Prime Numbers and Gaps: A Unified Approach to Goldbach's Conjecture Using the Difference of Squares

This study explores the Goldbach Conjecture and the Twin Prime Conjecture using novel approaches rooted in the difference of squares. By analyzing the equations x_1= 4uv-2u-2v+1 and x_1=(2g-1)^2-(2t)^2, we demonstrate the existence of gaps in the generation of odd composite numbers. These gaps inherently imply the presence of odd primes, leading to evidence supporting the infinitude of twin primes and the generalization of De Polignac's Conjecture. We introduce the "C002476 Goldbach's and Landau's Range Visualization Table," a structural tool to visualize the relationships between odd and even numbers, their sums, and their differences. This visualization provides a geometric perspective on the distribution of prime pairs and a pathway toward proving Goldbach's Conjecture through symmetry and bijection. Furthermore, the study discusses the Goldbach Zones in the MID framework, offering a unique lens for understanding prime pair generation in specific numeric intervals.This study, presented as a preprint, offers preliminary insights that are yet to undergo peer review. Future revisions may further refine the findings and conclusions drawn here.Keywords: Goldbach Conjecture, Twin Prime Conjecture, De Polignac's Conjecture, Difference of Squares, Prime Number Gaps, Quadratic Forms, Number Theory, Visualization Table, Goldbach Zones, Symmetry in Primes.2020 Mathematics Subject Classification: 11P32: Goldbach-type theorems; other additive questions involving primes; 11N05: Distribution of primes; 11A41: Primes; 11Y55: Calculation of integer sequences; 11D09: Quadratic and bilinear equations.