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Generalized Fermat Prime Search :
How many Genefer Primes?
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RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Found a web site that estimates the number of Generalized Fermat Primes in a sequence:
http://yves.gallot.pagespersoorange.fr/primes/stat.html
n = 524288, N = 2^19, bMin = 10, bMax = 700,000 (limit of BOINC testing)
Expected number of GF primes: 1.42
Number of GF primes found: 4 => error = +2.6
Poisson distribution:
Chance of no prime: 24.29% (100%)
Chance of 1 prime: 34.37% (76%)
Chance of 2 primes: 24.32% (41%)
Chance of 3 primes: 11.47% (17%)
Chance of 4 primes: 4.06% (6%)
Chance of 5 primes: 1.15% (1%)
Chance of 6 primes: 0.27% (0%)
n = 20, N = 2^20, bMin = 10, bMax = 600,000 (limit of GeneferX64)
Expected number of GF primes: 0.68
Poisson distribution:
Chance of no prime: 50.67% (100%)
Chance of 1 prime: 34.45% (49%)
Chance of 2 primes: 11.71% (15%)
Chance of 3 primes: 2.65% (3%)
Chance of 4 primes: 0.45% (1%)
n = 21, N = 2^21, bMin = 10, bMax = 495,000 (limit of GeneferX64)
Expected number of GF primes: 0.38
Poisson distribution:
Chance of no prime: 68.56% (100%)
Chance of 1 prime: 25.88% (31%)
Chance of 2 primes: 4.88% (6%)
Chance of 3 primes: 0.61% (1%)
I am sure some might have seen this previously but deserves a rerun.
Web site doesn't give option of n = 22.
Of course new programs could change the Blimits.
____________
 

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13648 ID: 53948 Credit: 285,444,507 RAC: 47,066

n = 524288, N = 2^19, bMin = 10, bMax = 700,000 (limit of BOINC testing)
Expected number of GF primes: 1.42
Number of GF primes found: 3 => error = +1.6
We've found 4 primes at 524288.
____________
My lucky number is 75898^{524288}+1  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13648 ID: 53948 Credit: 285,444,507 RAC: 47,066

To be a tiny bit more precise with the larger ranges where we've been testing:
N=262144:
Testing completed through b=864204
7 primes found so far (6 at PrimeGrId)
Expected number of primes: 3.41
Chance of no prime: 3.31% (100%)
Chance of 1 prime: 11.28% (97%)
Chance of 2 primes: 19.22% (85%)
Chance of 3 primes: 21.84% (66%)
Chance of 4 primes: 18.61% (44%)
Chance of 5 primes: 12.69% (26%)
Chance of 6 primes: 7.21% (13%)
Chance of 7 primes: 3.51% (6%)
Chance of 8 primes: 1.5% (2%)
Chance of 9 primes: 0.57% (1%)
Chance of 10 primes: 0.19% (0%)
N=524288:
Testing completed through b=706260
4 primes found so far (4 at PrimeGrid)
Expected number of primes: 1.43
Chance of no prime: 24.01% (100%)
Chance of 1 prime: 34.25% (76%)
Chance of 2 primes: 24.44% (42%)
Chance of 3 primes: 11.62% (17%)
Chance of 4 primes: 4.15% (6%)
Chance of 5 primes: 1.18% (2%)
Chance of 6 primes: 0.28% (0%)
N=1048576
Testing completed through b=2834 (not terribly informative)
No primes found
Expected number of primes: 0.01
Chance of no prime: 99.43% (100%)
Chance of 1 prime: 0.57% (1%)
If we increase that to where the leading edge of the search is (currently b=98298) we get this:
No primes found
Expected number of primes: 0.13
Chance of no prime: 87.72% (100%)
Chance of 1 prime: 11.49% (12%)
Chance of 2 primes: 0.75% (1%)
However, approximately 4000 of the 20,000 candidates with b<=98298 do not yet have a result returned.
We're also searching t 32768 and 65536, but those are a little trickier because unlike the higher ranges, some of those ranges were searched outside of PrimeGrid. So it's possible I'm missing a few primes.
N=32768
Completed through b=5984880
92 primes found (32 at PrimeGrid)
Expected # of primes found: 72.98
Chance of 51 primes: 0.14% (100%)
Chance of 52 primes: 0.19% (100%)
Chance of 53 primes: 0.27% (99%)
Chance of 54 primes: 0.36% (99%)
Chance of 55 primes: 0.48% (99%)
Chance of 56 primes: 0.62% (98%)
Chance of 57 primes: 0.79% (98%)
Chance of 58 primes: 1% (97%)
Chance of 59 primes: 1.24% (96%)
Chance of 60 primes: 1.5% (95%)
Chance of 61 primes: 1.8% (93%)
Chance of 62 primes: 2.12% (91%)
Chance of 63 primes: 2.45% (89%)
Chance of 64 primes: 2.8% (87%)
Chance of 65 primes: 3.14% (84%)
Chance of 66 primes: 3.47% (81%)
Chance of 67 primes: 3.78% (77%)
Chance of 68 primes: 4.06% (74%)
Chance of 69 primes: 4.3% (69%)
Chance of 70 primes: 4.48% (65%)
Chance of 71 primes: 4.6% (61%)
Chance of 72 primes: 4.67% (56%)
Chance of 73 primes: 4.66% (51%)
Chance of 74 primes: 4.6% (47%)
Chance of 75 primes: 4.48% (42%)
Chance of 76 primes: 4.3% (38%)
Chance of 77 primes: 4.07% (33%)
Chance of 78 primes: 3.81% (29%)
Chance of 79 primes: 3.52% (26%)
Chance of 80 primes: 3.21% (22%)
Chance of 81 primes: 2.89% (19%)
Chance of 82 primes: 2.57% (16%)
Chance of 83 primes: 2.26% (13%)
Chance of 84 primes: 1.97% (11%)
Chance of 85 primes: 1.69% (9%)
Chance of 86 primes: 1.43% (7%)
Chance of 87 primes: 1.2% (6%)
Chance of 88 primes: 1% (5%)
Chance of 89 primes: 0.82% (4%)
Chance of 90 primes: 0.66% (3%)
Chance of 91 primes: 0.53% (2%)
Chance of 92 primes: 0.42% (2%)
Chance of 93 primes: 0.33% (1%)
Chance of 94 primes: 0.26% (1%)
Chance of 95 primes: 0.2% (1%)
Chance of 96 primes: 0.15% (1%)
Chance of 97 primes: 0.11% (0%)
N=65536
Completed through b=2894432
24 primes found (12 at PrimeGrid)
(There were also 4 primes found elsewhere at a much higher B)
Expected number of primes: 35.85
Chance of 20 primes: 0.14% (100%)
Chance of 21 primes: 0.23% (100%)
Chance of 22 primes: 0.38% (99%)
Chance of 23 primes: 0.59% (99%)
Chance of 24 primes: 0.88% (99%)
Chance of 25 primes: 1.26% (98%)
Chance of 26 primes: 1.74% (96%)
Chance of 27 primes: 2.31% (95%)
Chance of 28 primes: 2.96% (92%)
Chance of 29 primes: 3.66% (89%)
Chance of 30 primes: 4.38% (86%)
Chance of 31 primes: 5.06% (81%)
Chance of 32 primes: 5.67% (76%)
Chance of 33 primes: 6.16% (71%)
Chance of 34 primes: 6.5% (64%)
Chance of 35 primes: 6.66% (58%)
Chance of 36 primes: 6.63% (51%)
Chance of 37 primes: 6.43% (45%)
Chance of 38 primes: 6.06% (38%)
Chance of 39 primes: 5.57% (32%)
Chance of 40 primes: 5% (27%)
Chance of 41 primes: 4.37% (22%)
Chance of 42 primes: 3.73% (17%)
Chance of 43 primes: 3.11% (13%)
Chance of 44 primes: 2.53% (10%)
Chance of 45 primes: 2.02% (8%)
Chance of 46 primes: 1.57% (6%)
Chance of 47 primes: 1.2% (4%)
Chance of 48 primes: 0.9% (3%)
Chance of 49 primes: 0.66% (2%)
Chance of 50 primes: 0.47% (1%)
Chance of 51 primes: 0.33% (1%)
Chance of 52 primes: 0.23% (1%)
Chance of 53 primes: 0.15% (0%)
Chance of 54 primes: 0.1% (0%)
____________
My lucky number is 75898^{524288}+1  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Michael, thanks for the clarifications. Bottom line is more WU's tested, more chance of finding a prime!
____________
 

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

N=1048576
Testing completed through b=2834 (not terribly informative)
No primes found
Expected number of primes: 0.01
Chance of no prime: 99.43% (100%)
Chance of 1 prime: 0.57% (1%)
If we increase that to where the leading edge of the search is (currently b=98298) we get this:
No primes found
Expected number of primes: 0.13
Chance of no prime: 87.72% (100%)
Chance of 1 prime: 11.49% (12%)
Chance of 2 primes: 0.75% (1%)
However, approximately 4000 of the 20,000 candidates with b<=98298 do not yet have a result returned.
Update for n = 20, N = 1048576
Testing completed through b = 201,920
No primes found
Expected number of primes: 0.25
Chance of no prime: 77.76% (100%)
Chance of 1 prime: 19.56% (22%)
Chance of 2 primes: 2.46% (3%)
Chance of 3 primes: 0.21% (0%)
The number in the bracket is the rollup (chance of at least that many primes).
We have a total of 126,418 candidates at n = 20.
41,102 of these have now been tested.
 

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Thanks to Yves for updating his stats calculator! I now covers 2^22, 2^23 and 2^24.
You might need to do an F5refresh to get the new version.
For n = 22, N = 4194304, bMin = 2, bMax = 475,000 (limit of BOINC testing)
Expected number of GF primes: 0.16
Chance of no prime: 84.82% (100%)
Chance of 1 prime: 13.97% (15%)
Chance of 2 primes: 1.15% (1%)
Testing completed through b = 10428
No primes found  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 260,600
No primes found
Expected number of primes: 0.32
Chance of no prime: 72.81% (100%)
Chance of 1 prime: 23.1% (27%)
Chance of 2 primes: 3.66% (4%)
Chance of 3 primes: 0.39% (0%)
We have a total of 126,418 candidates at n = 20.
52,156 of these have now been tested.  


Can we get an update on these stats?
b=>305928
http://www.primegrid.com/stats_genefer.php
The window is closing...  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 305,928
No primes found
Expected number of primes: 0.37
Chance of no prime: 69.27% (100%)
Chance of 1 prime: 25.44% (31%)
Chance of 2 primes: 4.67% (5%)
Chance of 3 primes: 0.57% (1%)
We have a total of 126,418 candidates at n = 20.
67,282 of these have now been tested.  

axnVolunteer developer Send message
Joined: 29 Dec 07 Posts: 285 ID: 16874 Credit: 28,027,106 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 305,928
No primes found
Expected number of primes: 0.37
Chance of no prime: 69.27% (100%)
Chance of 1 prime: 25.44% (31%)
Chance of 2 primes: 4.67% (5%)
Chance of 3 primes: 0.57% (1%)
We have a total of 126,418 candidates at n = 20.
67,282 of these have now been tested.
are these for tests completed so far or tests to be done? if the former, can we get the latter as well?  

HonzaVolunteer moderator Volunteer tester Project scientist Send message
Joined: 15 Aug 05 Posts: 1909 ID: 352 Credit: 4,397,220,652 RAC: 5,260,712

are these for tests completed so far or tests to be done? if the former, can we get the latter as well?
According to subproject Genefer page, they are already done.
In first post, you can see expected primes for bMax = 600,000 (limit of GeneferX64)
____________
My stats
Badge score: 1*1 + 5*1 + 8*3 + 9*11 + 10*1 + 11*1 + 12*3 = 186  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 402,080
No primes found
Expected number of primes: 0.47
Chance of no prime: 62.42% (100%)
Chance of 1 prime: 29.42% (38%)
Chance of 2 primes: 6.93% (8%)
Chance of 3 primes: 1.09% (1%)
Chance of 4 primes: 0.13% (0%)
We have a total of 126,418 candidates at n = 20.
76,245 of these have now been tested.  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 504,368
No primes found
Expected number of primes: 0.58
Chance of no prime: 55.99% (100%)
Chance of 1 prime: 32.47% (44%)
Chance of 2 primes: 9.42% (12%)
Chance of 3 primes: 1.82% (2%)
Chance of 4 primes: 0.26% (0%)
We have a total of 126,418 candidates at n = 20.
96,043 of these have now been tested.  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 605,258
No primes found
Expected number of primes: 0.69
Chance of no prime: 50.39% (100%)
Chance of 1 prime: 34.54% (50%)
Chance of 2 primes:11.84% (15%)
Chance of 3 primes: 2.70% (3%)
Chance of 4 primes: 0.46% (1%)
We have a total of 126,418 candidates at n = 20.
116,257 of these have now been tested.  

RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1138 ID: 120786 Credit: 268,621,444 RAC: 0

Update for n = 20, N = 1048576
Testing completed through b = 821,904
No primes found
Expected number of primes: 0.91
Chance of no prime: 40.34% (100%)
Chance of 1 prime: 36.62% (60%)
Chance of 2 primes:16.62% (23%)
Chance of 3 primes: 5.03% (6%)
Chance of 4 primes: 1.14% (1%)
Chance of 5 primes: 0.21% (0%)
A total of 150,346 WU's have now been completed at n = 20.
687 WU's are currently in progress, leading edge at b = 835,224.  

Message boards :
Generalized Fermat Prime Search :
How many Genefer Primes? 