The Prime Square-Difference Identity
[ (p_{n+2})^{2} = (p_n)^{2} + 12x ]
gives rise to an integer PSD Factor (x) with a rigid last-digit pattern {0,4,6}.
The factor admits both:
- a modular explanation, and
- a natural analytic approximation
[ x \approx \frac{g,p_n}{6}, ] where (g = p_{n+2} - p_n) is the skip-one prime gap.
In the structured case where the skip-one gap is (6), the PSD Factor recovers (up to ±1) the middle prime of a twin-prime pair and determines which adjacent prime forms the pair. This combination of algebraic rigidity, modular restriction, and prime-gap structure gives the PSD Factor a rich internal behavior and suggests several directions for further exploration.
paper— The full PSD manuscript (PDF).README.md— You are here.
Proxmire, A. (2025). The Prime Triangle and Prime Square-Difference Identity. Zenodo.
DOI: 10.5281/zenodo.17740447
This repository and accompanying materials are released under the MIT License.