8
$\begingroup$

I came across a problem in Niven's number theory text (problem 51 on page 20) that asks the following:

Show that if $(a, b) = 1$ and $p$ is an odd prime, then $$\left(a + b, \frac{a^p + b^p}{a + b}\right) = 1 \text{ or } p.$$

I am not asking for a solution to this problem; instead, I'm trying to understand why $a^p + b^p$ would always be divisible by $a + b$ given the above conditions. Does anyone have any insights as to why this would be true? Where (if at all) do we use the conditions that $(a, b) = 1$ and $p$ is an odd prime?

  • 2
    Hint: $p$ is odd - try with $p$=3 (or 5) and divide through to see what happens.2011-10-11
  • 0
    If the gcd of a and b is not 1 or p, then the statement is obviously wrong.2011-10-11

3 Answers 3