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a_n = n^n + (n - 1)^{n+1}, eventually periodic modulo m

Source: Finland 2019, p4

September 8, 2019

Periodic sequenceSequenceperiodicbir tinga qimmat masala

Problem Statement

Define a sequence

a_n = n^n + (n - 1)^{n+1}

when

n

is a positive integer. Define all those positive integer

m

, for which this sequence of numbers is eventually periodic modulo

m

, e.g. there are such positive integers

K

and

s

such that

a_k \equiv a_{k+s}

(

mod \,m

), where

k

is an integer with

k \ge K