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April 10, 20260 citationsOpen Access

The Prime Jump System: Nine Elementary Theorems and the Universal Termination Conjecture

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JRJose Luis Roman

Key Points

  • The aim is to establish foundational theorems about the Prime Jump System and explore its properties.
  • Developed a discrete additive dynamical system over natural numbers.
  • Coupled a Fibonacci recurrence with a primality-stopping condition.
  • Analyzed orbits for predictability, strict monotonicity, and parity.
  • Investigated the role of Mersenne primes as fixed points.
  • Formulated the Universal Termination Conjecture and employed CRT sieves.
  • Proved nine fundamental theorems about the orbits in the Prime Jump System.
  • Demonstrated that Mersenne primes serve as perfect absorbing fixed points.
  • Provided insights into the density of attractors via heuristics linked to CRT.

Abstract

This paper presents the “Prime Jump System” (PJS), a discrete additive dynamical system over N that couples a linear Fibonacci recurrence with a primality-stopping condition. We prove nine fundamental theorems governing the predictability, strict monotonicity, and parity of the orbits. We demonstrate that Mersenne primes act as perfect absorbing fixed points within the system. Finally, we formulate the Universal Termination Conjecture and provide heuristics regarding the density of attractors, supported by constructive Chinese Remainder Theorem (CRT) sieves for arbitrarily long composite sequences.

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Cite This Study

Jose Luis Roman (2026) studied this question.

synapsesocial.com/papers/69d895046c1944d70ce05effhttps://doi.org/10.5281/zenodo.19461661
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