Unified Approach to Major Conjectures and Fundamental Problems in Number Theory: Goldbach, Rarity, and Distribution of Prime Numbers

Number theory holds a central place in arithmetic and addresses fundamental problems such as Christian Goldbach’s conjecture, its weak form, as well as the rarity and distribution of prime numbers. Although these problems are formulated in relatively simple terms, they have resisted rigorous proof for centuries.This work proposes an original approach to simultaneously tackle these four major challenges through a structured and progressive methodology. In particular, we demonstrate that the accumulation of multiples of odd numbers within the set of natural numbers naturally leads to the rarity of prime numbers. This rarity, in turn, largely explains the distribution of prime numbers among the integers.By reorganizing the logical order of these issues, this work provides an innovative and accessible perspective that contributes to a better understanding of the structure of integers and the laws governing prime numbers. This approach sheds new light on phenomena long perceived as mysterious and has the potential to guide future research in number theory.