Congruence Identities Arising From Dynamical Systems

By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory. Let S be a nonempty set and let f be a map from S into itself. For every positive integer n, we define the n iterate of f by letting f 1 = f and f = f ◦ f for n ≥ 2. For y ∈ S, we call the set { f(y) : k ≥ 0 } the orbit of y under f . If f(y) = y for some positive integer m, we call y a periodic point of f and call the smallest such positive integer m the least period of y under f . We also call periodic points of least period 1 fixed points. It is clear that if y is a periodic point of f with least period m, then, for every integer 1 ≤ k ≤ m − 1, f(y) is also a periodic point of f with least period m and they are all distinct. So, every periodic orbit of f with least period m consists of exactly m points. Since distinct periodic orbits of f are pairwise disjoint, the number (if finite) of distinct periodic points of f with least period m is divisible by m and the quotient equals the number of distinct periodic orbits of f with least period m. Therefore, if there is a way to find the numbers of periodic points of all periods for a map, then we obtain infinitely many congruence identities in number theory. This is an interesting application of dynamical systems theory to number theory which is not found in [1, 2]. Let φ(m) be an integer-valued function defined on the set of all positive integers. If m = p1 1 p k2 2 · · ·p kr r , where the pi’s are distinct prime numbers, r and ki’s are positive integers, we let Φ1(1, φ) = φ(1) and let Φ1(m,φ) =