Problem 2

Let

m

be a fixed positive integer. The infinite sequence

\{a_n\}_{n\ge1}

is defined in the following way:

a_1

is a positive integer, and for every integer

n\ge1

a_{n+1}=a_n^2+2^m

a_n<2^m

, and

a_{n+1}=a_n/2

a_n\ge2^m

. For each

m

, determine all possible values of

a_1

such that every term of the sequence is an integer.

Step 3 of 7: The odd part stabilizes at 1

Previous step Next step

\{b_i\}\ \text{bounded, non-decreasing, odd}\ \Longrightarrow\ b_i=1\ \text{eventually}

Detailed analysis

By steps 1–2, $\{b_i\}$ is a non-decreasing sequence of odd positive integers bounded above by $2^m$ , so it takes only finitely many values and is eventually constant, say $b_i=B$ for $i\ge J$ . The large-rule case cannot apply forever (each use decreases $c_i$ by $1$ , and $c_i\ge0$ ), so the small rule applies infinitely often; for $b_i$ to stay constant at those steps, step 2 forces the borderline case each time, which requires $b_i=1$ . Hence $B=1$ : eventually every term is a power of two.