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 (Δ² = 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 $1.125p$.