Arithmetic Modulation of Maximal Prime Gaps: An Empirical and Theoretical Investigation

This dataset and associated research paper investigate the growth behavior of maximal gaps between consecutive primes within arithmetic progressions a \pmod q. While the Ford–Green–Konyagin–Maynard–Tao theorem establishes lower bounds for record gaps, the fine-scale empirical behavior—especially within specific residue classes—remains poorly understood. Using a segmented sieve algorithm, we analyzed primes up to $10^8$ for moduli $q \leq 150$. Our findings reveal a systematic growth of the normalized maximal gap $R_{a,q}(p) = G_{a,q}(p)/\log^2 p$, characterized by a positive slope $\beta_{a,q}$ relative to $\log p$. This observation stands in stark contrast to Random Matrix Theory (RMT) predictions, which suggest negative slopes for similar ranges. Key highlights of this study: Statistical Robustness: Bootstrap analysis on 710 progressions (prime moduli) confirms $\beta > 0$ with a significance level of $p \approx 1.58 \times 10^{-185}$. Arithmetic Scaling: We identify a logarithmic dependence of the growth rate on the modulus, $\beta_{a,q} \approx -0.233 + 0.158 \log q$. Data Content: The repository includes Python implementation of the segmented sieve, regression analysis scripts, and the processed record gap data for $q \leq 150$. This work suggests that arithmetic constraints dominate maximal gap growth in the finite range, necessitating a re-evaluation of probabilistic models when applied to specific progressions. Reproducibility: For ease of verification and further exploration, the primary analysis script arithmetic_modulation.ipynb is provided. This notebook is designed to be fully compatible with the Google Colab cloud environment. To run the analysis, upload the .ipynb file to Google Colab and follow the internal documentation. The script handles data generation, statistical regression, and bootstrap analysis automatically. Keywords: Prime numbers, Maximal prime gaps, Arithmetic progressions, Computational number theory, Random Matrix Theory, Cramér's conjecture, Python, Bootstrap analysis.