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 5 of 7: Rule out odd part 3 at the start

Previous step Next step

m=2:\quad a_1<4\ \Rightarrow\ b_1\in\{1,3\};\qquad b_1=3\ \Longrightarrow\ a_2=13,\ b_2=13>4\ (\text{contradicts step 1})

Detailed analysis

For $m=2$ , step 1 requires $b_1\le4$ , and $b_1$ odd gives $b_1\in\{1,3\}$ . Since the large rule preserves the odd part, if $b_1=3$ the sequence keeps odd part $3$ under repeated halving until it reaches the term $a=3$ itself (when $3<4$ ); then $a_{n+1}=3^2+4=13$ , whose odd part $13$ exceeds $2^m=4$ , contradicting step 1. Hence $b_1=1$ : $a_1$ must itself be a power of two, $a_1=2^\ell$ for some integer $\ell\ge0$ .