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 2014-03-16, 04:21 #1 wreck     "Bo Chen" Oct 2005 Wuhan,China 23×3×7 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?
 2014-03-16, 05:42 #2 LaurV Romulan Interpreter     Jun 2011 Thailand 22×17×139 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....
 2014-03-16, 05:52 #3 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·5·17·37 Posts
 2014-03-16, 06:14 #4 ewmayer ∂2ω=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 M-primes, 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.
2014-03-16, 20:31   #6
ewmayer
2ω=0

Sep 2002
República de California

3×3,877 Posts

Quote:
 Originally Posted by wreck If this conjecture is true, I think it should be a beautiful theorem.
Not without any actual mathematics behind it. All I see is a simple small-number curve fit turned into a wild "for all n" conjecture.

 2014-03-17, 03:57 #7 CRGreathouse     Aug 2006 598510 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.
2014-03-17, 11:53   #8
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by CRGreathouse 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.
Someone by the name of Zhou has posted an endless series of such
"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.

2014-03-17, 17:31   #9
R.D. Silverman

Nov 2003

22×5×373 Posts

Quote:
 Originally Posted by R.D. Silverman Someone by the name of Zhou has posted an endless series of such "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.
Apologies to everyone, including Zhou. I had him confused with
Zhi-Wei Sun.

 2014-05-04, 14:37 #10 Qubit     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 exponent-interval $(2^{2^n},2^{2^{n+1}})$ is asymptotically $e^\gamma\cdot 2^n$. According to Zhou there are 58 Mersenne primes below $2^{2^5}$, 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 $2^{2^5}$.). (Generally, it claims that asymptotically there are $e^\gamma\cdot 2^n$ Mersenne primes for exponents below $2^{2^n}$. Which means that the exponent-intervals $(0, 2^{2^n})$ and $(2^{2^n},2^{2^{n+1}})$ 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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