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k(k+1)/3 is square iff k is square

Source: 11th Brazilian Math Olympiad - Problem 2

December 27, 2017
Brazilian Math Olympiad 1989Brazilian Math Olympiadnumber theoryPerfect SquaresPell equations

Problem Statement

Let kk a positive integer number such that k(k+1)3\frac{k(k+1)}{3} is a perfect square. Show that k3\frac{k}{3} and k+1k+1 are both perfect squares.
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