The $k=3$ Multifactorial Operator: A Framework for the Systematic Generation of Congruent Numbers

Manuscript Description: The $k=3$ Multifactorial Operator Framework This manuscript introduces a novel algebraic framework for the systematic generation of congruent numbers using the multifactorial polynomial operator $C_{n}^{(k)}(m) = n \prod_{i=1}^{k} (in - m)$ for the specific case $k=3$. We demonstrate that this operator admits a universal Euclidean parametrization ($u = 2n - m, v = n$), enabling an exhaustive exploration of the double-entry parametric space $(n, m)$. Key Findings: * **Structural Identity:** The operator's output is intrinsically tied to Pythagorean geometry, with 100% of raw areas $N \equiv 0 \pmod 6$ and 75.02% of square-free cores $n_{sf} \equiv 0 \pmod 3$.* **Modular Distribution:** Analysis of $12.4 \times 10^6$ records reveals a systematic mapping of residue classes, where 66.65% of cores concentrate in $\{2, 6\} \pmod 8$, while maintaining a balanced distribution ($\approx 8.33\%$) across odd classes $\{1, 3, 5, 7\} \pmod 8$, consistent with Smith-Tunnell densities.* **Parametric Coverage:** The $m/n$ ratio exhibits a **platykurtic distribution** ($\gamma_2 \approx -1.20$), confirming an unbiased and uniform sweep of the search space ($1 < u/v < 2$).* **Scalability:** The framework efficiently identifies high-magnitude candidates up to $10^{16}$, establishing the $C_{n}^{(3)}(m)$ operator as a robust closed-form alternative for the discovery of rational points in high-height elliptic curves. ---**Open Access Repository:** [https://github.com/FranciscoLuceroBravo/k3-Operator](https://github.com/FranciscoLuceroBravo/k3-Operator) Support & Collaboration: As an independent researcher, this project is currently self-funded. I am open to sponsorship, academic affiliations, or grants to support the computational expansion of the k=3 Operator framework. If you or your institution are interested in supporting this research, please contact me at fjlucerob@gmail.com.