For , define oto be the highest exponent to which a prime divides it. For instance, . Prove that exists.
Previous Result: The cardinal of: is where is the floor function and is the Möbius function.
Proof
Consider: , the cardinal of this set is, by inclusion-exclusion:
But: thus note that we can write: but we want and the rest follows.
At this point, note that every number that is free of squares, is free of cubes ... and so on. Further, if a number is free of cubes, but not of squares, then
That is:
Thus we get: (This is already very suggestive)
Mmmm, here I've not been able to finish it formally -it gets quite nasty-, but I do think that: -remember and -