[quote=philmoore;176954]I'm pretty sure that Prime95 does a rounding, not a truncation. If Glucas does the same, Tony's results are still consistent with our suspicions.[/quote]
1000/500009=0.001999964000647988336209948221...
Prime95 v25.9 build 4:
[code][Jun 10 12:21] Iteration: 1000 / 500009 [0.19%]. Per iteration time: 0.001 sec.[/code]It definitely truncates in Prime95. What I'm not sure of is whether Glucas truncates or rounds. Let's see what Tony's new status brings to the table... (assuming truncation)
[code]25000000/.5863=42640286.54272556711581101825
25000000/.5864=42633015.00682128240109140518[/code]42633015 to 42640286. Compare to earlier 42632066 to 42639970 and 42635862 to 42647058, which together make 42635862 to 42639970. The new limits are completely outside of the previous approximation, so [B]42635862 to 42639970[/B] (recall that 42643801 does not fit this) is still the best estimate assuming truncation.

Now let's see what happens if we assume instead that it's rounded. Iteration 16240000 was 38.08%, iteration 23000000 was 53.94%, and iteration 25000000 was 58.63%.
So:
[code]16240000/.38075=42652659.22521339461588969140
16240000/.38085=42641459.89234606800577655245
23000000/.53935=42643923.24093816631130063966
23000000/.53945=42636018.16665121883399759014
25000000/.58625=42643923.24093816631130063966
25000000/.58635=42636650.46473949006566044172
42641459 to 42652659
42636018 to 42643923
42636650 to 42643923
merge to become:
[B]42641459 to 42643923[/B][/code]Which contains 42643801.
I'm thinking M47 = M42643801, Tony hasn't been giving us false info, and Glucas rounds the percentages while Prime95 truncates them. There are 2584335-2584200=135 primes in that range. There are 91 factors over 76 exponents, leaving 59 exponents for LL, one of which is M42643801.

[QUOTE]Using the formula I found here:
[url]http://en.wikipedia.org/wiki/Poisson_distribution[/url]
the probability works out to be about 0.00157, quite small, but on the other hand, since there are 40 8-element sequences so far known, I'm guessing that the probability of at least one 8-element clustering this close is about 40 times this probability, or about 6.3%. Please correct any errors, but to me, this seems to support the idea that this clustering is unusual.[/QUOTE]

To me it seems unusual as well, but certainly possible since we are considering an average distribution and this is just one range. HOWEVER, I am not the person to ask on this as my expertise in statistics is rudimentary.

[QUOTE=philmoore;176956]So I conclude that we should have expected to find about 2.215 primes in this range (about 20 to 48 million) where we actually found 8. How unlikely is that?[/QUOTE]

Doing the calculation slightly differently (prime-by-prime, splitting by value mod 4) I get an average of 2.411 Mersenne primes in that range.

The naive Poisson probability that there would be 8 or more primes in that range is then about 1 in 300:
0: 8.97%
1: 21.6%
2: 26.1%
3: 21.0%
4: 12.6%
5: 6.09%
6: 2.45%
7: 0.844%
8+: 0.343%

Edit: This would make the chance of seeing this at least once in an 8-element sequence 12.9%, not quite so remarkable.

[QUOTE=Batalov;176878]babelfish.yahoo.com does a decent job.
Let's see if Tony like the translation:
[COLOR=green]Une personne est allée réparer l'antenne de télévision et, ayant glissé, est tombé du toit.
En volant devant la fenêtre, il a répondu à la question de la femme "Comment ça va?" : "Jusqu'ici, tout va bien".[/COLOR][/QUOTE]Perfect !
[COLOR="RoyalBlue"]"A man was repairing a TV aerial when he felt down the roof.
Passing in front of a window, he answers to a lady asking "How are you ?" : "Up to now, everything's fine".[/COLOR]

[QUOTE]Edit: This would make the chance of seeing this at least once in an 8-element sequence 12.9%, not quite so remarkable. [/QUOTE]

True. But how does that percent change if say, we found two more primes below 50M?

[QUOTE=Primeinator;176978]True. But how does that percent change if say, we found two more primes below 50M?[/QUOTE]

It goes to 0.792% (individual block chance 0.0209%).

[quote=Primeinator;176978]True. But how does that percent change if say, we found two more primes below 50M?[/quote]

I've read that they (Mersenne) Primes are distributed pseudorandomly. That means at the actual moment they can come in clusters or you have big gaps where nothing happens.

With our actual knowledge it's almost impossible to say how much M. Primes are still in that area ...

I think that we are just playing pin the tail on the donkey for the moment for the reason that we only have 46, possibly 47 data points. As we find more Mersenne primes we may get a more accurate prediction as to their distribution.

[QUOTE=CRGreathouse;176966]Doing the calculation slightly differently (prime-by-prime, splitting by value mod 4) I get an average of 2.411 Mersenne primes in that range.

The naive Poisson probability that there would be 8 or more primes in that range is then about 1 in 300:
0: 8.97%
1: 21.6%
2: 26.1%
3: 21.0%
4: 12.6%
5: 6.09%
6: 2.45%
7: 0.844%
8+: 0.343%

Edit: This would make the chance of seeing this at least once in an 8-element sequence 12.9%, not quite so remarkable.[/QUOTE]

I didn't consider the probabilities of more than 8, which does raise the figures somewhat. However, we can't consider the 40 possible sequences as independent, so I would guess that 12.9% is an overstatement of the true probability, but I have no idea how much.

[quote=Primeinator;176998]I think that we are just playing pin the tail on the donkey for the moment for the reason that we only have 46, possibly 47 data points. As we find more Mersenne primes we may get a more accurate prediction as to their distribution.[/quote]

I don't think that this will help. We have a lot of prime numbers and still can't predict how much there are in an interval. We have some probability for big intervals but thats it.

Well, something went wrong with my test, so I'm out. Mprime didn't mention any errors, but the residues don't lie, so at this point I'm just trying to figure out what went bad. The good news is the more seasoned double-checkers are still matching residues, so things are just back to the way they would normally be if I wasn't lucky enough to get an early start.