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.