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Old 2018-12-12, 12:04   #23
henryzz
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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 p-value 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.
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Old 2018-12-12, 13:00   #24
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Quote:
Originally Posted by henryzz View Post
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 p-value 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.
log(log(log(Mn))) wasn't tried ?
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Old 2018-12-12, 13:01   #25
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Quote:
Originally Posted by GP2 View Post
No doubt, I didn't zoom properly before saving the image.That made things needlessly difficult.

This time, after zooming to the maximum in the PDF and saving the image, I then zoomed some more in the paint program to get nice big pixels. It is this second process that makes the antialiasing more readily visible.

I suspect that if I redid the calculations with the new pixel counts it would give a more accurate estimate, to an extra decimal place. Perhaps I'll try that tomorrow.

What estimate do you get?
When I did the same, viewing the PDF actual size on my screen and using a screen ruler, I got 1370 pixels for the new exponent, 1365 for M50 and 1362 for M49. That implies a tight range of 3841 exponents.

There is no anti-aliasing 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.
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Old 2018-12-12, 15:00   #26
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Just for fun, I looked at the slopes between the points (i, log(pi) and (j, log(pj) where j > i and log is the natural log, and pi is the i-th 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.
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Old 2018-12-12, 16:42   #27
kriesel
 
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Quote:
Originally Posted by GP2 View Post
What estimate do you get?
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.
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Old 2018-12-12, 17:22   #28
chalsall
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Quote:
Originally Posted by kriesel View Post
Need MOAR PIXELS. (Or more patience.) (Courteous) suggestions welcome.
Why not simply wait for the official announcement?

IMO, this leak (the original PDF) was ill-advised....
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Old 2018-12-12, 18:41   #29
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Quote:
Originally Posted by chalsall View Post
Why not simply wait for the official announcement?

IMO, this leak (the original PDF) was ill-advised....
Very much so.
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Old 2018-12-12, 22:42   #30
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Quote:
Originally Posted by Batalov View Post
That's the one I criticized in #14. (The product of inverse probabilities, but only to a limit.)

I later googled a free version of Wagstaff (1983) (below) and with all due respect the UTM simplified explanation has more holes than the original paper. I don't think the https://primes.utm.edu/notes/faq/NextMersenne.html quite matches what the paper does.
Someone should try to follow Wagstaff's recipe from that paper and do the same chi-squared test with all the factors known today, with access to the GIMPS factor database.

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
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Old 2018-12-12, 23:10   #31
ewmayer
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Quote:
Originally Posted by JeppeSN View Post
Someone should try to follow Wagstaff's recipe from that paper and do the same chi-squared test with all the factors known today, with access to the GIMPS factor database.

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
I met SamW at a West Coast NT Conference ~20 years ago, along with the (now) late John Selfridge and a half-dozen others of similar stature.

I don't see an updated re-do 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 M-prime occurrences would seem to be tied to a corresponding change in factor-occurrence statistics, so perhaps some deep TF DB data-mining could be useful.

---------
*"Real" in terms of the underlying statistical odds, not the M-prime exponents themselves, which are what they are.
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Old 2018-12-13, 01:11   #32
Prime95
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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.
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Old 2018-12-13, 03:03   #33
Prime95
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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 2018-12-13 at 03:09
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