a squence question

• Dec 26th 2006, 08:32 PM
Jenny20
a squence question
Please see the " Microsoft word" attachment.
• Dec 27th 2006, 01:44 PM
ThePerfectHacker
There seems to be something wrong.
If $\displaystyle n=5$
Then,
$\displaystyle 2=2$
$\displaystyle 2^2=4$
$\displaystyle 2^4=16=1$
$\displaystyle 2^1=2$
This is non-constant.
• Dec 27th 2006, 11:45 PM
CaptainBlack
Quote:

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

$\displaystyle 2^{2^4}=2^{16}=65536\equiv 1\ \mod\ 5$

Also

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

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

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