20180719, 01:37  #12  
Einyen
Dec 2003
Denmark
110001101010_{2} 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}·269 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
2585_{10} 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}×269 Posts 
Didn't think of having it factored.
2983954661377  F12 2983954661377 = 114689 x 26017793 
20180719, 03:09  #16  
Sep 2003
5·11·47 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
5·11·47 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
11·439 Posts 

20180719, 12:26  #19  
Einyen
Dec 2003
Denmark
2×7×227 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
2·3^{2}·83 Posts 
Quote:


20180719, 13:25  #21  
Feb 2017
Nowhere
3^{3}×5×37 Posts 
Quote:
;) Quote:


20180719, 20:49  #22  
Einyen
Dec 2003
Denmark
2×7×227 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 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Prime95 potential race on WORKER_THREADS_ACTIVE and WORKER_THREADS_STOPPING  Explorer09  Software  2  20170309 08:17 
Would you give up your cat ownership if it increased your potential virility?  jasong  jasong  21  20160528 09:59 
A potential cause of Windows lowmemory messages  cheesehead  Software  14  20130516 00:45 
LowStress Job with High Potential? Mathematician  cheesehead  Lounge  20  20090605 20:24 
Formula to calculate number of potential factors?  Fusion_power  Miscellaneous Math  13  20051024 17:58 