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Prove that c_n = 0 implies n = 0

Source: IMO Shortlist 1989, Problem 9, ILL 22

September 18, 2008
algebrairrational numbernumber theoryIMO Shortlistequation

Problem Statement

∀n>0,n∈Z, \forall n > 0, n \in \mathbb{Z}, there exists uniquely determined integers an,bn,cn∈Z a_n, b_n, c_n \in \mathbb{Z} such \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}. Prove that c_n \equal{} 0 implies n \equal{} 0.
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