let's define a function on binary numbers to be $f(x)=x\oplus(2x+1)$ if $x$ is odd. And $f(x)=\frac{x}{2}$ if $x$ is even. Where $\oplus$ is bitwise xor. For example if $x=010001$ then $f(x)$ would be: $\space\space\space\space010001$ $\oplus\space100011$ $=110010$ And $f(110010)=11001$ Define a ...

I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of choice in there. I'm interested in whether the power of ultrafilters adds anything to this probl...