 
I think the conjecture can be extended to 1G, by just simple sieving. Find a factor for 2^p+1 or 2^p1. Either way the p will be eliminated. It probably can be incorporated into the LMH program code, if anyone is interested.
Secondly, If you assume that gaussian mersennes and mersennes have the same distribution of primes and you extend the above to them then the conjecture fails for p=29. Anyway these are the only known p so far for which Gaussian mersenne and their counterpart produce a prime. p = 3, 5, 7, 11, 29 and 283. Or can a new conjecture be made about Gaussian mersennes? I am at present trying to extend the list Citrix Last fiddled with by Citrix on 20051015 at 18:17 
20051015, 20:01  #9  
∂^{2}ω=0
Sep 2002
República de California
5·2,351 Posts 
Quote:


20051015, 22:41  #10  
Jun 2003
3×5×107 Posts 
Quote:
very low, will take very very long. Citrix 

20051016, 03:45  #11  
Jun 2003
3·5·107 Posts 
Quote:
But it can be incorporated into the LMH code at no extra computational cost. So if you are interested they can look into it. Citrix 

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