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p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1), perfect square

Source: 2011 Romania District VIII p2

September 1, 2024
number theoryPerfect Square

Problem Statement

a) Show that m2−m+1m^2- m +1 is an element of the set {n2+n+1∣n∈N}\{n^2 + n +1 | n \in N\}, for any positive integer m m.
b) Let pp be a perfect square, p>1p> 1. Prove that there exists positive integers rr and qq such that p2+p+1=(r2+r+1)(q2+q+1).p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).
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