What is the length of the smallest period of the following sequence $$f[n] = \left< \left< n \right>_N \right>_M$$ where $\left<n\right>_N$ represents $n \pmod N$. Is there a special term for performing nested modulo operations?
For $N$ even (say 8) and $M = 2$, it appears that the sequence is periodic with length 2. But for $N$ odd, it seems the least period length is $N$. So this leads me to believe that if $N$ is coprime to $M$ then the least period length is $N$.
- How do I go about showing this to be the case if it is indeed true?
- How do I, in general, calculate the least period for $f[n]$ for arbitrary cases of $N$ and $M$?