TITLE: Large integer help
QUESTION [19 upvotes]: The integer 
$5685858885855807765856785858569666876865656567858576786786785^{22}$
has 6436343 divisors. Using only a scientific calculator, find a way to show it has exactly 5 prime divisors.

REPLY [5 votes]: Im not really answering my own question, but its to large to put in a comment, I was thinking of using the fact that $$\lim_{s\to\infty}\frac{\ln(d(k^s))}{\ln(1+s)}=\text{number of prime divisors $k$ has}$$ $d(k)=$ number of divisors of $k$
And then using the fact $s$ is very large, I could say $$\frac{\ln(d(k^{22}))}{\ln(23)}\sim \text{number of prime divisors $k$ has}$$
It gives me $5$ also, but this isn't really a proof, it works with $s=2$, also btw.
Also I was just experimenting it looks like $\dfrac{\ln(d(k))}{\ln(2)}=$ (number of prime diviors of $k$) very often.