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 6 of 7: Rule out the exponent zero

Previous step Next step

\ell=0:\ a_1=1<4\ \Rightarrow\ a_2=5,\ b_2=5>4\ (\text{contradicts step 1})\ \Longrightarrow\ \ell\ge1

Detailed analysis

If $\ell=0$ , i.e. $a_1=1<4$ , the small rule gives $a_2=1^2+4=5$ , whose odd part $5$ already exceeds $2^m=4$ , contradicting step 1. So $\ell=0$ is impossible, leaving $\ell\ge1$ .