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Periodicity of Remainder Sequence

Source:

August 8, 2024

number theory2022

Problem Statement

For an integer

n

and positive integers

n\leq 100

and

k

, let

f_k(a, n)

be the remainder when

a^k

is divided by

n

. Find the largest composite integer

n\leq 100

that guarantees the infinite sequence

f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots

to be periodic for all integers

a

(i.e., for each choice of

a

, there is some positive integer

T

such that

f_k(a,n) = f_{k+T}(a,n)

for all

k

).

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