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 7 of 7: Confirm every ℓ≥1 gives an infinite integer cycle

Previous step Next step

a_1=2^\ell\ (\ell\ge1)\ \longrightarrow\ 2\to8\to4\to2\to8\to4\to\cdots

Detailed analysis

For $\ell\ge2$ , $a_1=2^\ell\ge4$ , so the large rule repeatedly halves the exponent by $1$ (each term an integer power of two) until it reaches $a=2^1=2$ . From $a=2$ (true also directly when $\ell=1$ ): since $2<4$ , $a_{n+1}=4+4=8$ ; since $8\ge4$ , halving gives $4$ ; since $4\ge4$ , halving gives $2$ , returning to the start of the cycle $2\to8\to4\to2\to\cdots$ . Every term in this cycle, and every term in the initial descent from $2^\ell$ , is a positive integer. Hence $a_1=2^\ell$ with $\ell\ge1$ produces an all-integer sequence.