Kusniec's Theorem (expanded)

This work presents a comprehensive study on the distribution of prime numbers through geometric and arithmetic methodologies, offering new insights into their behavior and verifying several conjectures, including Legendre’s, Oppermann’s, Brocard’s, and Andrica’s conjectures. By introducing tools such as the Parabolic Sieve of Primes with Offset (PSO), the Periodic Table of Odd Prime Numbers, and the Map of Odd Numbers with Their Divisors (MOND), this study demonstrates the structured and periodic nature of prime distributions. It introduces novel numerical zones between odd squares and oblong numbers to systematically analyze prime densities and sieve methods. Additionally, implications for other prominent conjectures like Bertrand’s Postulate, Cramér’s, Firoozbakht’s, and Ramanujan primes are explored, further affirming the non-sparse distribution of primes across the number line. The findings offer a unifying perspective on prime gaps, density, and divisors, providing a robust foundation for future research in analytic and computational number theory.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: Prime Numbers, Legendre’s Conjecture, Oppermann’s Conjecture, Brocard’s Conjecture, Andrica’s Conjecture, Prime Gaps, Parabolic Sieve, Numerical Zones, Oblong Numbers, Ramanujan Primes, Bertrand’s Postulate.2020 Mathematics Subject Classification: 11A41: Primes, 11N05: Distribution of primes, 11A07: Congruences; primitive roots; residue systems, 11P32: Gaps between numbers represented by polynomials or linear forms, 11Y55: Computational number theory, 11A25: Arithmetic functions; related numbers, 11B75: Combinatorial number theory, 11A51: Factorization; primality.