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Turkey TST 1997 Problem 5, there exist n, x_i and y_i

Source: Turkey TST 1997 Problem 5

November 30, 2011

modular arithmeticabstract algebranumber theory proposednumber theory

Problem Statement

Show that for each prime

p \geq 7

, there exist a positive integer

n

and integers

x_{i}

y_{i}

(i = 1, . . . , n)

, not divisible by

p

, such that

x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}

where

x_{n+1} = x_{1}