Resolving Legendre's Conjecture: An Analytic-Computational Approach Using Weighted Prime Density

Key features: Novel methodology combining Gaussian-weighted prime density L(n), rigorous asymptotic bounds, and computational verification; Full verification for all integers n ≤ 10^7 (i.e., n less than or equal to ten million), using optimized parallel algorithms (open-source code included); Explicit inequality proving that L(n) > 0 for all n ≥ 10^6 (i.e., for all n greater than or equal to one million); see Appendix B.