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Old 2018-12-11, 23:30   #12
kriesel
 
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Quote:
Originally Posted by GP2 View Post
Assuming the plot is accurate, analysis of the pixels places the new exponent between 82.60M and 82.75M 82.65M.

We could do better than that of course. Simply identify the software that created the plot and start plotting points, varying the exponent value until you come up with a pixel-by-pixel reproduction of the plotted box pattern, including the pale shading and blur (sorry, I don't know the correct graphics terminology).
How close does your method get to the other, known p values nearby, say M49 and M50?
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Old 2018-12-11, 23:37   #13
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Quote:
Originally Posted by ewmayer View Post
I would be interested in some discussion of the statistical significance ascribable to the apparent trendline break of the last dozen M-prime exponents. Chris Caldwell's "this graph is amazingly linear" is getting more untenable with each new sooner-than-expected find.
It does look less and less like a fit. On the other hand, pretty much anything that is first magnitude sorted, then run through a log function and then graphed on a log scale will look pretty linear. I did the same with the set of populations of Wisconsin villages, and got a rather linear looking plot. ln(ln(x)) is very flattening. Plot the actual ordered Mersenne prime values (not exponents) on a linear scale, and, HOCKEY STICK!
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Old 2018-12-11, 23:41   #14
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Quote:
Originally Posted by GP2 View Post
Assuming the plot is accurate, analysis of the pixels places the new exponent between 82.60M and 82.75M 82.65M.
Your former range and other clues (based upon at least 1 being interpreted) point to 5 exponents. Your updated range limits that to <5.

Last fiddled with by Uncwilly on 2018-12-11 at 23:42
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Old 2018-12-12, 00:00   #15
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Methodology for my estimate:

Magnify the graph about 5x in a Paint program.

Under magnification, we see that the lines that draw the square-box plot points are not always sharp, but rather they are often blurred and smeared into two or even three adjacent lines of varying intensity. This reflects the fact that the exact center of the desired plot point usually does not map exactly onto an exact pixel point, so the plotting program uses blurring to compensate.

So we can use the relative paleness/brightness of the components of the blurred/smeared lines to assign a fractional vertical pixel position to both the top horizontal line and the bottom horizontal line of each square-box plot point, and then average them to find the center.

For the relative vertical pixel positions, with the zero reference point at the bottom horizontal bar of the square representing M46, I get:

Code:
      bottom   top         center            Mn            log(Mn)

M46   -0.2      9.1         4.45          42643801        17.56839
M47    0.3     10.0         5.15          43112609        17.57933
M48   20.2     30.0        25.1           57885161        17.87397
M49   37.05    46.7        41.875         74207281        18.12237
M50   39.8     49.4        44.6           77232917        18.16234
M51   44.3     54.0        49.15          xxxxxxxx        18.2296 to 18.2301

Last fiddled with by GP2 on 2018-12-12 at 00:04
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Old 2018-12-12, 00:26   #16
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Quote:
Originally Posted by ewmayer View Post
I would be interested in some discussion of the statistical significance ascribable to the apparent trendline break of the last dozen M-prime exponents. Chris Caldwell's "this graph is amazingly linear" is getting more untenable with each new sooner-than-expected find.
I would like such a discussion too.

I'd also like to see your graph along with a line with the slope predicted at
https://primes.utm.edu/notes/faq/NextMersenne.html. "we should get approximately a straight line with slope 1/egamma (about 0.56145948)"
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Old 2018-12-12, 00:34   #17
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I think it (that discussion) has started (since post like #10 and #14 )

It reminds me of that anecdote where someone heard about the prime and the only question that they had was "but what was the sign of the penulitmate iteration?"
Numbers, shnumbers... Ideas!

Found it --
Quote:
Originally Posted by L.Welsh (2003)
After M29 was discovered, that was the very first question Dick Lehmer asked. I think it interested him more than the value of p!!
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Old 2018-12-12, 00:52   #18
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Quote:
Originally Posted by Batalov View Post
I think it (that discussion) has started (since post like #10 and #14 )
I'm looking for analysis along the lines that Chris did here: https://primes.utm.edu/notes/faq/NextMersenne.html. The math is above my pay grade.

We have a theory that Mersenne primes "follow" a logarithmic line with a specific slope. We have data on the first 51 Mersennes. If someone with a good probability/statistics background could weigh in with an analysis of how well the 51 data points are matching the theory, then I and others here could hopefully learn a thing or two.
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Old 2018-12-12, 02:43   #19
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Quote:
Originally Posted by Prime95 View Post
I'm looking for analysis along the lines that Chris did here: https://primes.utm.edu/notes/faq/NextMersenne.html. The math is above my pay grade.

We have a theory that Mersenne primes "follow" a logarithmic line with a specific slope. We have data on the first 51 Mersennes. If someone with a good probability/statistics background could weigh in with an analysis of how well the 51 data points are matching the theory, then I and others here could hopefully learn a thing or two.
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.
Attached Files
File Type: pdf 1983_Wagstaff.pdf (1.35 MB, 69 views)
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Old 2018-12-12, 03:56   #20
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Quote:
Originally Posted by GP2 View Post
Methodology for my estimate:

Magnify the graph about 5x in a Paint program.

Under magnification, we see that the lines that draw the square-box plot points are not always sharp, but rather they are often blurred and smeared into two or even three adjacent lines of varying intensity. This reflects the fact that the exact center of the desired plot point usually does not map exactly onto an exact pixel point, so the plotting program uses blurring to compensate.

So we can use the relative paleness/brightness of the components of the blurred/smeared lines to assign a fractional vertical pixel position to both the top horizontal line and the bottom horizontal line of each square-box plot point, and then average them to find the center.

For the relative vertical pixel positions, with the zero reference point at the bottom horizontal bar of the square representing M46, I get:

Code:
      bottom   top         center            Mn            log(Mn)

M46   -0.2      9.1         4.45          42643801        17.56839
M47    0.3     10.0         5.15          43112609        17.57933
M48   20.2     30.0        25.1           57885161        17.87397
M49   37.05    46.7        41.875         74207281        18.12237
M50   39.8     49.4        44.6           77232917        18.16234
M51   44.3     54.0        49.15          xxxxxxxx        18.2296 to 18.2301
When I zoom Ernst's plot, there's no antialiasing, all the way up to 1600%.
Zoom first, then screen copy paste, and there's little or no antialiasing in the paint program input. A spreadsheet using pixel locations of X marks the center gives method error estimates provided by the knowns. You seem to be using the already known Mn values above, rather than computing estimates from the pixel locations for comparison to the knowns. What did I miss?
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Old 2018-12-12, 05:38   #21
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Quote:
Originally Posted by kriesel View Post
When I zoom Ernst's plot, there's no antialiasing, all the way up to 1600%.
Zoom first, then screen copy paste, and there's little or no antialiasing in the paint program input. A spreadsheet using pixel locations of X marks the center gives method error estimates provided by the knowns. You seem to be using the already known Mn values above, rather than computing estimates from the pixel locations for comparison to the knowns. What did I miss?
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?
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Old 2018-12-12, 06:09   #22
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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.
Wagstaff's analysis involves estimating the expected number of prime factors of a Mersenne number between two bounds A and B (on the second page, page 386).

If A = 2p and B = sqrt of the Mersenne number, then you can try to estimate the total number of factors a Mersenne number of a given size would be expected to have. I tried to do that in a post a few months ago, maybe someone who actually understands the math can correct it.

However, thanks to user TJAOI and his very systematic "by k" factoring, we can be quite certain that we know all factors of bit-length 65 or less of all Mersenne numbers with prime exponent less than 1 billion. As I posted last year in the "User TJAOI thread", empirically we can see that he does not miss any factors, except for one very minor glitch in 2014 that he very quickly corrected.

So we can use our factor data to check if the actual exact number of factors smaller than B = 2^65 matches what Wagstaff's heuristic predicts, for various ranges of p.

If our factor data is in fact consistent with the Poisson process implied by Wagstaff's analysis, then we should expect that the predicted number of Mersenne primes is also consistent with it.

I really think someone here with the required math knowledge could do the analysis and maybe make a paper out of it.
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