20180719, 01:37  #12  
Einyen
Dec 2003
Denmark
7×419 Posts 
Quote:
Quote:
Fermat numbers grows very very fast (double exponential function) so 1/log(2^(2^N)) = 1/(2^N*log (2)) grows very very slowly so even if there are infinite number of terms it will still be a small finite value. If you sum 1/(2^N*log(2)) from N=33 to 1000 or higher in PARI/GP you get ~ 3.359*10^(10) which is about 1 in 2.98 billion. If we only count those above n=33 with no known factor the chance is about 1 in 3.4 billion. Last fiddled with by ATH on 20180719 at 01:44 

20180719, 02:34  #13 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}·3·79 Posts 
Barring my usual miscalculation, I found a factor for F12 which is not listed here:
https://oeis.org/A023394/list Is there somewhere I can have it submitted or otherwise included? Thank you in advance. Last fiddled with by a1call on 20180719 at 02:37 
20180719, 02:51  #14  
Sep 2003
2^{2}·3·5·43 Posts 
Quote:
The known factors of F12 are: 7 × 2^{14} + 1 = 114689 397 × 2^{16} + 1 = 26017793 973 × 2^{16} + 1 = 63766529 11613415 × 2^{14} + 1 = 190274191361 76668221077 × 2^{14} + 1 = 1256132134125569 17353230210429594579133099699123162989482444520899 × 2^{15} + 1 = 568630647535356955169033410940867804839360742060818433 The largest was discovered in 2010. People have been looking for a larger one since then, without success. If you find one that's not in the above list, chances are it's a composite factor consisting of two or more of the above multiplied together. If you've really found a new one, just post it here and become famous. But almost certainly you haven't. Last fiddled with by GP2 on 20180719 at 02:56 

20180719, 03:06  #15 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}·3·79 Posts 
Didn't think of having it factored.
2983954661377  F12 2983954661377 = 114689 x 26017793 
20180719, 03:09  #16  
Sep 2003
2^{2}·3·5·43 Posts 
Quote:
I wonder if there could possibly be easy additional factors to be found for F29 and higher, where little or no ECM can be done. I already mentioned the only known factors for F31 and F32, for the others they are: F29 has one known factor: 1120049 × 2^{31} + 1 F30 has two known factors: 149041 × 2^{32} + 1 and 127589 × 2^{33} + 1 

20180719, 03:53  #17 
Sep 2003
2^{2}·3·5·43 Posts 
As already mentioned, the paper by Boklan and Conway cited in the Wikipedia article on Fermat numbers estimates the odds of a new Fermat prime at less than one in a billion.
Interestingly, it also conjectures that there is only a finite number of Mersenne primes whose exponent p is a Sophie Germain prime (i.e., 2p + 1 is also prime). The known SophieGermainprime exponents of Mersenne primes are: 2, 3, 5, 89, 9689, 21701, 859433, 43112609. Edit: this is actually OEIS sequence A065406, except the last term needs to be added. The paper actually makes a more general conjecture: "There are only finitely many Mersenne primes 2^{p} − 1 where p is a prime and ap + b is also prime (for some fixed integers a and b where (b, p) = 1)." For the special case of Sophie Germain primes, a = 2 and b = 1. Last fiddled with by GP2 on 20180719 at 04:08 
20180719, 07:19  #18 
Banned
"Luigi"
Aug 2002
Team Italia
2·2,383 Posts 

20180719, 12:26  #19  
Einyen
Dec 2003
Denmark
2933_{10} Posts 
Quote:
So the tested limits for F32 is the limits for k*2^34, k*2^35, etc. 

20180719, 12:57  #20  
"Robert Gerbicz"
Oct 2005
Hungary
11·127 Posts 
Quote:


20180719, 13:25  #21  
Feb 2017
Nowhere
2^{2}×5×173 Posts 
Quote:
;) Quote:


20180719, 20:49  #22  
Einyen
Dec 2003
Denmark
2933_{10} Posts 
Quote:
Code:
21: 3p+10: 3 7 17 19 31 61 89 107 607 1279 2203 3217 9689 9941 110503 132049 216091 3021377 24036583 25964951 42643801 25: 3p+50: 3 7 13 17 19 61 89 127 521 607 2203 4253 9941 23209 44497 86243 132049 859433 1257787 1398269 6972593 20996011 24036583 25964951 30402457 12p+25: 3 7 13 17 31 61 89 127 521 607 1279 2281 3217 4253 4423 9689 11213 44497 86243 132049 216091 756839 1257787 1398269 43112609 42p+55: 2 3 7 13 17 19 61 89 107 521 1279 2203 2281 4253 4423 9689 9941 19937 86243 859433 1398269 6972593 32582657 37156667 43112609 27: 15p+532: 3 5 13 17 31 61 89 107 127 1279 2203 2281 3217 4253 4423 44497 86243 132049 859433 1398269 3021377 13466917 25964951 32582657 37156667 42643801 77232917 21p+880: 13 17 19 31 61 89 127 521 607 1279 2203 2281 3217 4423 19937 86243 110503 216091 756839 859433 1257787 1398269 2976221 6972593 20996011 37156667 74207281 52p+525: 13 17 31 61 89 107 127 521 607 1279 3217 9689 9941 19937 21701 44497 132049 216091 756839 859433 1257787 6972593 13466917 30402457 37156667 74207281 77232917 504p+55: 2 3 7 13 17 19 31 127 1279 2203 2281 3217 4253 4423 9689 11213 19937 21701 44497 110503 756839 859433 1257787 24036583 30402457 43112609 74207281 31: 33p+2590: 3 13 19 31 61 89 107 127 607 1279 2203 2281 3217 4253 4423 9689 9941 21701 44497 86243 132049 216091 756839 859433 1257787 6972593 13466917 24036583 37156667 42643801 57885161 

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