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a^m+a^n=a^p+a^q, a^{3m}+a^{3n}=a^{3p}+a^{3q}, prove mn=pq

Source: Lusophon 2015 CPLP P4

August 29, 2018
algebrasystem of equations

Problem Statement

Let aa be a real number, such that a≠0,a≠1,a≠−1a\ne 0, a\ne 1, a\ne -1 and m,n,p,qm,n,p,q be natural numbers . Prove that if am+an=ap+aqa^m+a^n=a^p+a^q and a3m+a3n=a3p+a3qa^{3m}+a^{3n}=a^{3p}+a^{3q} , then m⋅n=p⋅qm \cdot n = p \cdot q.
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