Power Tower Help Please.

[ilml]

Hello, I was randomly browsing sites and I came upon this math question which I am stuck on (and desperately want to solve!): Find the last two digits of 3^3^3^3. Can you find the last three digits?

I'm stuck on this problem because I really don't know where to begin. I figure that you need to find a pattern or period of some sort, but the numbers grow too fast and therefore there aren't that many readily available samples. Any help is appreciated! THANKS!

[arildno]

Instead of tackling the original problem, you should try first to find the answer to the following problem:
What is the last digit of the tower?
Note the following:
We have the recurrence pattern for the last digit:
3->3
3*3->9
3*3*3->7
3*3*3*3->1

And back again!
Hence, we have a "4-periodicity" for the last digit, and you should be able to solve the problem.

Clearly, you may grit your teeth together and find similar reccurrence patterns for the 2 last digits and 3 last digits problem.

However, I would believe that such recurrence patterns are well known in number theory; since I'm ignorant of that field, I can only suggest that you post your question in the "Number Theory" subforum.