mersenneforum.org > Math Anybody know about where the next prime should be?
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 2002-08-19, 14:38 #1 Kevin     Aug 2002 Ann Arbor, MI 1B116 Posts Anybody know about where the next prime should be? The whole point of this is project is to study distribution of mersenne primes. I read on some old page that they had a rough formula worked out that pretty well approximates the mersenne prime distribution. But the guesses that they had figured out were pre M(13466917). Anybody have updates on about where the next prime should show up, both if it were to be the 40th (exponent bigger than 13466917) or 39th (exponent lower than 13466917)?
 2002-08-19, 17:11 #2 David   Aug 2002 25 Posts Re: Anybody know about where the next prime should be? I think the web page you're talking about is http://www.utm.edu/research/primes/n...tMersenne.html This is pre M(13466917). The graph is very linear. However, I don't thinks it's very useful for determining what the next Mersenne prime will be. I doubt that future primes will lie exactly on the best-fit line. On another page http://www.utm.edu/research/primes/n...917/index.html, the author points out that, "Mersenne primes appear to be scattered randomly." Oh well. It is probably very hard to predict what the next prime is when we don't know that M(6972593) and M(13466917) are the 38th and 39th Mersenne primes. I too would still like to see a prediction made using M(13466917) in the analysis. Anyone else have any ideas?
 2002-08-19, 22:26 #3 binarydigits   Aug 2002 22×13 Posts Mersenne primes are almost certainly scattered randomly, but it is still possible to ascertain the overall behavior of random systems. Unfortunately, the sample of data points (39) is much too small, especially since we still don't know if that last point is the 39th or 40th (or even 41st). Just looking at the first twelve would give the expectation that the later points would all lie much closer to the line than they actually do, and it is possible that it will get even worse in the future. It is also possible that the line could start curving up or down, but the (limited) evidence at hand gives no reason to expect that. It is even possible that there aren't any more, although I doubt anyone believes that to be the case. About all anyone can say at this point is that there is about an X% probability of finding a Mersenne prime between M(Y) and M(Z), but there will be disagreements about even those probabilities. I originally expected another one to show up below M(13466917), but we are rapidly running out of untested candidates. If that turns out to be #39 then there is a good chance (and no, I refuse to define "good") that the next will be between M(17M) and M(21M). Or maybe not...;~)
 2002-08-20, 17:43 #4 David   Aug 2002 25 Posts Did you use the graph of the first 38 Mersenne primes to come up with your prediction for #40? While there is a "good" chance that there may be a new prime between M(17M) and M(21M), that range is very large (4M). I think it is impractical to use a graph to try to guess where the next Mersenne prime might be, especially since so few Mersenne primes are known. It is necessary to test every prime exponent; there are probably more surprises like M(2976221) and M(3021377) (two Mersenne primes very close to each other).
 2002-08-21, 09:45 #5 binarydigits   Aug 2002 22·13 Posts David, a guess is a guess, although there is a difference between an educated guess and a wild guess. It may be impractical to use the graph to make a guess or prediction, but there is nothing else to use. My range was large precisely because Mersenne prime distribution is so unpredictable, which was the whole point of my last message.
2002-08-21, 17:51   #6
David

Aug 2002

25 Posts

Quote:
 Mersenne prime distribution is so unpredictable, which was the whole point of my last message.
My point too. :D We really don't have enough data to make better predictions.

 2002-08-21, 18:28 #7 Kevin     Aug 2002 Ann Arbor, MI 433 Posts I know the predictions weren't supposed to be accurate. I was wondering more along the lines of whether the very rough trend showed if your odds were slightly better for first time LL tests (16M range) or million digit prime (33M), or maybe some range inbetween. I mean, the prediction from before just said the next one would be near 10 million, and it turned out to be 13 million, so I'm not expecting anything too accurate. With people testing almost to 80 million (though not assigned through Primenet), could you maybe extrapolate something like which ten million range would be most likely to conatin a new mersenne prime under various circumstances (i.e. we find 0 , 1 , or 2 new mersenne primes below the current highest)?
 2002-08-21, 18:35 #8 Tasuke     Aug 2002 23·32 Posts there was a prediction for mid 10 million and one for the 13 million. The 13 million was dead on(something like a .3 million spread) So it is entirely possible that there is a 10 million one that has slipped through as a first time error, or has not been checked yet. And I am one of the nuts checking a 77.9 million(2 actually)
2002-08-23, 10:25   #9
[CZ]Pegas

Aug 2002

7 Posts
Guess: 40th and 41th prime close to each other?

Quote:
 Originally Posted by David It is necessary to test every prime exponent; there are probably more surprises like M(2976221) and M(3021377) (two Mersenne primes very close to each other).
I guess 40th and 41th would be close to each other . It is very easy and probably false guess, but:

32th and 33th are close (M(756839) and M(859433), diff=102594)
34th and 35th are not so close (M(1257787) and M(1398269))
36th and 37th are close (M(2976221) and M(3021377), diff=45156)
38th and 39th are not close (M(6972593) and M(13466917))
40th and 41th can be close :D

I know it is not really so much MATH ops: as this forum says :( but if there is no mathematical apparatus to predict next prime, why not play this little game? ;)

 2002-08-23, 16:45 #10 Complex33     Aug 2002 Texas 5·31 Posts At this juncture one cannot say with certainty that M39 is M(1398269)... Only 128,947 LL tests to go Complex
2002-08-23, 17:20   #11
[CZ]Pegas

Aug 2002

7 Posts

Quote:
 Originally Posted by Complex33 At this juncture one cannot say with certainty that M39 is M(1398269)... Only 128,947 LL tests to go Complex
Ok, I forgot to say "as far as we know" :D
my post was not build on any solid brick so this uncertainity perfectly fit :(

Now suddenly one question came to me... was there a situation that prime was found between two others primes (I mean: in not double LL exponents below prime)? How high is probability that we find prime somewhere bewlow M(13466917)?

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