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gcd of {a^n+(a+1)^n+(a+2)^n | a in N}

Source: Itamo 2014 - p3

November 20, 2014
number theorygreatest common divisormodular arithmeticnumber theory unsolved

Problem Statement

For any positive integer nn, let DnD_n denote the greatest common divisor of all numbers of the form an+(a+1)n+(a+2)na^n + (a + 1)^n + (a + 2)^n where aa varies among all positive integers.
(a) Prove that for each nn, DnD_n is of the form 3k3^k for some integer k≥0k \ge 0. (b) Prove that, for all k≥0k\ge 0, there exists an integer nn such that Dn=3kD_n = 3^k.
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