Prove that where is the floor of
Very nice! I can prove it, but I can't give any "deep" reason why it should be true. I'm sure there is a deep idea behind this formula.
By induction on : it's trivial for , so suppose it holds for . Write in binary as , with . We want to show that
.
Now for any positive , we have
Thus it suffices to notice that there is only one for which , namely .
I don't know if is necessarily set to be prime ,
We have
First , write in base-p representation , ie :
(that means , the coefficient of is )
Let
Then
if
if
Consider
We have first terms belonging to the first case and other belonging to the second ( because if , then but the next term so is true , the corresponding term should be in the second case . )
Therefore , we have terms that belong to the second case and we can conclude that
Finally ,