A New Arithmetic Structure Matrix Related to the Hardy–Littlewood F-Conjecture

This study introduces a novel matrix defined over the nonzero natural numbers, whose entries are governed by a rigorous closed-form expression. The matrix architecture replicates the topological properties of the Ulam spiral, mapping the integer sequence onto a structured lattice with a well-defined formulation. We investigate the interplay between the matrix’s linear algebraic properties and its number-theoretic implications. A primary focus is the established connection between the matrix’s lines, rows, diagonals, and antidiagonals, and the Hardy–Littlewood F-conjecture. By analyzing the matrix’s internal structure, this work provides a new analytical framework for further study of the conjecture. The matrix links its visual characteristics to quadratic polynomials, offering fresh insights into the distribution of prime numbers.