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Find least integer such that 2006 divides x_{2006}

Source: Turkey, TST D2, P1

May 10, 2006
inductionquadraticsGaussmodular arithmeticnumber theoryprime numbersnumber theory proposed

Problem Statement

For all integers n≥1n\geq 1 we define xn+1=x12+x22+⋯+xn2x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2, where x1x_1 is a positive integer. Find the least x1x_1 such that 2006 divides x2006x_{2006}.
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