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x=a^2+ab+b^2 with a,b \in Z

Source: China Northern MO 2012 p3 CNMO

May 4, 2024
number theorydivides

Problem Statement

Suppose S={x∣x=a2+ab+b2,a,b∈Z}S= \{x|x=a^2+ab+b^2,a,b \in Z\}. Prove that:
(1) If m∈Sm \in S, 3∣m3|m , then m3∈S\frac{m}{3} \in S
(2) If m,n∈Sm,n \in S , then mn∈Smn\in S.
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