20040531, 18:08  #1 
Sep 2002
2×331 Posts 
What is the probability distribution for M42 ?
The following page, Where is the next Mersenne prime?
http://www.utm.edu/research/primes/n...tMersenne.html has a graph, Probability distribution for t = log2(log2(41st Mersenne)) http://www.utm.edu/research/primes/gifs/M41_none.gif just after the text: If we had no idea what the first 38 Mersennes were, we would predict values for t=log2(log2(Mn)) as follows. M41 was found very near the highest probability. What is the graph for M42 ? How is it calculated ? 
20040601, 16:58  #2  
Nov 2003
2^{2}·5·373 Posts 
Quote:
I'm not sure that you really know what it is that you are asking for. "near the highest probability" is mathematical gibberish. It is *believed* that the Mersenne primes are Poisson distributed with respect to their exponents. IN THE LONG RUN. But asking for a "graph for M42" (i.e. a specific number) is meaningless, especially in the SHORT RUN. If instead you ask: "given that we know M41, p = 24036583 and that the probability that a randomly chosen q, for q > p has M_q a prime, what value of q is most likely to make M_q prime", the answer clearly is "the smallest untested value of q greater than p". If instead you ask: "given that the primes q that make M_q prime" are Poisson distributed, and given that M_24036583 is prime, what is the expected value for the next prime M_q", that is an entirely different question. There are heuristics which suggest that the expected number of Mersenne primes in the interval [2^n, 2^2n] is exp(gamma), independent of n as n>oo. But there is no proof. Thus, we expect about 1.7 Mersenne primes in the exponent interval [24M, 48M], but trying to guess which one is the "most likely" is meaningless. All this guessing as to the next "most likely place to search" is meaningless. 

20040602, 02:16  #3  
"Mark"
Feb 2003
Sydney
3×191 Posts 
Quote:
However, we know more than that  we know the first 41, give or take a bit of uncertainty. After the graph referred to, Chris Calwell's page goes on to explain (with another graph) that the best place to look is just after the largest one we already know. Basically it is because the distribution is modelled so well by a Poisson process, which has "no memory" of what came before. I hope that helps! Mark 

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