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
13·433 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
197 Posts 
Quote:


20160315, 23:28  #5 
"Forget I exist"
Jul 2009
Dartmouth NS
2·3·23·61 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
197 Posts 

20160315, 23:45  #7 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
25213_{8} Posts 

20160316, 00:07  #8  
Einyen
Dec 2003
Denmark
2·17·101 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 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Mersenne prime exponent not randomly distributed?  alpertron  Math  78  20191002 14:31 
prime distribution near mersenne primes  Unregistered  Homework Help  43  20090816 14:27 
Fun with the new Mersenne prime exponent  ewmayer  Lounge  4  20060906 20:57 
Distribution of Mersenne prime factors mod 6  alpertron  Math  0  20060623 20:07 
Mersenne composites (with prime exponent)  Dougy  Math  4  20050311 12:14 