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Binomial expansion of cube root 2-1

Source: Romania TST 1 P1, 2013

April 5, 2013
geometry3D geometrymodular arithmeticsymmetryalgebrapolynomialnumber theory unsolved

Problem Statement

Given an integer n≥2,n\geq 2, let an,bn,cna_{n},b_{n},c_{n} be integer numbers such that (23−1)n=an+bn23+cn43. \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. Prove that cn≡1(mod3)c_{n}\equiv 1\pmod{3} if and only if n≡2(mod3).n\equiv 2\pmod{3}.
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