Experimental Observations on Cycle Structures of the Mappings 2x-(n+1)⌊x/n⌋ and Possible Connections with Prime Numbers

This paper introduces a new parameterized family of discrete dynamical systems defined by the iteration of a generating function fn ∶ ℕ → ℕ based on Euclidean division: fn(x) = 2x −(n+1)⌊x/n⌋ Where n is a fixed positive integer. The present work focuses on a large-scale computational exploration of the arithmetic properties exhibited by these mappings. An exhaustive Python program was developed to determine the periodic cycles generated by fn for every parameter 1 ≤ n ≤ 2.10 ^4. The numerical experiments reveal several highly regular patterns governing both the number of periodic cycles and their lengths. These regularities are particularly striking for specific arithmetic families of parameters, including values of the form P − 1 ,2^P,2^P −2 and P^m −1, where P is a prime number.The observed phenomena suggest unexpected connections between the dynamics of the mapping and several classical topics in number theory, including Fermat's little theorem, Mersenne numbers, and the twin prime conjecture. Although the results reported here are entirely experimental and no general proofs are currently known, they naturally lead to a collection of conjectures that may provide new directions for the study of arithmetic properties of discrete dynamical systems generated by floor-function mappings.