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Sequence involving 2011 gives pefect square

Source: Vietnamese TST 2012 Problem 4

May 9, 2012
algebrapolynomialDiophantine equationnumber theory unsolvednumber theory

Problem Statement

Consider the sequence (xn)n≥1(x_n)_{n\ge 1} where x1=1,x2=2011x_1=1,x_2=2011 and xn+2=4022xn+1−xnx_{n+2}=4022x_{n+1}-x_n for all n∈Nn\in\mathbb{N}. Prove that x2012+12012\frac{x_{2012}+1}{2012} is a perfect square.
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