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f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} & a_{k+1} is residue of a_k^2 mod n

Source: Mexican Mathematical Olympiad 2001 OMM P4

July 30, 2018

number theoryNumber theoretic functionsresidue

Problem Statement

For positive integers

n, m

define

f(n,m)

as follows. Write a list of

2001

numbers

a_i

, where

a_1 = m

, and

a_{k+1}

is the residue of

a_k^2

mod \, n

(for

k = 1, 2,..., 2000

). Then put

f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}

. For which

n \ge 5

can we find m such that

2 \le m \le n/2

and

f(m,n) > 0