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Let

\{b_n\}_{n\geq 1}^{\infty}

be a sequence of positive integers. The sequence

\{a_n\}_{n\geq 1}^{\infty}

is defined as follows:

a_1

is a fixed positive integer and

a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.

Find all positive integers

m\geq 3

with the following property: If the sequence

\{a_n\mod m\}_{n\geq 1 }^{\infty}

is eventually periodic, then there exist positive integers

q,u,v

with

2\leq q\leq m-1

, such that the sequence

\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}

is purely periodic.

Problem Statement