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An Investigation into a Putative 16-adic Correlation in Goldbach Partitions

Authors: Ramirez Bochard, Enrique A.;

An Investigation into a Putative 16-adic Correlation in Goldbach Partitions

Abstract

This record contains the complete set of materials for a computational number theory investigation into a hypothesized link between the Goldbach Conjecture and a simplified, 16-adic Collatz-like dynamical system. The research initially explored whether the density of Goldbach partitions for an even number n showed a statistically significant correlation with its residue class modulo 16. Preliminary computational results suggested a strong, intriguing pattern, which prompted the development of the framework detailed in the included paper. However, a subsequent rigorous verification, based on a corrected partition-counting algorithm (also included), revealed that the observed correlation was an artifact of a software bug in the original code. The corrected data, for both small (n < 10^5) and large (n between 2^20 and 2^21) numbers, showed no significant partition variance across residue classes. This deposit serves as a complete scientific record of that investigation. It is published as a valuable case study on the importance of algorithmic correctness and reproducibility in computational research. The null result is a definitive finding that invalidates the initial hypothesis. Contents of this Record: ramirez-bochard_goldbach_16adic_investigation_v1.pdf: The final manuscript detailing the hypothesis, methodology, and null result. ramirez-bochard_goldbach_16adic_investigation_v1.tex: The LaTeX source code for the manuscript. goldbach_partition_counter.cpp: The corrected C++ program used to generate the partition count data. This program is robust and can be used for further Goldbach-related explorations. calculate_weights.py: The Python script used to analyze the output of the C++ program and calculate the empirical weights. goldbach_partitions_1M_2M.csv.zip: A sample dataset generated by the C++ program for the range [1,048,576, 2,097,152]. goldbach_activated_sums_v7.py: (Note) This Python script is included as supplementary material. It represents a separate, valid research direction for exploring the Goldbach Conjecture by analyzing "activation sets" (Δₖ) and using machine learning to predict partition difficulty. It is distinct from the 16-adic investigation. Related Works: This record documents a concluded investigation with a null result. A separate and ongoing research project, based on the supplementary file goldbach_activated_sums_v7.py, explores a machine learning approach to predicting Goldbach partition difficulty. That work will be archived in a separate Zenodo record upon completion. Link to related work: https://doi.org/10.5281/zenodo.15869753 

Keywords

16adic, Collatz, Null Result, Machine Learning, Computational Number Theory, Goldbach Conjecture, Collatz Conjecture, Reproducibility of Results, 16-adic, Reproducibility, Goldbach

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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