20121012, 03:35  #1 
P90 years forever!
Aug 2002
Yeehaw, FL
5^{2}×311 Posts 
How unlucky?
I know we are dealing with large k values, but have we been inordinately unlucky in not finding a single factor yet?

20121012, 03:46  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5×7×277 Posts 
Unlucky?! I found 20!
The source works! But seriously unlucky for Fermats, I agree. 
20121012, 03:47  #3 
Aug 2002
2^{3}×1,051 Posts 
We don't see how luck plays in at all. It is what it is. (Right?)
Definition of LUCK a : a force that brings good fortune or adversity b : the events or circumstances that operate for or against an individual 
20121012, 04:31  #4 
Romulan Interpreter
"name field"
Jun 2011
Thailand
9870_{10} Posts 
I also say it works. It finds all (findable) known factors, why it should be missing unknown factors? (if this was the question, but I smell here the question is more about math, if you are in fact asking about expectancy to find any factor, like "were we unlucky, or the expectancy is really so low?", well, it is quite low too, for the ranges we are testing, only very few factors should be expected). I just think there are pure and simple no factors in the ranges we tested, except one factor which managed to stay hidden, but hiding is bad because now he is cornered somewhere and can't get out and I will put my paw on it soon...
Last fiddled with by LaurV on 20121012 at 04:35 
20121012, 06:00  #5  
"Åke Tilander"
Apr 2011
Sandviken, Sweden
2×283 Posts 
Quote:
I did quite some work for OBD and I thought for awhile that I was unfortunate, but then I found 2 factors so now I am more fortunate then I should be according to statistics I think. Someone could maybe do a little statistics here? Last fiddled with by aketilander on 20121012 at 06:02 

20121012, 13:45  #6 
P90 years forever!
Aug 2002
Yeehaw, FL
5^{2}×311 Posts 
Yes, it was a math question. With frmky churning out tons of work and several others contributing, I was wondering if the expected number of factors found was less than 1? 1 to 2? above 2? I guess I'm too lazy to go back through the posted results to come up with an exact figure.

20121013, 14:45  #7  
Einyen
Dec 2003
Denmark
3×5×7×31 Posts 
Quote:
I don't know how to calculate the odds of factors within a certain k range, but here is the completed ranges from results thread up to Batalov "N=25 to 2e15" Oct 13th (excluding the few fermat results with version 0.20). If someone knows the formula for the odds, I'll be happy to try to calculate it. Code:
Fermat: n k 25 500T2000T 28 550T1000T 29 550T1000T 33 700T1000T 34 700T1000T 37 4500T5000T 40 600T1000T 41 600T1000T 42 600T1000T 43 600T1000T 44 400T700T 45 500T1000T 50 300T1000T 51 350T1000T 52 300T1000T 53 200T1000T 54 200T1000T 55 200T1000T 56 200T1000T 57 280T1000T 58 280T1000T 60 200T1500T 61 200T1000T 62 200T1000T 63 200T1000T 71 300T1000T 72 300T1000T 73 300T1000T 74 300T1000T 83 35T281T 84 35T281T 85 35T281T 86 25T281T 87 25T281T 88 25T281T 89 25T281T 90 100T1000T 100 4T100T 101 16T100T 102 16T100T 103 16T100T 104 16T100T 105 16T100T 106 16T100T 107 16T100T 108 16T100T 109 16T100T 110 30T100T 111 30T100T 112 30T100T 113 30T100T 114 30T100T 115 30T100T 116 30T100T 117 30T100T 118 30T100T 119 30T100T 120 30T100T 121 30T100T 122 30T100T 123 30T100T 124 30T100T 125 30T100T 126 30T100T 127 30T100T 128 30T100T 129 30T100T 130 12T50T 131 12T50T 132 12T50T 133 12T50T 134 12T50T 135 12T50T 136 12T50T 137 12T50T 138 12T50T 139 12T50T k MM31 18000T25000T MM61 3573T10000T MM89 7T1300T MM107 4T1000T MM127 563T3700T and 4000T4600T and 5500T5800T Last fiddled with by ATH on 20121013 at 14:58 

20121013, 19:01  #8 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5×7×277 Posts 
It should be the probability of f=k*2^{N}+1 being prime (which is C/ln f) times probability of dividing a Fermat number which is 1/k, then sum (intergrate is fine) over the range. There's Bjorn/Riesel (1998) with a detailed treatment:

20121013, 23:37  #9 
Bemusing Prompter
"Danny"
Dec 2002
California
11·13·17 Posts 
I think it's more unlucky that we haven't found a new Mersenne prime since April 2009.

20121014, 11:15  #10 
"Åke Tilander"
Apr 2011
Sandviken, Sweden
2·283 Posts 
Well, between M[10,000,000 digits] and M[100,000,000 digits] we are supposed to find 6 Mersenne primes according to the theory (about 6 between M[10^{n} digits] and M[10^{n+1} digits]) so I would say we have been extremely lucky that have found 3 already.

20121014, 11:38  #11 
Einyen
Dec 2003
Denmark
3·5·7·31 Posts 
In your screenshot they have the number of primes k*2^{n}+1 for k<K is G(K)=K/(ln(K*2^{n})1).
So trying this for example on n=25 and 500T<k<2000T G(2000T)G(500T) = 2.88*10^{13} primes Now each of these you say has a 1/k chance of dividing a fermat number, so if we divide the number of primes with the average k in the interval which is 1250T it should be an ok estimate? 2.88*10^{13}/1250*10^{12} = 0.0231, so 2.31% chance of finding a fermat factor in that interval. Doing this for all the ranges gives expected fermat factors in the ranges done so far at 0.95, unless my calculations are all wrong? Code:
Fermat: n k 25 500T2000T 0.02305923714 28 550T1000T 0.01081348535 29 550T1000T 0.01067572583 33 700T1000T 0.006163534161 34 700T1000T 0.006089842254 37 4500T5000T 0.001704400774 40 600T1000T 0.008058379498 41 600T1000T 0.00796937526 42 600T1000T 0.007882315112 43 600T1000T 0.007797136063 44 400T700T 0.008464224672 45 500T1000T 0.0101876489 50 300T1000T 0.01566992576 51 350T1000T 0.01386193619 52 300T1000T 0.01536014979 53 200T1000T 0.01886175148 54 200T1000T 0.01867863009 55 200T1000T 0.01849902961 56 200T1000T 0.01832284948 57 280T1000T 0.01529457624 58 280T1000T 0.01515181805 60 200T1500T 0.02016286432 61 200T1000T 0.01748999027 62 200T1000T 0.01733242097 63 200T1000T 0.01717766503 71 300T1000T 0.01293148238 72 300T1000T 0.01282475399 73 300T1000T 0.01271977269 74 300T1000T 0.01261649593 83 35T281T 0.01727941477 84 35T281T 0.01714751685 85 35T281T 0.01701761711 86 25T281T 0.01816263451 87 25T281T 0.01802700773 88 25T281T 0.01789339132 89 25T281T 0.01776174093 90 100T1000T 0.0170126916 100 4T100T 0.01833508944 101 16T100T 0.01425457045 102 16T100T 0.01415798841 103 16T100T 0.01406270624 104 16T100T 0.01396869789 105 16T100T 0.01387593798 106 16T100T 0.0137844018 107 16T100T 0.01369406531 108 16T100T 0.01360490506 109 16T100T 0.01351689825 110 30T100T 0.009970885797 111 30T100T 0.009907309386 112 30T100T 0.009844538526 113 30T100T 0.009782558006 114 30T100T 0.009721352994 115 30T100T 0.009660909025 116 30T100T 0.009601211995 117 30T100T 0.009542248144 118 30T100T 0.009484004048 119 30T100T 0.009426466609 120 30T100T 0.009369623046 121 30T100T 0.009313460883 122 30T100T 0.009257967941 123 30T100T 0.009203132331 124 30T100T 0.009148942443 125 30T100T 0.009095386938 126 30T100T 0.009042454743 127 30T100T 0.00899013504 128 30T100T 0.008938417258 129 30T100T 0.008887291072 130 12T50T 0.01012143297 131 12T50T 0.01006383816 132 12T50T 0.01000689509 133 12T50T 0.009950592732 134 12T50T 0.009894920352 135 12T50T 0.00983986743 136 12T50T 0.009785423686 137 12T50T 0.009731579065 138 12T50T 0.009678323732 139 12T50T 0.009625648066  Total 0.9522655104 
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