20140316, 04:21  #1 
"Bo Chen"
Oct 2005
Wuhan,China
2^{3}×3×7 Posts 
About Zhou's conjecture
This day, I search about the conjectures referenced to Mersenne prime.
And find a conjecture named "Zhou's speculate", it says, When the prime p is between 2^(2^n) and 2^(2^(n+1), there are 2^(n+1)1 Mersenne primes Mp. I set the n to 0,1,2,3 and found this conjecture is true. There are 21 Mersenne primes have been found larger than M_{2^(2^4)}, and this conjecture says there are 31 Mersenne primes between M65536 and M4294967296. I search this conjecture on wikipedia and mathworld.wolfram.com, but find no reference about it. Has anbody heard about this conjecture before? or you just think this conjecture is negligible? 
20140316, 05:42  #2 
Romulan Interpreter
Jun 2011
Thailand
2^{2}×17×139 Posts 
I pray to all my gods this to be false!
(and most probable it is false, just another case of Guy's "law of small numbers", those "3 cases" is like saying "3 is prime, 5 is prime, 7 is prime, so all odd numbers are prime"). Otherwise we have a lot of work to do here, for just 10 more primes.... 
20140316, 05:52  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3·5·17·37 Posts 

20140316, 06:14  #4 
∂^{2}ω=0
Sep 2002
República de California
3·3,877 Posts 
Note that the conjecture is roughly in line with Wagstaff's heuristic, according to which each doubling of the p upper limit yields roughly 2 new Mprimes, on average. The Wagstaff heuristic at least has some mathematically plausible reasoning behind it.
The "precise" number of primes predicted by Zhou is almost assuredly an example of Guy's SLoSN. 
20140316, 08:22  #5 
"Bo Chen"
Oct 2005
Wuhan,China
2^{3}×3×7 Posts 
Some more informations
Here is some more informations, I translate this article's tilte into English.
Zhou Haizhong, "Distribution regularity of Mersenne prime", ACTA Scientiarum Naturalium Universitatis Sunyatseni, Vol.31, No.4, 1992 This article is rather old, it even does not have English abstract. There is a link to this article online, http://www.cnki.com.cn/Article/CJFDT...Z199204018.htm But unfortunately it is in Chinese and is on charge. I read the review of this article, the author said he found the connection by "watching, analysis and research", but I only see "watching", it says, There is 1 Mersenne number between M2 and M4 (M3); There are 3 Mersenne numbers between M4 and M16 (M5,M7,M13); There are 7 Mersenne numbers between M16 and M256 (M17,...); ... This paper only have 2 pages. The second page is for charge. Anyway, we could know whether it is true when n=4 after all the exponents below 4294967296 have been verified or there are more than 31 Mersenne primes between M65536 and M4294967296. If this conjecture is true, I think it should be a beautiful theorem. 
20140316, 20:31  #6 
∂^{2}ω=0
Sep 2002
República de California
3×3,877 Posts 

20140317, 03:57  #7 
Aug 2006
5985_{10} Posts 
Is the conjecture that there are at least 2^(n+1)1 Mersenne primes in the associated range, or exactly that many? Either way it seems likely to be false, but the latter is much stronger.

20140317, 11:53  #8  
Nov 2003
2^{2}·5·373 Posts 
Quote:
"conjectures" on Victor Miller's number theory discussion/mailing list. I do not know whether it is the same person.... But I know how I would bet. 

20140317, 17:31  #9  
Nov 2003
2^{2}×5×373 Posts 
Quote:
ZhiWei Sun. 

20140504, 14:37  #10 
Jan 2014
2·19 Posts 
Maybe this "conjecture" was inspired by what Wikipedia calls the "Lenstra–Pomerance–Wagstaff conjecture", which states that the number of Mersenne primes in the exponentinterval is asymptotically .
According to Zhou there are 58 Mersenne primes below , and "according to" the second conjecture there are about 57 (The quotation marks are there because the second conjecture is asymptotic and claims nothing about .). (Generally, it claims that asymptotically there are Mersenne primes for exponents below . Which means that the exponentintervals and asymptotically hold the same amount of Mersenne primes.) Anyway, I'm not a big fan of such conjectures. They seem like wishful observations, without too much meat behind them I suspect. 
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