IT-CESENATICO 2014 Problem 3

For any positive integer $n$ , let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.

(a) Prove that for each $n$ , $D_n$ is of the form $3^k$ for some integer $k \ge 0$ . (b) Prove that, for all $k\ge 0$ , there exists an integer $n$ such that $D_n = 3^k$ .