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D_n as the largest integer that is a divisor of a^n + (a + 1)^n + (a + 2)^n

Source: Dutch IMO TST 2015 day 1 p5

August 30, 2019
number theorydivisormaximumexponential

Problem Statement

For a positive integer nn, we de ne DnD_n as the largest integer that is a divisor of an+(a+1)n+(a+2)na^n + (a + 1)^n + (a + 2)^n for all positive integers aa. 1. Show that for all positive integers nn, the number DnD_n is of the form 3k3^k with k≥0k \ge 0 an integer. 2. Show that for all integers k≥0k \ge 0 there exists a positive integer n such that Dn=3kD_n = 3^k.
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