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20181212, 12:04  #23 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
1011001011010_{2} Posts 
I created a plot with a linear fit along with a 95% confidence interval and a 95% prediction interval.
The linear fit was: log10p = 0.3941 + 0.1612 * n We expected a slope of 0.1690. I tested the significance of the difference by subtracting the expected slope from log10p in the linear regression. (log10p  0.1690*n) = 0.394139  0.007793*n This difference was significant with p = 0.01843 I don't necessarily trust that pvalue as linear regression assumptions have been violated. The points aren't independently distributed. A different analysis is needed. edit: I just looked at the differences rather than the values themselves. The mean difference was 0.15232. This was 0.01670 away from the expected value with p=0.393. I believe this value a lot more. Last fiddled with by henryzz on 20181212 at 12:17 
20181212, 13:00  #24  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:


20181212, 13:01  #25  
"/X\(‘‘)/X\"
Jan 2013
5462_{8} Posts 
Quote:
There is no antialiasing going on, so that implies the program plotted the square to the nearest pixel. If I try adjusting each square by one pixel, using 1364 for M50 and 1361 for M49, gives three tighter ranges of ~1430 exponents for M51. SImilarly with 1366 and 1363 I get three broader ranges of ~6200 exponents, but the first two are already LLed/PRPed in this case. Every other combination produces very broad ranges. 

20181212, 15:00  #26 
Feb 2017
Nowhere
3483_{10} Posts 
Just for fun, I looked at the slopes between the points (i, log(p_{i}) and (j, log(p_{j}) where j > i and log is the natural log, and p_{i} is the ith Mersenne prime exponent. It looked to me like there was a bit of "flattening out" around i = 20, and wanted to see if this showed up in the numbers. It sort of does. I sorted the results by slope. I then had Pari barf out the smallest and largest 20 slope values.
smallest 20: [46, 47, 0.010933588993086605929236533212337927955] [36, 37, 0.015058312767044147536611550816584524628] [21, 22, 0.025676397827141005981103996135256385826] [16, 17, 0.034793875482044907297702626994768438687] [19, 20, 0.039193582614693223608816298872297140257] [49, 50, 0.039963480120547020974157673862692869195] [25, 26, 0.067181792047021455026137524472709077295] [43, 44, 0.069256725434041762077583373565027573737] [21, 23, 0.073041299037938294276941847434381522933] [45, 47, 0.074336144912190375732533484497051675994] [24, 26, 0.075981416310133894011686361025839074128] [41, 42, 0.077170628677447021804128473078153561680] [24, 25, 0.084781040573246332997235197578969070961] [43, 47, 0.087323019753545325799141055225272292646] [44, 47, 0.093345117860046513706326949112020532282] [41, 47, 0.097373090844593132651365708342253314346] [43, 45, 0.10030989459490027586574862595349290930] [41, 44, 0.10140106382913975159640446757248609641] [42, 47, 0.10141358327802235482081315539507326488] [40, 47, 0.10278329131851350911045130597557776243] "top 20" [14, 16, 0.64452327787142243068139560148537830217] [37, 40, 0.64620659318861586785147333085784551100] [26, 27, 0.65088163668835737514358422787027049108] [26, 28, 0.65631437935830886839481195317493619818] [38, 39, 0.65824890669055576523747600675488898133] [27, 28, 0.66174712202826036164603967847960190527] [8, 9, 0.67688665968816500282222477888325753587] [31, 33, 0.69028662711326196252073174581645872462] [12, 16, 0.71334706508593796261633065752528899399] [14, 15, 0.74530501051934442788455893559983294112] [37, 39, 0.74726169875087284224397164596278205239] [35, 36, 0.75541933192501273070920427115554428144] [12, 15, 0.76988223837341713899569678757674410425] [12, 14, 0.78217085230045349455126571356519968582] [20, 21, 0.78417302251667736495562227981108795705] [11, 13, 0.79146060364573037239224844474727973098] [37, 38, 0.83627449081118991925046728517067512349] [30, 32, 0.87298874167702370682362907132731772743] [31, 32, 1.2534509339193628306681175436443297564] [12, 13, 1.4115629552947756450414548988015440687] The smallest slope is for the 46th and 47th exponents 42643801 and 43112609. The top value occurs for the 12th and 13th exponents 127 and 521. 
20181212, 16:42  #27 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2^{2}·3·7·53 Posts 
Evaluated for M41 through M51 with fitting at ln(x)=17,18,19.
I get large estimation errors on the ten Mp knowns M41  M50*, magnitudes ~5100 to 169,000. At this zoom, a half pixel digitization error is ~430,000. Estimated Mp51* ~82861679.44 +/ ~220,000. Need MOAR PIXELS. (Or more patience.) (Courteous) suggestions welcome. 
20181212, 17:22  #28 
If I May
"Chris Halsall"
Sep 2002
Barbados
2·4,657 Posts 

20181212, 18:41  #29 
"/X\(‘‘)/X\"
Jan 2013
5462_{8} Posts 

20181212, 22:42  #30  
"Jeppe"
Jan 2016
Denmark
5×31 Posts 
Quote:
Or do something even better if it is possible (I cannot tell). Does anyone know Lenstra, Pomerance or Wagstaff personally? All three appear to be alive. We could ask them if they (too) are worried about the fate of the conjecture regarding the Mersenne prime occurrence. These people came up with it. /JeppeSN 

20181212, 23:10  #31  
∂^{2}ω=0
Sep 2002
República de California
2·13·443 Posts 
Quote:
I don't see an updated redo of the Wagstaff approach as particularly promising, because to me the crux of the issue is "is the 'hockey stick' real^{*}, and if so, what accounts for it?" IOW what might possibly be behind the qualitative behavioral *change* starting around p = 10M, if there is in fact such a change? OTOH such a behavioral change in frequency of Mprime occurrences would seem to be tied to a corresponding change in factoroccurrence statistics, so perhaps some deep TF DB datamining could be useful.  ^{*}"Real" in terms of the underlying statistical odds, not the Mprime exponents themselves, which are what they are. 

20181213, 01:11  #32 
P90 years forever!
Aug 2002
Yeehaw, FL
5×1,433 Posts 
For fun, (y'all are ruining my getaway and why are .c files illegal to upload), I wrote a simulation of wagstaff's conjecture. It is attached for you to find bugs in or run.
In a run of 1000 simulations of testing all exponents to 85000000, 51+ Mersennes is quite normal  occurring 463 times. 13+ over 10 million is abnormal occurring only 5 times in 1000. I think the proposed analysis of TF factors is a great idea  just needs a volunteer. BTW, someone might want to modify my simulation to break down the expected 1 mod 4 vs. 3 mod 4 primes. Last fiddled with by Prime95 on 20181213 at 01:11 
20181213, 03:03  #33 
P90 years forever!
Aug 2002
Yeehaw, FL
5·1,433 Posts 
Chris Caldwell's summary of Lenstra and Pommerance's conjecture is there are (e^gamma/log 2) * log log x Mersenne primes less than x.
Plugging in 85000000 for x, I get 46 expected Mersenne primes, whereas my simulator is averaging about 50.5. Something is amiss. Maybe my simulator is buggy or maybe we haven't reached "asymptotically" yet. Last fiddled with by Prime95 on 20181213 at 03:09 
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