TITLE: Elementary results with p-adic numbers
QUESTION [56 upvotes]: I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience that includes probabilists, computer scientists and so on.
I'd rather like to do an introduction to $p$-adic numbers and $p$-adic analysis. I think these kind of things come up to be really cool when you work on it even just for a short time, but I have the ambitious aim to show them something nice and elementary whose statement will be understood by everyone (of course the proof may also be really hard, but there I could give them just its general idea).
In other words the question is: if I had prepared something about Galois theory I would have finished with the application to resolubility of polynomial or compass and straightedge constructions; if it had been something about modular forms, it would have been for sure Fermat's last theorem; with 3-surfaces it would have been Poincaré conjecture and so on. What if it's about p-adic numbers or p-adic analysis?
I thought about results on valuations of roots of polynomials, but it seems to me already too complicated (par ailleurs since I'm introducing valuations at the beginning of the talk, it won't turn out to be an application to something they already knew).

REPLY [4 votes]: I'm a bit surprised that nobody mentions the interesting fact that $\log(-1)=0$ in $\mathbb{Q}_2$.
This result could be stated without involving $p$-adic numbers: the partial sums of the series $\sum_{i=1}^\infty 2^i/i$ have $2$-adic valuations tending to infinity.
Same for the dilogarithm $\log_2$: one has $\log_2(-1)=0$ in $\mathbb{Q}_2$.