4
votes
3answers
162 views

$\log(n)$ is what power of $n$?

Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small ...
66
votes
4answers
2k views

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
1
vote
2answers
61 views

Proof of a lower bound $\lambda(n)$ of the smallest number of multiplications $\ell(n)$ needed to compute $a^n$ for an integer $a$

Let $\ell(n)$ be the smallest number of multiplications needed to compute $a^n$ for any integer $a$. Here, a multiplication is $a_i := a_j \cdot a_k$ for $j, k < i$ and $a_0 := a$, e.g. $\ell(8) = ...