On a Primorial-Based Sequence and Its Prime-Generating Properties up to n = 1000

This repository contains the full research note and computational assets for the study: "On a Primorial-Based Sequence and Its Prime-Generating Properties up to n = 1000". Abstract:We perform a computational exploration of the sequence defined by the second consecutive prime following the n-th primorial, denoted as q_2(P_n #), and analyze the displacement sequence r_n = q_2(P_n #) - P_n # for n <= 1000. We present formal proofs regarding the absence of prime factors p <= p_n in r_n and explore the gap dynamics beyond traditional bounds. Additionally, empirical linear bounds for the gap lower envelope are established alongside heuristic predictions based on the Prime Number Theorem. Files included:- Full research note (PDF)- Computational source code link and replication data Source Code & Repository:The complete source code and generation pipeline are publicly available on GitHub:https://github.com/JCHernandezT/primorial-gap-recurrence