Prove that they are no integers a,b,n>1 such that (a^n - b^n) | (a^n + b^n).
The Attempt at a Solution
Do I solve this by contradiction? If so, how do I start it?
I think it is by contradiction (i suppose you could show the gcd of [itex] (a^n - b^n , a^n + b^n ) [/itex] is 1 or 2 ) viz,
let d be the gcd clearly then , d must divide the sum (and the difference) of the two , [tex] d | a^n + b^n + a^n - b^n [/tex]
[tex] d | 2a^n [/tex]
which implies, [itex] d|2, [/itex] [itex] d|a^n [/itex] this last result shows d is either 1 or 2 , thus if the gcd of the two is 2 or 1, no integers a,b, c >1 can exist to satisfy the requirement (there are no numbers a,b, c> 1 that can divide in that manner) though im only into number theory as a hobby, i might not be quite on the mark, ;-) ,