20120129, 22:15  #1 
Mar 2010
2^{6}·3 Posts 
Fermat number F6=18446744073709551617 is a composite number. Proof.
Recently I did something similar for sixth Fermat number F6=18446744073709551617 as I did before for F5.
(see my thread "F5=4294967297 is a product of two integers."). Using elementary methods I proved that 274177 is a factor of F6. Proof is on my www page http://www.literka.addr.com/mathcoun...mth/proof3.htm. Proof of mentioned thread about F5 is on the web page http://www.literka.addr.com/mathcoun...mth/proof2.htm. My web page http://www.literka.addr.com/mathcoun...mth/proof1.htm contains basic proof (not mine), which inspired me to do this job. 
20120130, 04:14  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
22640_{8} Posts 
Kinda makes you wonder if F7 is composite!
But I guess we just have to learn to live with the fact that some things we will never know  these proofs become increasingly complex. ^_^ But wait! Code:
Let F7=2^{128}+1=340282366920938463463374607431768211457. Let p=59649589127497217. We'll show that p is a factor of F7. 340282366920938463463374607431768211457  59649589127497217  298247945637486085 (5)  5704689200685129054721 420344212834523784  417547123892480519 (7) 27970889420432656 (0) 279708894204326563  238598356509988868 .................... ................... ................... .................. 59649589127497217  59649589127497217 0  OMG!! It divides! Last fiddled with by Batalov on 20120130 at 04:16 Reason: :P 
20120130, 04:57  #3  
Mar 2010
2^{6}·3 Posts 
Quote:
Thank you for a beautiful proof. I will print it and I will hang it in my bedroom. It will always remind me how excellent science may be. 

20120130, 09:18  #4 
Einyen
Dec 2003
Denmark
2×7×229 Posts 
There are 185 fermat numbers known to be composite with a total of 218 factors:
http://homes.cerias.purdue.edu/~ssw/fdub.html 
20120130, 09:32  #5 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{5}·7·43 Posts 
253 composite and 293 prime factors now, 10 years later.
As for the proof, I tried to travel back 150 years to the times of Thomas Clausen and Le Lasseur. They've proven 11111111111111111 composite about the same way (they arguable knew the restricted residue classes, btu the rest is simply dividing away during long winter evenings). For 1111111111111111111, some gentleman divided away until he'd proven it prime; he submitted the works to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. See the Proceedings of the Society for 14th February, 1918. 
20120130, 12:28  #6 
Einyen
Dec 2003
Denmark
2×7×229 Posts 
I didn't even notice it was an old version of the page, didn't look at the dates. I just wondered why the address was different than usual.

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