Ergodic and Arithmetical Properties of Geometrical Progression’s Dynamics and of Its Orbits

The multiplication by a constant (say, by 2) acts on the set Z/nZ of residues (mod n) as a dynamical system, whose cycles relatively prime to n all have a common period T (n) and whose orbits consist each of T (n) elements, forming a geometrical progression or residues. The paper provides many new facts on the arithmetical properties of these periods and orbits (generalizing the Fermat’s small theorem, extended by Euler to the case where n is not a prime number). The chaoticity of the orbit is measured by some randomness parameter, comparing the distances distribution of neighbouring points of the orbit with a similar distribution for T randomly chosen residues (which is binominal). The calculations show some kind of repulsion of neighbours, avoiding to be close to other members of the same orbit. A similar repulsion is also observed for the prime numbers, providing their distributions nonrandomness, and for the arithmetical progressions of the residues, whose nonrandomness degree is similar to that of the primes. The paper contains also many conjectures, including that of the infinity of the pairs of prime numbers of the form (q, 2q + 1), like (3, 7), (11, 23), (23, 47) on one side and that on the structure of some ideals in the multiplicative semigroup of odd integers—on the other. 2000 Math. Subj. Class. 11A07 (11N69 37A45 37B99).