Let $B(n)$ denote the Bell's number which is the number of the equivalence relation which can be defined on a set of cardinality $n$.


While I was trying to solve a problem, I reached another result;


$$B(p^k)\equiv k+1 \ mod \ p$$


Is this result evident or trivial ? Any comments and remark are welcome.


**Note:** My field is not Number theory, that is why I am not famialiar with the tools and result in number theory. If this question is something trivial, please excuse me. 



Is this property of the Bell's number evident?
This follows from the well-known Touchard's congruence ([here](http://arxiv.org/abs/0906.0696) for a random reference). Following your notation, the congruence is:


$$B(n+p^k)\equiv kB(n)+B(n+1) \mod \ p$$


Taking $n=0$:


$$B(p^k)\equiv kB(0)+B(1) \mod \ p$$


And since $B(0)=B(1)=1$,


$$B(p^k)\equiv k+1 \mod \ p$$


