Relative Primality of Fermat Numbers
Proof:
Let d=hcf (Fm, Fn) [Fm, Fn distinct Fermat numbers][d clearly odd]
Let x=2**(2**n) and k=2**(m-n)
(Fm - 2)/Fn = ((2**(2**n))**(2**(m-n)) - 1)/(2**(2**n) + 1)
=(x**k - 1)/(x+1)
=x**(k-1) - x**(k-2) + … -1
Therefore Fn|(Fm -2)
But d|Fn, so d|(Fm - 2)
Also, d|Fm, so d|2, so d=1