A Simple Method Which Generates Infinitely Many Congruence Identities

On the other hand, let S be a subset of the real numbers and let / be a function from S into itself. For every positive integer n, we let f denote the n iterate of f : f =f and f=f<> f~ for n > 2. For every #0 G s > w e call the set {f(xQ)\k > 0} the orbit of xQ under /. If XQ satisfies f{x$) = XQ for some positive integer 77?, then we call XQ a periodic point of / and call the smallest such positive integer m the minimal period of XQ and of the orbit of #0 (under f). Note that, if XQ is a periodic point of / with minimal period 77?, then, for every integer 1 < k < 777, f(x§) is also a periodic point of / with minimal period 77? and they are all distinct, so every periodic orbit of f with minimal period m consists of exactly 777 distinct points. Since it is obvious that distinct periodic orbits of / are pairwise disjoint, the number (if finite) of distinct periodic points of / with minimal period 77? is divisible by ??? and the quotient equals the number of distinct periodic orbits of f with minimal period m. This observation, together with a standard inclusion-exclusion argument, gives the following well-known result.