Enhanced Computational Efficiency in Paley Type I Hadamard Matrix Construction via Triangular Residue Optimization

Paley Type I Hadamard matrices are traditionally generated using quadratic residues, requiring iterative modular arithmetic. Leveraging the isometric equivalence between polygonal and quadratic residues, an optimized construction method using triangular residues (k=3) is proposed. The minimal second difference (² = 1) inherent in triangular numbers reduces the modular reduction overhead by exactly 50% compared to the quadratic baseline. Numerical verification up to p = 100, 000 confirms a reduction in total arithmetic complexity from 1. 25p to 1. 125p.