[quote=Prime95;177017]and a second prime95 cannot serve as the "official" double-check because it doesn't pass the "independent software" requirement.[/quote]
Pity, because Kevin (with 4 cores) was due to complete at the same
time as Tony.
Can you shed any light on the interim residue discrepancy, or would
that let the cat out of the bag?

[QUOTE=davieddy;177070]Can you shed any light on the interim residue discrepancy[/QUOTE]Well it would be easy to test for yourself. Just use a very small number like p=31 or something and print residues every iteration for first and double check modes. You can then compare to a manually computed set of residues. Easy.

Why all the guessing? It can be proved in a matter of minutes!

[QUOTE=Mr. P-1;176184]Can "slightly better" be quantified?

Could this be tested empirically by seeing if those p where p-1 is smooth really do have fewer factors?[/QUOTE]

I have no idea how to quantify this. An empirical test is the best I can think of. Only relatively small divisors should be affected, so one might check if those 2[SUP][I]p[/I][/SUP] where [I]p[/I]-1 has at least [I]n[/I] divisors are more likely to survive trial division to, say, 2[SUP]40[/SUP].

Alex

[QUOTE=philmoore;177001]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]

I think it is better to do simulation, using [URL="http://primes.utm.edu/notes/faq/NextMersenne.html"]http://primes.utm.edu/notes/faq/NextMersenne.html[/URL] conjecture for the probability that Mp is prime. 2000 simulations for the [2,5*10^7] interval, gives:
[code]
The most number of Mersenne primes in a [x,2.06*x] interval (for the exponent),
what we have after the verification for M40-M47 are 8 primes.
3 Mersenne primes in 'small' interval: 5
4 Mersenne primes in 'small' interval: 110
5 Mersenne primes in 'small' interval: 536
6 Mersenne primes in 'small' interval: 710
7 Mersenne primes in 'small' interval: 432
8 Mersenne primes in 'small' interval: 162
9 Mersenne primes in 'small' interval: 40
10 Mersenne primes in 'small' interval: 5
[/code]

So it isn't very rare 8 or more primes, the probability is 10.35%, close to the easy Poission estimation.
I've also counted the number of Mersenne primes in the runs. Note that using the conjecture the expected number of primes is about 48.9 (I've included p=2 in every cases).
[code]
26 Mersenne primes: 1
27 Mersenne primes: 0
28 Mersenne primes: 2
29 Mersenne primes: 0
30 Mersenne primes: 1
31 Mersenne primes: 4
32 Mersenne primes: 2
33 Mersenne primes: 3
34 Mersenne primes: 4
35 Mersenne primes: 13
36 Mersenne primes: 15
37 Mersenne primes: 13
38 Mersenne primes: 39
39 Mersenne primes: 37
40 Mersenne primes: 41
41 Mersenne primes: 66
42 Mersenne primes: 72
43 Mersenne primes: 81
44 Mersenne primes: 94
45 Mersenne primes: 100
46 Mersenne primes: 117
47 Mersenne primes: 95
48 Mersenne primes: 134
49 Mersenne primes: 121
50 Mersenne primes: 113
51 Mersenne primes: 134
52 Mersenne primes: 130
53 Mersenne primes: 106
54 Mersenne primes: 93
55 Mersenne primes: 86
56 Mersenne primes: 62
57 Mersenne primes: 56
58 Mersenne primes: 36
59 Mersenne primes: 36
60 Mersenne primes: 27
61 Mersenne primes: 23
62 Mersenne primes: 11
63 Mersenne primes: 10
64 Mersenne primes: 9
65 Mersenne primes: 3
66 Mersenne primes: 5
67 Mersenne primes: 2
68 Mersenne primes: 1
69 Mersenne primes: 1
70 Mersenne primes: 0
71 Mersenne primes: 0
72 Mersenne primes: 0
73 Mersenne primes: 1
[/code]

[quote=R. Gerbicz;177080]I think it is better to do simulation, using [URL]http://primes.utm.edu/notes/faq/NextMersenne.html[/URL] conjecture for the probability that Mp is prime. 2000 simulations for the [2,5*10^7] interval, gives:
[code]
The most number of Mersenne primes in a [x,2.06*x] interval (for the exponent),
what we have after the verification for M40-M47 are 8 primes.
3 Mersenne primes in 'small' interval: 5
4 Mersenne primes in 'small' interval: 110
5 Mersenne primes in 'small' interval: 536
6 Mersenne primes in 'small' interval: 710
7 Mersenne primes in 'small' interval: 432
8 Mersenne primes in 'small' interval: 162
9 Mersenne primes in 'small' interval: 40
10 Mersenne primes in 'small' interval: 5
[/code]

So it isn't very rare 8 or more primes, the probability is 10.35%, close to the easy Poission estimation.
[/quote]

Since you have chosen x to 2.06x as your interval,
and 2.06 is M47/M40 (exponents of) I don't think we should
count both M40 [I]and [/I]M47 in that interval. So we should call it
a run of 7 rather than 8 primes, much more likely (32%)..

[QUOTE=davieddy;177085]Since you have chosen x to 2.06x as your interval,
and 2.06 is M47/M40 (exponents of) I don't think we should
count both M40 [I]and [/I]M47 in that interval. So we should call it
a run of 7 rather than 8 primes, much more likely (32%)..[/QUOTE]

For x=20996011, after the verification we'll know 8 primes in [x,2.06*x],
2.06*x>43112609.

[quote=R. Gerbicz;177086]For x=20996011, after the verification we'll know 8 primes in [x,2.06*x],
2.06*x>43112609.[/quote]

Not unless x is chosen carefully i.e. very close to M40.
This is bias.
I still think you are "double counting" like saying the are 8 primes in
the interval M1-M8, 8 in the interval M8-M16...etc.

Another way of looking at it is to ask for the probability of finding
8 or more primes AFTER M39 all <= 43112609, giving us anterval of [x,3*x]
(roughly)

PS I might not have understood exactly what you did in your simulation.
Did you generate mock sequences of "M Primes" based on the probability, then
take x to be each prime in turn, then count the number of primes
>=x and < 2.06*x ?

.

[QUOTE=davieddy;177090]Not unless x is chosen carefully i.e. very close to M40.
This is bias.
I still think you are "double counting" like saying the are 8 primes in
the interval M1-M8, 8 in the interval M8-M16...etc.

Another way of looking at it is to ask for the probability of finding
8 or more primes AFTER M39 all <= 43112609, giving us anterval of [x,3*x]
(roughly)

PS I might not have understood exactly what you did in your simulation.
Did you generate mock sequences of "M Primes" based on the probability, then
take x to be each prime in turn, then count the number of primes
>=x and < 2.06*x ?

.[/QUOTE]

OK, I understand you, choosing r as ratio of two exponents of Mersenne primes increase the count in an "unfair" way. What I wanted to point out that even for r=2.06 the 8 Mesenne primes isn't a big surprise, a 10% event isn't very rare.

Yes, the simulation was that: by probab(p)=if(p%4==3,a=2,a=6);return(1.781*log(a*p)/(p*log(2))) probability I've choosen Mp as prime for every p>2. Note that the interseting fact that probab(3) and probab(5)
are higher than 1 so M2,M3,M5 were always in the sequence as primes.

The another way would be: simulate also the sequence of primes: if n>1 then by 1/log(n) probability choose n as prime and if it is choosen prime then by probab(p) probability Mp is prime. However this simulation is slower than the previous and I think in theory it gives the same distribution.

[quote=R. Gerbicz;177094]What I wanted to point out that even for r=2.06 the 8 Mesenne primes isn't a big surprise, a 10% event isn't very rare.
[/quote]
That is exactly what I meant by "No need to invoke divine intervention"
in my post which initiated this discussion.
Many thanks for your contribution.

[QUOTE=davieddy;177070]Pity, because Kevin (with 4 cores) was due to complete at the same
time as Tony.
Can you shed any light on the interim residue discrepancy, or would
that let the cat out of the bag?[/QUOTE]

Apparently there wasn't supposed to be any interim residue discrepancy (besides the +2 shift between glucas/mlucas and mprime/prime95, which was accounted for). No point in finishing the test if the interim residues show that something already went wrong.

[quote=Kevin;177105]Apparently there wasn't supposed to be any interim residue discrepancy (besides the +2 shift between glucas/mlucas and mprime/prime95, which was accounted for). No point in finishing the test if the interim residues show that something already went wrong.[/quote]
Are you sure, though, that you weren't simply testing the wrong exponent from the start? If that's the case, then the interim residuals would be different no matter what, and you may as well finish the test rather than letting the already completed work go to waste. No harm in doing a plain old doublecheck on an exponent that needs it. :smile: