Quadratic Relations and Oblong Parabolic Prime Sieve: A New Approach to Goldbach's Conjecture

This study expands the geometric tool known as the Parabolic Sieve of Primes (PSO) 12 by introducing the Oblong Parabolic Sieve of Primes with Offset (Oblong PSO). This technique, applying an offset to the oblong D-parabola instead of the square D-parabola, facilitates the isolation of primes in the Cartesian plane. We demonstrated that principles applicable to square numbers can equivalently apply to oblong numbers, resulting in the discovery of the C003515 Table of Odd Integers Distributed Along the Center-Aligned Bases of f-Triangles. To address the limitation of non-vertically aligned primes, we developed the C003518 Expanded Table, suggesting consistent vertical prime pairs, thereby supporting Goldbach's Conjecture. The C003519 Quadratic Relations of Prime Pairs Table further explores prime pairs within quadratic sequences. These findings, combined with the study on The Quadratic Distribution of Integers Relative to Squares and Oblongs 13, provide robust evidence for Goldbach's Conjecture. This study adopts the terms 'primals' and 'compounds' for clarity and is a continuation of the Mersenne Forum thread The proof of Goldbach's conjecture and Landau's problem, accessible at https://www.mersenneforum.org/showthread.php?t=28200.