Algebraic Cycle Structures and Bridges to Fermat's Little Theorem and the Twin Prime Conjecture for the Mappings 2x-(m+1)⌊x/m⌋

This paper introduces a new parameterized family of discrete dynamical systems generated by iterating an integer-valued function based on Euclidean division. For each fixed positive integer parameter m, the iteration starts from an arbitrary positive integer x. At each step, the Euclidean division of the current integer x by m is performed to obtain its quotient denoted by q and remainder denoted by r. The next iterate is then obtained by adding the difference between the remainder and the quotient to the current value x+(r-q). Repeating this procedure generates a deterministic discrete dynamical system. We prove that every orbit generated by this family eventually reaches either a fixed point or a periodic cycle. An extensive computational investigation reveals striking structural regularities in the organization of the resulting phase spaces. These observations lead to several conjectures concerning cycle duplication, period invariance, and rigid translation mechanisms within specific subfamilies. Finally, we identify unexpected connections between the arithmetic structure of these periodic cycles and classical topics in number theory, including Fermat's little theorem and the twin prime conjecture. These results reveal a rich arithmetic structure underlying this family of parameterized discrete dynamical systems and provide a new framework for investigating the interplay between discrete dynamics and number theory.