A unified way for obtaining dividing formulas \(n| Q(n)\)
Using a function \(f(x)= f^{[1]}(x)\) and defining \(f^{[n]}(x)= f(f^{[n- 1]}(x))\), \(n= 2,3,\dots\) the period-\(n\) points of \(f\) are given by \(f^{[n]}(x)= x\) and \(f^{[i]}(x)\neq x\) for \(i= 2,\dots, n-1\). It is shown that \(n\mid N(n)\), where \(N(n)\) denotes the number of period-\(n\) points of \(f\). By this general result, special dividing formulas, e.g. (1) \(n\mid a^n- a\) for \(n\) prime and \(a= 2,3,\dots\), (2) \(n\mid a^{\varphi(n)- 1}\), \((a,n)= 1\), where \(\varphi(n)\) is Euler's \(\varphi\)-function, (3) \(n\mid (F_{n+ 1}+ F_{n- 1}- 1)\), \(n\) prime and \(F_n\) is the \(n\)th Fibonacci number can be derived.
scientific article; zbMATH DE number 1269916