A Framework for the Bouniakowsky Conjecture: Odd Numbers as Differences of Opposite Parity Squares and Their Role in Polynomial Prime Generation

Abstract: This study explores a novel approach to the Bouniakowsky Conjecture, leveraging the fundamental property that all odd integers can be expressed as the difference of two squares with opposite parity. By generalizing this property, we examine the behavior of polynomial sequences in the generation of prime numbers and provide a framework for understanding divisibility patterns with respect to odd prime and composite divisors. We demonstrate how the interaction between irreducible polynomials and the algebraic structure of numbers leads to insights into prime density and quadratic residues. Our analysis extends to the interplay between polynomial coefficients and Taylor shifts to ensure irreducibility, offering a pathway to validating the conjecture in broader cases. This study highlights the intricate relationship between parity, prime generation, and the structure of polynomial sequences.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: Bouniakowsky Conjecture, Polynomial Prime Generation, Opposite Parity Squares, Prime Density, Divisibility Patterns, Taylor Shifts, Quadratic Residues.2020 Mathematics Subject Classification: 11A41: Primes; 11N32: Primes represented by polynomials; 11D09: Quadratic and bilinear equations; 11B13: Patterns of numbers; other sequences and sets.