18
$\begingroup$

I'm working through the problems in Niven's number theory book, and problem 46 in section 1.2 (page 19) has me stumped.

Prove that there are no positive integers $a, b, n > 1$ such that $(a^n - b^n) | (a^n + b^n)$.

I've tried playing around a bit (e.g. noticing that $a^n - b^n$ must divide $2a^n$ and $2b^n$), but in general I've just been going around in circles. Can anyone please provide a (small) hint in the right direction?

  • 1
    See also [Prove that there are no positive integers $a, b$ and $n >1$ such that $a^n – b^n$ divides $ a^n + b^n$.](http://math.stackexchange.com/q/1124645)2017-01-06
  • 0
    Here's another duplicate with a solution: http://math.stackexchange.com/questions/2085924/an-bn-anbn-does-not-have-a-solution2017-01-06

3 Answers 3