Clearly the identity function f=id satisfies fn=id for any n∈N. However, there are also other functions with this property. For instance, with n=2 we have self-inverse functions like x↦1−x. For functions f:R→R, we can use roots of unity e.g. f(x)=exp(2πin)×x.
I am interested in functions f:N→N, and in particular n=3. Are there non-identity functions such that f∘f∘f=id? If so, how many functions of this type? Can you give an example, or an infinite family of examples?
I am not looking for rigorous proofs here, but just examples. Probably there is a name for functions of this type but I have searched and cannot find anything. This is not part of any formal teaching, but just an interesting question I came up with.
Answer
Any partition of N into ordered triples gives a function f that acts on each triple by cycling (x,y,z)↦(y,z,x) (and conversely, every function f can be described fully by such a list of triples) This gives you an uncountably infinite family of functions. If you want an explicit function of this type, consider f(n)={n−2n is a multiple of 3n+1otherwise
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