A dynamical systems proof of Euler's generalization of the little theorem of Fermat (Q2721276)

This paper gives an elementary proof of the Euler-Fermat theorem by counting points of period \(p^r\) for a map with \(k^n\) points of period \(n\) (for then there are \(k^{p^r}-k^{p^{r-1}}\) points of least period \(p^r\), which must comprise a union of orbits of length \(p^r\), so \(k^{p^r}\) is congruent to \(k^{p^{r-1}}\) mod \(p^r\), which implies Euler-Fermat since Euler's function is multiplicative). More general results of this type appear in a recent paper by \textit{Y. Puri} and \textit{T. Ward} [J. Integer Seq. 4, Article 01.2.1 (2001; Zbl 1004.11013].NEWLINENEWLINEFor the entire collection see [Zbl 0948.00027].