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 2020-10-09, 07:02 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 11548 Posts Why does a prime p divide a Fermat Number? Hi all, There is an interesting article in Mathematics Magazine (Vol 92, No 4, October 2020). The title of the article is "Why does a prime p divide a Fermat Number?" They reference fermatsearch.org. They state that a Fermat prime divisor divides one and only one Fermat number. Regards, Matt
2020-10-09, 08:27   #2
LaurV
Romulan Interpreter

Jun 2011
Thailand

22·23·97 Posts

Quote:
 Originally Posted by MattcAnderson Why does a prime p divide a Fermat Number?
Because he can, and he wants to do so.
The "unicity" of factors is known, and trivial to prove.

Last fiddled with by LaurV on 2020-10-09 at 08:28

2020-10-09, 08:52   #3
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

23·3·59 Posts

Quote:
 Originally Posted by MattcAnderson They state that a Fermat prime divisor divides one and only one Fermat number.
I've learnt in this nice way: F(n)=2+F(0)*F(1)*F(2)*...*F(n-1), so if we have a common divisor d of F(n) and F(m), where m<n then d|2, hence d=1 because all F(k) is odd.

 2020-10-09, 16:11 #4 LaurV Romulan Interpreter     Jun 2011 Thailand 100010110111002 Posts Yep, that's what wikipedia used to say for a while
2020-10-09, 20:12   #5
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

23×3×59 Posts

Quote:
 Originally Posted by LaurV Yep, that's what wikipedia used to say for a while
Really not read, in general I know the shortest proofs. Another way:
If 2^(2^m)==-1 mod d then for n>m taking this to the 2^(n-m)-th power:
2^(2^n)==1 mod d from this F(n)==2 mod d and the rest is the same. This could be even shorter proof when you write down, but needs more knowledge.

2020-10-09, 22:01   #6
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

5×1,831 Posts

Quote:
 Originally Posted by LaurV Because he can, and he wants to do so.
Are you sure that this is not the answer for the 'Why did the chicken cross the road?"

Then again, this answer is quite universal, just like "42". I like it.

 2020-10-13, 15:31 #7 LaurV Romulan Interpreter     Jun 2011 Thailand 213348 Posts That's not a "show off" for the fact I know the math. I may know it, I may not know it. I was barking more in that direction that people come and ask questions whose answers (and more details about subject) could be found by a simple search. Last fiddled with by LaurV on 2020-10-13 at 15:31
 2020-10-13, 22:50 #8 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 22×5×31 Posts We know that the Fermat numbers are defined as F(n) = 2^(2^n) + 1. Also, F(0) to F(4) are prime numbers. F(5) to F(32) are composite numbers. F(33) to F(35) are of unknown character. F(36) is composite. This data is available at - http://www.prothsearch.com/fermat.html I spent several years using trial division programs trying to find the smallest prime factor of F(34). So far we know that the smallest prime divisor of F(34) is greater than 7*10^17. Maybe someone else wants to work on this. The program (mmff) can be downloaded from fermatsearch.org Regards, Matt
2020-10-14, 01:05   #9
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

5·1,831 Posts

Quote:
 Originally Posted by MattcAnderson I spent several years using trial division programs trying to find the smallest prime factor of F(34).
If you used the programs that most people use (didn't write your own?) - then you don't know which F(n) you will find a factor for.

The stats show that you mostly reserved ranges for N=36.
This means that you can just as well find a factor of F(33) or F(32), if you keep at it.
You are not guaranteed to find the factor of F(34)!

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