20160315, 23:00  #1 
"NOT A TROLL"
Mar 2016
California
197 Posts 
Mersenne Prime Exponent Distribution
This question has never been considered before, so is there a way to determine the number of Mersenne prime Expoenents (p such that 2^p1 is prime) less than n. A second similar problem is what is the chance 2^p1 is prime for a random prime p. For example, is 2^713242075252104680411 prime. If not, what is the nearest prime p such that 2^p1 is prime for. (I don't really expect specific answers to these examples, but this is just to show the types of problems no one has really encountered before.)

20160315, 23:10  #2  
"Forget I exist"
Jul 2009
Dartmouth NS
2·3·23·61 Posts 
Quote:
Last fiddled with by science_man_88 on 20160315 at 23:13 

20160315, 23:17  #3 
"Curtis"
Feb 2005
Riverside, CA
2×3×5^{2}×37 Posts 
How would you know? You can't even say it hasn't been considered on this forum before.
To find the number of mersenne primes less than n, count them. There's a list readily available of known mersenne primes, so for any n below the doublecheck line your question is trivial. 
20160315, 23:24  #4  
"NOT A TROLL"
Mar 2016
California
C5_{16} Posts 
Quote:


20160315, 23:28  #5 
"Forget I exist"
Jul 2009
Dartmouth NS
10000011100010_{2} Posts 
right but you asked about the odds and I'm just saying that there's one case where it could be 0.

20160315, 23:33  #6 
"NOT A TROLL"
Mar 2016
California
C5_{16} Posts 

20160315, 23:45  #7 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{4}×3^{3}×5^{2} Posts 

20160316, 00:07  #8  
Einyen
Dec 2003
Denmark
D55_{16} Posts 
Quote:
If we knew exactly where and how many Mersenne primes there are then where would be no point of GIMPS, but there are of course conjectures: https://primes.utm.edu/mersenne/heuristic.html https://primes.utm.edu/notes/faq/NextMersenne.html 

20160316, 00:15  #9 
"NOT A TROLL"
Mar 2016
California
197 Posts 
Using that information, there are probably 2 undiscovered primes p such that 2^p1 is prime between 74,000,000 and 600,000,000, roughly speaking.

20160316, 01:37  #10 
"Forget I exist"
Jul 2009
Dartmouth NS
10000011100010_{2} Posts 
really I get 5 potentially. 600/74> 3 <1.5^3>1.47...^3 okay technically I changed one thing late.point still stands.
Last fiddled with by science_man_88 on 20160316 at 01:53 
20160316, 01:47  #11 
"NOT A TROLL"
Mar 2016
California
197 Posts 

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