I came across this problem.
I have to find the last two digits ,and if I can , the last three digits of this number (which I believe is Infinity.)
$$n={2017}^{({2018)}^{{(2019)}^{{(•)}^{{(•)}^{(•)}}}}}$$
I started by computing $n \pmod {10} $ and which I think is $$7^{2k} \equiv 1 {\pmod {10}}$$ where $k$ is even …
Next , I tried calculating $n{\pmod {100}}$ and I believe the answer is either $21,41,61,81$ but I don't know for sure..It was lot of trial and error…
So Could you please check my answer and ,if wrong, provide me a hint in the right direction?
Best Answer
Let's assume the power tower is finite but includes at least the $2020$ term and possibly many more
We can say:
suggesting the final three digits are $241$