1) Show that a prime divisor p of the Fermat number must be of the form .
(Hint: Show that . Then show that is congruent to 1 (mod p). Conclude that (p-1)/2)
Well, if has order modulo , i.e. is the least positive integer such that , then for any exponent such that , we must have .
Hence if , this means that the order of must divide ...
Extra hint : to show that , you can use the "second supplement" to the quadratic reciprocity law, which states that for any odd prime .