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

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