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Old 2016-01-12, 07:49   #111
ewmayer
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Quote:
Originally Posted by Batalov View Post
On the contrary, this is a ridiculously small jump by Poisson statistics.
Not necessarily - 1.4*57885161 ~= 81M, and 70M ~= 1.2*57885161, so at the bottom of the 7******* range we're still 1.2x larger than the last one - not at all extreme - and at the top of the range we're very close to the average ratio for the purported Poisson distribution.

On the other hand, going back 10 previous M-primes we get 13466917*1.4^10
= 389,536,843 which is much larger than the latest one.

OTOOH, such distributions are well-known to be 'clumpy', so much depends on the chosen starting point: going back 20 M-primes we get 110503*1.4^20 = 92,455,932, which is again quite close (given the largeness of the logarithmic interval) to the latest one.

Quote:
Originally Posted by Dubslow View Post
Edit: For AVX/Prime95, 4000K (!= 4M) is for exponents 76210000-76790000, 4032K for exponents up to 77990000, then 4096K=4M for exponents through 85200000.
You sure about that last one? Based on my estimates p ~= 85M should require FFT of ~4500K.

Last fiddled with by ewmayer on 2016-01-12 at 07:55
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Old 2016-01-12, 07:54   #112
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1.4 is a rather dubious number. Is that a new conjecture? ;-)

There is a large difference between 1.4^10 and 1.47576^10; that's roughly 28.93 and 49, so (quote:)
going back 10 previous M-primes we get 13466917*1.47576^10
= 389,536,843 659,820,084 which would have been monstrous.

Not much better going back 20 previous M-primes:
110503*1.47576^20 = 92,455,932, 265,270,382, which is again quite close monstrous.

Last fiddled with by Batalov on 2016-01-12 at 08:02
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Old 2016-01-12, 07:58   #113
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Quote:
Originally Posted by ewmayer View Post

You sure about that last one? Based on my estimates p ~= 85M should require FFT of ~4500K.
Considering I read those values from the jump table in George's mult.asm, yeah I'm pretty confident. Unless I somehow misinterpreted it, which honestly is probably far more likely than this prime.

Edit: I two different jump tables after the AVX comment. The top one is directly below the AVX comment, and I'm not sure what the second jump table is.
Code:
        PRCSTRT 76210000, 4096000, 0.028093
        PRCENTRY2               yfft_r4dwpn_4000K_12800_2, 8005632
        PRCENTRY2               yfft_r4dwpn_4000K_6400_2, 6982912, I7_64
        DD                      0
        PRCSTRT 76790000, 4128768, 0.027915
        PRCENTRY2               yfft_r4dwpn_4032K_9216_2, 6665728
        PRCENTRY2               yfft_r4dwpn_4032K_4608_2, 5633280
        PRCENTRY2               yfft_r4dwpn_4032K_2304_4, 6786304
        DD                      0
        PRCSTRT 77990000, 4194304, 0.026160
        PRCENTRY2               yfft_r4dwpn_4M_14_4, 9644800
        PRCENTRY2               yfft_r4dwpn_4M_14_2, 9644800
        PRCENTRY2               yfft_r4dwpn_4M_14_1, 9644800
        PRCENTRY2               yfft_r4dwpn_4M_13_4, 8500736
        PRCENTRY2               yfft_r4dwpn_4M_13_2, 8500736, I7 + FMA3_64
        PRCENTRY2               yfft_r4dwpn_4M_13_1, 8500736
        PRCENTRY2               yfft_r4dwpn_4M_12_4, 5320704
        PRCENTRY2               yfft_r4dwpn_4M_12_2, 5320704
        PRCENTRY2               yfft_r4dwpn_4M_12_1, 5320704
        PRCENTRY2               yfft_r4dwpn_4M_11_4, 6458368
        DD                      0
        PRCSTRT 85200000, 4587520, 0.031930
        PRCENTRY2               yfft_r4dwpn_4480K_10240_2, 7414272
        PRCENTRY2               yfft_r4dwpn_4480K_5120_2, 6263040, I7
        DD                      0
        PRCSTRT 87400000, 4718592, 0.030538
        PRCENTRY2               yfft_r4dwpn_4608K_12288_2, 8322304
        PRCENTRY2               yfft_r4dwpn_4608K_12288_1, 8322304
        PRCENTRY2               yfft_r4dwpn_4608K_9216_2, 9464320, I7_64
        PRCENTRY2               yfft_r4dwpn_4608K_9216_1, 9464320
        PRCENTRY2               yfft_r4dwpn_4608K_6144_2, 7122944, I7_32 + FMA3_64
        PRCENTRY2               yfft_r4dwpn_4608K_4608_4, 5932032
        PRCENTRY2               yfft_r4dwpn_4608K_3072_4, 8450560
        PRCENTRY2               yfft_r4dwpn_4608K_2304_4, 7235072

...

        PRCSTRT 76050000, 4096000, 0.029047
        PRCENTRY2               yfft_r4dwpn_4000K_ac_6400_2, 4583424, I7_64
        DD                      0
        PRCSTRT 77950000, 4194304, 0.027172
        PRCENTRY2               yfft_r4dwpn_4M_ac_14_4, 9577984
        PRCENTRY2               yfft_r4dwpn_4M_ac_14_2, 9577984
        PRCENTRY2               yfft_r4dwpn_4M_ac_14_1, 9577984
        PRCENTRY2               yfft_r4dwpn_4M_ac_13_4, 5068288
        PRCENTRY2               yfft_r4dwpn_4M_ac_13_2, 5068288, I7 + FMA3_64
        PRCENTRY2               yfft_r4dwpn_4M_ac_13_1, 5068288
        PRCENTRY2               yfft_r4dwpn_4M_ac_12_4, 3998720
        PRCENTRY2               yfft_r4dwpn_4M_ac_12_2, 3998720
        PRCENTRY2               yfft_r4dwpn_4M_ac_12_1, 3998720
        DD                      0
Either way I'm pretty certain this was (and still is by a decent margin) a Cat 4 LL assignment; quite lucky.

Last fiddled with by Dubslow on 2016-01-12 at 08:07
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Old 2016-01-12, 08:18   #114
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Quote:
Originally Posted by Batalov View Post
1.4 is a rather dubious number. Is that a new conjecture? ;-)
That's the approximate average-ratio as I recall it from way-back-when ... of course new data help us continue to refine it.

Perhaps someone with a handy-dandy stats best-fit suite can input the known M-prime exponents and compute the best-fit params for a Poisson distribution, and - if possible - 'goodness of fit' to the given distribution.
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Old 2016-01-12, 08:23   #115
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Quote:
Originally Posted by Dubslow View Post
Considering I read those values from the jump table in George's mult.asm, yeah I'm pretty confident. Unless I somehow misinterpreted it, which honestly is probably far more likely than this prime.
These are the maximum exponents testable by the FFT.
If 77990000 is the maximum for 4096, then 77990000 * 4480/4096 = 85301562.5 would be the maximum for 4480 (with some reduction due to fewer bits/limb).
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Old 2016-01-12, 08:31   #116
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Quote:
Originally Posted by axn View Post
These are the maximum exponents testable by the FFT.

Last fiddled with by Dubslow on 2016-01-12 at 08:33
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Old 2016-01-12, 10:55   #117
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The Xeon run with a larger FFT and a separate CUDA run also came up positive. I think we have a winner here.

The clLucas run and mLucas runs are both on track to back those up right now.
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Old 2016-01-12, 10:59   #118
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Cool some statistics for the weary eyes

If we take the first 40 M-primes and do a linear regression of log2 log2 Mn ~ n, then the 95% confidence interval for the slope's estimate is (0.557, 0.587) and both Wagstaff's and Eberhart's conjectured slopes are within this CI.
Recall that Wagstaff's \hat{\beta_1} = {1 \over {e ^ \gamma }} = 0.561458..., while Eberhart's = log2 3/2 = 0.5849625...

If we take the first 48 known M-primes, then the 95% confidence interval for the slope's estimate is (0.5366, 0.5659) and Wagstaff's is in 95% CI, but Eberhart's is not (in other words, Eberhart's, of "K-Y", conjecture can be rejected at 95% confidence level).

If we take the first 49 known M-primes, then the confidence interval slides slightly lower still, but Wagstaff's is still in the 95% CI, and Eberhart's can be rejected with slighly higher confidence level. (Note that if there exist "missed" M-primes, the slope will be lower still.)

The elementary implementation in R (following B.Caffo's class notation, if someone wants to check):
Code:
x <- 1:48 
y <- c(1,1.584962501,2.321928095,2.807354922,3.700439718,4.087462841,4.247927513,4.95419631,5.930737338,6.475733431,6.741466986,6.988684687,9.025139562,9.245552706,10.32080055,11.10525378,11.15545073,11.65150022,12.05426514,12.11080953,13.24213206,13.27917527,13.4528847,14.28316072,14.4054739,14.50239674,15.44142045,16.39611974,16.75372601,17.01071385,17.72127946,19.52962691,19.71302565,20.2624562,20.4152105,21.50505023,21.52677478,22.73326384,23.68291627,24.32361192,24.51872848,24.63006217,24.85768459,24.95760092,25.14711776,25.3458327,25.36160654,25.78669022) 

fit <- lm(y ~ x) 
sumCoef <- summary(fit)$coefficients 
sumCoef[2,1] + c(-1, 1) * qt(.975, df = fit$df) * sumCoef[2, 2] 
[1] 0.5365694 0.5658668

> summary(fit)

Call:
lm(formula = y ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.71476 -0.45148 -0.07631  0.55816  1.24279 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.042977   0.204827   5.092 6.45e-06 ***
x           0.551218   0.007277  75.743  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6985 on 46 degrees of freedom
Multiple R-squared:  0.992,    Adjusted R-squared:  0.9919 
F-statistic:  5737 on 1 and 46 DF,  p-value: < 2.2e-16
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Old 2016-01-12, 11:47   #119
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Quote:
Originally Posted by Batalov View Post
If we take the first 49 known M-primes, then the confidence interval slides slightly lower still, but Wagstaff's is still in the 95% CI, and Eberhart's can be rejected with slighly higher confidence level. (Note that if there exist "missed" M-primes, the slope will be lower still.)
If M50 turns out to be the next unverified exponent after M49 (and there are no other smaller primes), will that pull down the CI enough to reject Wagstaff's as well. Alternately, how high should M50 be (with enough precision to not compromise the identity of M49) to keep Wagstaff in play?
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Old 2016-01-12, 12:35   #120
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Code:
> fit <- lm(y - 0.5849625*x ~ x)
> summary(fit)
Call:lm(formula = y - 0.5849625 * x ~ x)
Residuals:     Min       1Q   Median       3Q      Max 
-1.71476 -0.45148 -0.07631  0.55816  1.24279 
Coefficients:
Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.042977   0.204827   5.092 6.45e-06 ***
x           -0.033744   0.007277  -4.637 2.94e-05 ***
---Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 
Residual standard error: 0.6985 on 46 degrees of freedom
Multiple R-squared:  0.3185,    Adjusted R-squared:  0.3037 
F-statistic:  21.5 on 1 and 46 DF,  p-value: 2.94e-05
I get a p-value of 2.94e-05.
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Old 2016-01-12, 13:06   #121
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Quote:
Originally Posted by Madpoo View Post
I crunched some data myself because I wondered if it were easily guessable from the known info.

In the 4M FFT size (AVX2 table) there are 60,100 currently unverified exponents or with no known factors. Not only that but over 22K are unassigned, so it's kind of a big pool. Had it not been for the 4M FFT slip, it'd be even more obscure. LOL

And besides, the method(s) used to hide it on the website seem to be just fine. I did uncover one odd quirk that may have tipped someone off if they were *really* diligent, which I'll bring up with George to make sure that's not the case next time, but otherwise, yeah, I think it's doing a good job of keeping it under wraps.

I don't think anyone would stumble on it, but you're a bunch of smart folks and some of you are very persistent. Well, not too much longer before it's public anyway.
Aha, I think I know what you mean. In that case the new M-prime exponent is 78,xxx,xxx ? But I may be wrong as so many times before!
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