Prime Geometry as a Bundle of Lines — The Geometric Structure of the Composite‑Number Generating Formula 𝑛 = 𝑝 ^2 + 2 𝑝 ( 𝑑 − 1 )

1. Introduction

Prime numbers have long been studied through analytic, algebraic, and combinatorial methods, yet geometric perspectives have increasingly attracted attention. These approaches attempt to visualize the hidden structure of primes by embedding them into spatial or dynamical frameworks. Such methods resonate with ideas from fractal geometry, discrete dynamical systems, and information‑structural analysis, all of which seek to reveal patterns that remain obscure in purely symbolic or analytic treatments.
This paper examines a simple but surprisingly rich construction: for each prime number, one can define a linear rule that generates a sequence of composite numbers associated with that prime. When these sequences are plotted on a two‑dimensional plane whose horizontal axis represents a depth parameter and whose vertical axis represents the generated number, each prime produces a straight line. Collectively, these lines form a geometric object that can be described as a bundle or fan of rays. Although the underlying rule is elementary, the resulting geometry captures essential features of the multiplicative structure of the integers.
The analysis focuses on primes up to fifty and depth values up to one hundred. Within this bounded region, the line bundle reveals a clear hierarchy: small primes generate gentle, slowly rising lines, while large primes generate steep, rapidly rising ones. The lines never intersect within the region of interest, and their spacing increases as the depth grows. These properties reflect the independence of prime factors, the growth rates of their associated composite sequences, and the layered structure of factorization.

2. The Structure of the Generating Rule

The composite‑number generating rule assigns to each prime a straight line on the plane. The horizontal coordinate represents a depth parameter, which can be interpreted as the position within a sequence of composite numbers divisible by that prime. The vertical coordinate represents the composite number produced at that depth.
Each prime determines a line with two essential characteristics: a slope and an intercept. The slope is proportional to the prime itself, meaning that larger primes produce steeper lines. The intercept grows quadratically with the prime, placing the starting point of each line higher on the vertical axis as the prime increases.
This linear structure is crucial. Because the slope depends directly on the prime, the lines diverge from one another as the depth increases. Because the intercept also grows with the prime, the lines begin at different vertical positions even when the depth is minimal. Together, these features ensure that the lines form a coherent bundle that expands outward like a fan.

3. Constructing the Line Bundle for Small Primes

To illustrate the structure concretely, consider the fifteen primes less than or equal to fifty. For each of these primes, one draws a straight line determined by the generating rule. The result is a set of fifteen lines, each representing a distinct sequence of composite numbers associated with a particular prime.
The slopes of these lines range from four for the smallest prime to nearly one hundred for the largest. The intercepts range from zero to more than two thousand. These values determine the overall shape of the bundle: the smallest primes produce lines that rise slowly and begin near the origin, while the largest primes produce lines that rise sharply and begin far above the lower region of the plane.
The contrast between small and large primes becomes increasingly pronounced as the depth increases. At small depths, the lines lie relatively close together, because the intercepts are not yet overshadowed by the differences in slope. But as the depth grows, the slopes dominate, and the lines spread apart rapidly.

4. Geometric Features of the Line Bundle

4.1 Fan‑like Expansion
The most striking feature of the line bundle is its fan‑like expansion. When all lines are plotted together, they appear to radiate outward from the left side of the plane. The lines are relatively dense near the starting region, where the depth is small, but they diverge quickly as the depth increases.
This expansion reflects the fact that larger primes generate composite numbers at a faster rate. The geometry thus encodes a fundamental property of the integers: the multiplicative sequences associated with larger primes grow more quickly than those associated with smaller primes. The fan‑like structure is a visual manifestation of this arithmetic reality.
4.2 Non‑intersection of Lines
Another important feature is that the lines never intersect within the region of interest. Although the lines are not parallel, their slopes and intercepts are arranged in such a way that any potential intersection point lies outside the domain where the depth is positive. As a result, each line remains entirely separate from the others.
This non‑intersection property reflects the independence of prime factors. Each prime generates its own sequence of composite numbers, and these sequences do not overlap in a way that would cause the lines to cross. The geometry thus mirrors the uniqueness of prime factorization: each prime contributes its own independent branch to the structure of the integers.
4.3 Density and Divergence
The density of the line bundle varies across the plane. Near the left side, where the depth is small, the lines cluster closely because the intercepts are relatively similar. As the depth increases, the slopes dominate, and the lines diverge rapidly. This divergence creates a wide spacing between the lines in the upper‑right region of the plane.
This pattern reflects the shifting influence of different primes. At small depths, the contributions of small primes are most significant, and the structure is dominated by their gentle lines. At larger depths, the contributions of large primes become more prominent, and the structure is dominated by their steep lines. The geometry thus captures the hierarchical nature of prime distribution.

5. Interpreting the Geometry of Composite‑Number Generation

5.1 Lines as Prime‑Rooted Branches
Each line in the bundle can be interpreted as a branch rooted at a prime. The generating rule produces a sequence of composite numbers divisible by that prime, and these numbers lie along the line associated with the prime. Although the rule does not generate all multiples of the prime, it generates a structured subset that forms a coherent trajectory.
This interpretation highlights the role of primes as the fundamental building blocks of the integers. Each prime gives rise to an infinite branch extending upward and to the right, and the geometry of these branches reflects the multiplicative structure of the integers.
5.2 The Line Bundle as a Forest of Factorization
When all lines are overlaid, the result resembles a forest of branches. Each branch corresponds to a prime, and the composite numbers appear as points lying on one or more branches. This forest provides a visual representation of the factorization structure of the integers.
The geometry makes it possible to see how composite numbers are distributed among the primes, how quickly each prime generates composite numbers, and where the sequences cluster densely. It also reveals the hierarchical nature of factorization: small primes generate dense, slowly growing branches, while large primes generate sparse, rapidly growing ones.

6. The Meaning of Depth and the Growth of Lines

6.1 Depth as a Hierarchical Parameter
The depth parameter can be interpreted as a measure of hierarchical position within the sequence of composite numbers associated with a prime. It represents the level of the branch rooted at the prime. As the depth increases, the corresponding composite numbers grow larger.
This interpretation aligns with the idea of factorization as a layered process. Each prime contributes a sequence of composite numbers, and the depth indexes the levels of this sequence.
6.2 Growth with Increasing Depth
As the depth increases, each line extends upward to the right. The rate of growth is determined by the slope, which is proportional to the prime. Larger primes produce steeper lines, causing the bundle to open widely on the right side of the plane.
This geometric behavior reflects the arithmetic fact that multiples of larger primes grow more quickly than multiples of smaller primes. The geometry thus provides a visual representation of the growth rates associated with different primes.

7. Number‑Theoretic Implications

7.1 Coverage of Composite Numbers
Each line represents a subset of composite numbers divisible by a given prime. When all lines are combined, they cover a significant portion of the composite numbers within the range of interest. Although the generating rule does not produce all composite numbers, it produces enough to reveal the underlying structure of factorization.
The line bundle thus provides a geometric framework for understanding how composite numbers are distributed among the primes.
7.2 Hierarchy in Prime Distribution
The geometry of the line bundle reflects the hierarchical nature of prime distribution. Small primes generate gentle lines that form the foundation of the structure. Large primes generate steep lines that dominate the higher regions of the plane.
This hierarchy mirrors classical results in number theory, such as the density of primes and the distribution of composite numbers. The geometric representation makes these relationships visually apparent.
7.3 Fractal‑like Properties Although the line bundle is not formally a fractal, it exhibits several fractal‑like properties. The branches extend independently to infinity, the structure expands hierarchically, and the arrangement of branches displays a form of global self‑similarity. These properties reflect the self‑similar nature of prime factorization.

8. Conclusion

The line bundle generated by the composite‑number rule provides a simple yet powerful geometric representation of the multiplicative structure of the integers. It reveals a fan‑like expansion determined by the slopes associated with the primes, non‑intersecting trajectories that reflect the independence of prime factors, and a hierarchical arrangement corresponding to the distribution of primes.
This geometric framework offers a new way to visualize prime number theory. It highlights the relationships between primes, composite numbers, and the multiplicative structure of the integers. Future research may explore the density of the line bundle, the coverage of composite numbers, the geometric dimension of the bundle, and potential connections with analytic objects such as the zeta function.