20100125, 20:23  #1 
Banned
"Luigi"
Aug 2002
Team Italia
4800_{10} Posts 
New Fermat factor!
Today, January 25th 2010 Sergei Maiorov found the first Fermat factor of 2010 using Fermat.exe 4.4: 84977118993.2^520+1 divides F_517 !
Congratulations go to Sergei and the 100 other FermatSearch followers! Luigi 
20100125, 21:45  #2 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17×251 Posts 
Congrats to all!
The size of these numbers and factors is truly astounding compared even to the largest Mersenne numbers being worked on. The number of digits in the number of digits in F_517 is 156, (if I understand this page at Wolfram Alpha correctly) and the factor is 168 digits, or 557 bits, long! Incredible. (I was going to ask if anybody had tried proving the cofactor composite/prime before I noticed how big F_517 is...now I see that'll probably have to wait a few centuries) Last fiddled with by MiniGeek on 20100125 at 21:48 
20100125, 22:11  #3  
∂^{2}ω=0
Sep 2002
República de California
2·7·829 Posts 
Nice  but ...
Quote:
k * n * (effort to multiply a pair of nbit integers). I don't mean to kill anyone's enthusiasm here, just to suggest that the excitement be somewhat proportional to the magnitude of the accomplishment, rather than the Fermat number in question, whose magnitude is a very misleading measure in this respect. Your friendly local neighborhood buzzkill, Ernst 

20100128, 09:44  #4 
Sep 2004
215_{16} Posts 
I read no new largest composite fermat has been found for almost 10 years. It seems like it should be easy to test if arbitrary large numbers are divisible by primes up to 30 and show that almost all fermats are composite? or do they not follow regular rules. I'd like to break that record if it would just be less than a few cpu weeks.

20100128, 10:39  #5 
"William"
May 2003
New Haven
100100111001_{2} Posts 
Fermat Numbers belong the class of numbers a^b +/ 1. It's known that factors either divide b or are b*k+1. b is pretty large pretty quickly for Fermat numbers.
William 
20100128, 17:05  #6  
Aug 2005
Seattle, WA
1,669 Posts 
Quote:
If you meant 30 digits, I can only suggest that you try it since it's so easy. The smallest Fermat number of unknown character (i.e. prime vs. composite) is F33. Exercise: how many digits does that number have? How much memory is required to store it? How much memory does it take to run one ECM curve on it with a B1 of 250000 (the 30digit level)? How long does that one curve take? What does "almost all" mean here? There are after all an infinite number of Fermat numbers. How do expect showing one of them to be composite will affect the rest? I'm not sure what you mean by "regular rules" here, but it may be useful to know that no prime can divide more than one Fermat number. E.g. 2424833 divides F_9, so it can't possibly divide any other Fermat number. Is that an example of not following regular rules in some way? Does it affect your opinion on whether it's feasible to show that "almost all" Fermat numbers are composite? < Note: I'm not trying to be snippy here; I really don't know what you meant by "regular rules" and "almost all" Fermat numbers, and I'm wondering if this addresses that in some way. What record do you mean? 

20100128, 17:38  #7  
Oct 2004
Austria
2×17×73 Posts 
Quote:
I don't think that it is possible to prove that "almost all" Fermat numbers are composite by mere trial factoring. 

20100128, 17:43  #8  
Nov 2003
2^{2}·5·373 Posts 
Quote:
form a set of density 0. However, the exceptions can still form an infinite set. Example. Almost all integers are composite. But there are an infinite number of exceptions. 

20100128, 17:45  #9  
Nov 2003
16444_{8} Posts 
Quote:
and "almost all". 

20100128, 17:47  #10 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
5818_{10} Posts 
I thought that "almost all" would be a too imprecise phrase for you.

20100128, 19:15  #11 
Aug 2006
2^{2}·1,493 Posts 

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