# Number Theory

## Homework Statement

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?

## Answers and Replies

I think it is by contradiction (i suppose you could show the gcd of $(a^n - b^n , a^n + b^n )$ is 1 or 2 ) viz,
let d be the gcd clearly then , d must divide the sum (and the difference) of the two , $$d | a^n + b^n + a^n - b^n$$
$$d | 2a^n$$
which implies, $d|2,$ $d|a^n$ 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, ;-) ,
good luck

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I think it is by contradiction (i suppose you could show the gcd of $(a^n - b^n , a^n + b^n )$ is 1 or 2 ) viz,
let d be the gcd clearly then , d must divide the sum (and the difference) of the two , $$d | a^n + b^n + a^n - b^n$$
$$d | 2a^n$$
which implies, $d|2,$ $d|a^n$ 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, ;-) ,
good luck

I consulted with my study partners and we agree that what you have is correct. But we didn't get exactly what you got so we had to make some corrections. thanks for your help.

(a^n - b^n) | (a^n + b^n)

what does that mean?? does the bar symbol mean "given"

Dick
Science Advisor
Homework Helper
(a^n - b^n) | (a^n + b^n)

what does that mean?? does the bar symbol mean "given"

The bar symbol means "divides" in the sense of integer divisibility.

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