1. ## a squence question

Please see the " Microsoft word" attachment.

2. There seems to be something wrong.
If $n=5$
Then,
$2=2$
$2^2=4$
$2^4=16=1$
$2^1=2$
This is non-constant.

3. Originally Posted by ThePerfectHacker
There seems to be something wrong.
If $n=5$
Then,
$2=2$
$2^2=4$
$2^4=16=1$
$2^1=2$
This is non-constant.
$2^{2^4}=2^{16}=65536\equiv 1\ \mod\ 5$

Also

$
2^{4k}=(2^4)^k \equiv 1 \ \mod \ 5
$

and since the $\log_2$ of all the numbers in the sequence after $2^{2^2}$ are divisible by $4$ the result follows for $\mod\ 5$.

By a similar argument we can show that if any element in the sequence is ever congurent to 0 or 1 so
are all subsequent terms.

RonL