20031204, 02:02  #1 
Nov 2003
European Union
150_{8} Posts 
probability of finding a Mersenne prime
The Status command of the Prime95 client outputs: "The chance that the exponent you are testing will yield a Mersenne prime is about 1 in 278494".
How is this probability (1 in 278494) calculated? 
20031204, 04:04  #2 
Sep 2003
29·89 Posts 
Some of the math is discussed here:
http://www.utm.edu/research/primes/m...heuristic.html I'm not sure if this is the actual calculation that Prime95 uses. 
20031206, 19:03  #3  
Nov 2003
3·5·11 Posts 
Quote:
"What are the chances that the LucasLehmer test will find a new Mersenne prime number? A simple approach is to repeatedly apply the observation that the chance of finding a factor between 2X and 2X+1 is about 1/x. For example, you are testing 2100001391 for which trial factoring has proved there are no factors less than 264. The chance that it is prime is the chance of no 65bit factor * chance of no 66 bit factor * ... * chance of no 5000070 bit factor. That is: 64 65 5000069  *  * ... *  65 66 5000070 This simplifies to 64 / 5000070 or 1 in 78126. This simple approach isn't quite right. It would give a formula of how_far_factored divided by (exponent divided by 2). However, more rigorous work has shown the formula to be (how_far_factored1) / (exponent times Euler's constant (0.577...)). In this case, 1 in 91623." Doing the calculation on both my computers, I come up with a rather close number using the formula with Euler's constant. 

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