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If (a-1)^3 + a^3 + (a+1)^3 is a cube, then 4 | a

Source: All-Russian Olympiad 2006 finals, problem 10.2

May 7, 2006
geometry3D geometrymodular arithmeticnumber theoryrelatively primenumber theory proposed

Problem Statement

If an integer a>1a > 1 is given such that (a−1)3+a3+(a+1)3\left(a-1\right)^3+a^3+\left(a+1\right)^3 is the cube of an integer, then show that 4∣a4\mid a.
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