20181207, 02:47  #1 
"Steve Sybesma"
May 2012
Brighton, CO USA
3^{4} Posts 
predicting approximate location of Mersenne primes
I saw a thread called "Simple graph to predict Primes" but it went into a discussion of sieves which is not what I was thinking of.
Has there ever been an attempt to plot the locations of known Mersenne primes so as to detect a pattern that would help predict an approximate location where the next one might be? Thank you. Last fiddled with by ssybesma on 20181207 at 02:48 Reason: correction 
20181207, 03:08  #2 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{6}×31 Posts 

20181207, 03:13  #3 
Sep 2003
3^{2}×7×41 Posts 
The only thing we think we know is that the average ratio between the exponents of successive Mersenne primes is approximately 1.47576
See for instance https://primes.utm.edu/mersenne/heuristic.html However, this doesn't tell us exactly where to expect the next Mersenne prime. No one can predict that. For instance, you can measure the average traffic flow on a street, but that won't tell you exactly when the next car will drive by. 
20181207, 15:54  #4  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,333 Posts 
Quote:
Quote:
In April 2009, these were known: Code:
# p 1 2 2 3 3 5 4 7 5 13 6 17 7 19 8 31 9 61 10 89 11 107 12 127 13 521 14 607 15 1,279 16 2,203 17 2,281 18 3,217 19 4,253 20 4,423 21 9,689 22 9,941 23 11,213 24 19,937 25 21,701 26 23,209 27 44,497 28 86,243 29 110,503 30 132,049 31 216,091 32 756,839 33 859,433 34 1,257,787 35 1,398,269 36 2,976,221 37 3,021,377 38 6,972,593 39 13,466,917 40 20,996,011 41 24,036,583 42 25,964,951 43 30,402,457 44 32,582,657 45 37,156,667 46 43,112,609 

20181207, 17:21  #5  
"Steve Sybesma"
May 2012
Brighton, CO USA
3^{4} Posts 
Quote:
That logarithmic graph is exactly what I had in mind, thank you. The series does look very linear when plotted that way. 

20181207, 17:26  #6  
"Steve Sybesma"
May 2012
Brighton, CO USA
1010001_{2} Posts 
Quote:
Of course ultimately all of the prime exponents still have to be checked in case one is missed. I'm just saying that in the future as more and more of these are discovered, it may be possible to zero in on them a little more accurately. 

20181207, 17:28  #7  
"Steve Sybesma"
May 2012
Brighton, CO USA
51_{16} Posts 
Quote:


20181207, 18:21  #8  
"Curtis"
Feb 2005
Riverside, CA
3×1,559 Posts 
Quote:
It's the same fallacy as rolling a 6sided die many times, observing that 6 comes up 1/6th of the time, and saying "well, we should expect every 6th roll to be a 6; I've had 5 rolls in a row that weren't 6, so my next one is more likely to be 6." Each roll of the die is an independent event, just as each primality test is an independent event. Prior results can give you a sense for average frequency of outcomes, but you cannot use averages to refine specific probabilities for specific tests any more than you can profitably claim "I am due for a 6!" when rolling a die. 

20181207, 20:44  #9  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,333 Posts 
Quote:
Finally, there is always a remote chance that the car has passed but was invisible to you  your sensory organs and your video cameras could not detect it. (that is: that the limitations of the computational procedure as implemented currently produce a false negative result. The chances of that are very low, but not zero.) 

20181208, 13:57  #10 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{6}·31 Posts 
I don't think the car analogy applies here because prime numbers are not as random as passing cars. Prime numbers occur simply at missjunctions points (vs hit points) of harmonics of all the primes less than the squareroot of the point. Take the harmonics of any finite small number of the primes and the misses become absolutely predictable. What makes the prediction of primes infinitely complex is that the harmonics become infinitely numerous as you progress higher.
There are expected patterns of concentration that can be observed. Take M1279 which is a prime number next to M1277 which is the smallest Composite Mersenne number without any know factors. A confidence? I would say not. 
20181208, 14:02  #11  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:


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