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All n which divide one of the two numbers

Source: International Zhautykov Olympiad 2013 - D1 - P2

January 17, 2013

modular arithmeticnumber theory proposednumber theory

Problem Statement

Find all odd positive integers

n>1

such that there is a permutation

a_1, a_2, a_3, \ldots, a_n

of the numbers

1, 2,3, \ldots, n

where

n

divides one of the numbers

a_k^2 - a_{k+1} - 1

and

a_k^2 - a_{k+1} + 1

for each

k

1 \leq k \leq n

(we assume

a_{n+1}=a_1