THE DISTRIBUTION OF PRIME NUMBERS AND THE DISTANCE BETWEEN THEIR OCCURRENCES

THE DISTRIBUTION OF PRIME NUMBERS AND THE DISTANCE BETWEEN THEIR OCCURRENCES Author: Cláudio Vicente da SilvaAffiliation: Independent Researcher — Londrina, Paraná, BrazilDate: November 2026 Description This study investigates the distribution of prime numbers and the distances separating their consecutive occurrences, developing a relational approach to the arithmetic, geometric, and conceptual structures underlying the sequence of prime numbers. The analysis focuses on prime gaps, represented by (g_n=p_{n+1}-p_n), examining how the intervals between consecutive primes vary throughout the natural number sequence. The study incorporates three important conjectures in number theory: Andrica's Conjecture, Oppermann's Conjecture, and the Twin Prime Conjecture, using them as mathematical references for examining the occurrence and distribution of prime numbers. A central contribution of the study is the articulation of three interconnected elements: occurrence, absence, and distance. The occurrence identifies the position of a prime number; the absence characterizes the numerical space in which no prime occurs; and the distance expresses the interval separating consecutive prime occurrences. Together, these elements provide a relational framework for examining prime distribution. The investigation also explores the midpoint between consecutive prime numbers as a geometric reference within the interval. For consecutive primes greater than two, the midpoint is an integer but cannot itself be prime, since no prime number can occur between two consecutive primes. This observation provides a basis for examining the relationship between numerical centrality and the discrete organization of prime numbers. Another fundamental aspect concerns the distinction between concrete continuity and abstract mathematical discreteness. While concrete structures allow proportional division, intermediate positions, and equilibrium between quantities, the sequence of prime numbers belongs to a discrete arithmetic domain governed by the properties of integers and divisibility. Within this distinction, the proportion 1/2 is examined as a reference of equilibrium in continuous relations, rather than as an additional prime occurrence within the discrete sequence. By bringing together prime gaps, numerical intervals, geometric centrality, and the distinction between continuity and discreteness, the study proposes an integrated perspective on the organization of prime numbers. Its approach emphasizes that the mathematical structure of prime distribution involves not only the numbers that occur, but also the intervals and absences that define the relationships between them. The work contributes to the mathematical and philosophical investigation of prime numbers by presenting a relational interpretation of their distribution and by examining the role of absence and distance in the architecture of numerical sequences. Keywords Prime Numbers; Distribution of Prime Numbers; Prime Gaps; Andrica's Conjecture; Oppermann's Conjecture; Twin Prime Conjecture; Numerical Intervals; Mathematical Discreteness; Geometric Centrality; Equilibrium 1/2; Occurrence and Absence; Philosophy of Mathematics.