Unified Modular Geometry: Prime Constellation Patterns

Visualizing twin, cousin, and sexy prime constellations across modular bases (7, 9, 10, 12) using digital root patterns.

Abstract

This paper develops a Unified Modular Geometry framework for analyzing prime constellations through the lens of digital roots across multiple numeral bases. The digital root is formally defined as a modular compression function:

\[ \mathrm{DR}_b(n) \equiv n \mod (b - 1) \]

We apply this framework to prime constellations—including twin, cousin, sexy, triplet, and quadruplet primes—by translating their characteristic gap structures into deterministic orbits within modular spaces. The analysis demonstrates that the modulus \( b - 1 \) of each base governs both the range and the stability of digital root patterns.

For composite moduli (e.g., 9 in Base 10, 15 in Base 16), the compression enforces strict constraints that yield a finite taxonomy of patterns (such as the three admissible digital root configurations for twin primes in Base 10). Conversely, prime moduli (e.g., 11 in Base 12) produce a less compressed but structurally richer spectrum of patterns.

These findings formalize the intuition that prime sequences display strong local regularities despite their global irregularity. The Unified Modular Geometry framework not only provides a novel visualization tool for prime distributions but also offers structural insights relevant to major conjectures in analytic number theory, including Hardy–Littlewood’s prime pair conjectures and sieve-theoretic models of prime gaps.

Keywords: Modular arithmetic, Prime constellations, Digital root, Modular orbits, Prime gaps, Sieve theory

Introduction

MSC Classification: Primary 11A07; Secondary 11N05, 11B83, 11K99

The distribution of prime numbers across the number line has long fascinated mathematicians, representing a delicate interplay between apparent chaos and hidden order. The Prime Number Theorem offers a statistical approximation for prime density (Hadamard, Vallée-Poussin, 1896), yet the precise location of primes remains unpredictable.

Despite this unpredictability, striking regularities emerge when primes are examined in structured configurations. These configurations, termed prime constellations, have been studied since the 19^th century, beginning with Alphonse de Polignac’s conjecture (1849), which asserts that every even number occurs infinitely often as the difference of two consecutive primes. This broad conjecture encompasses the famous Twin Prime Conjecture, which proposes the infinitude of prime pairs separated by a gap of two.

The present study seeks to formalize these hidden regularities by introducing a modular framework grounded in digital roots. Traditionally, research on prime distribution has relied on probabilistic or asymptotic analysis (Hardy, 1979). In contrast, this work demonstrates that a simple compression mechanism—the digital root, defined modulo \( b - 1 \)—can uncover a deterministic geometric structure underlying prime constellations.

\[ \mathrm{DR}_b(n) \equiv n \mod (b - 1) \]

By recasting prime gaps in the language of modular arithmetic, we reveal stable and predictable patterns that are dictated not by randomness, but by the inherent properties of the base itself.

This perspective builds upon foundational results in analytic number theory. The concept of prime constellations is central to the Hardy–Littlewood k-tuple conjecture (1923), which predicts the asymptotic densities of admissible prime patterns such as twin \((p, p+2)\), cousin \((p, p+4)\), and sexy \((p, p+6)\) primes.

More recently, the field advanced dramatically with Yitang Zhang’s breakthrough (2014), which proved for the first time that infinitely many primes exist within bounded gaps (initially less than 70 million). This milestone was refined through the collaborative Polymath Project (2014), involving James Maynard, Terence Tao, and others, which reduced the bound to 246.

The analytic machinery behind these advances—particularly sieve theory and admissible sets—directly informs the present study, since these methods illuminate the structural constraints that govern prime constellations.

The goal of this work is not only descriptive but classificatory. By examining prime constellations through digital roots across multiple bases (7, 9, 10, 12, and 16), we establish a comparative taxonomy of modular patterns. This analysis demonstrates how the arithmetic structure of each base’s modulus \( b - 1 \) governs the range, predictability, and stability of digital root behaviors.

In doing so, it reframes prime constellations not as isolated curiosities, but as manifestations of a deeper unified modular geometry—an approach that both visualizes and constrains prime distributions in ways that connect to longstanding conjectures about their gaps.

Theoretical Foundations and Methodology

Formal Definition of Digital Roots and Modular Arithmetic

The concept of a digital root is traditionally introduced as an iterative process of summing a number’s digits until a single-digit value is obtained. While often seen as elementary, it is a direct consequence of modular arithmetic.

Let a number \( N \) be expressed in base \( b \) with digits \( d_k, d_{k-1}, \dots, d_0 \):

\[ N = d_k b^k + d_{k-1} b^{k-1} + \cdots + d_1 b^1 + d_0 \]

Since \( b \equiv 1 \mod (b - 1) \), it follows that:

\[ N \equiv d_k + d_{k-1} + \cdots + d_0 \mod (b - 1) \]

Thus, any integer is congruent to the sum of its digits modulo \( b - 1 \). The iterative digit summation is equivalent to repeatedly applying this congruence until stabilization. By convention, if \( N \equiv 0 \mod (b - 1) \), the digital root is defined as \( b - 1 \).

\[ \mathrm{DR}_b(n) \equiv n \mod (b - 1) \]

This principle enables digital root analysis across any base \( b > 2 \).

Admissible Prime Constellations

Prime constellations are finite sets of primes with fixed gaps:

• Twin primes: \( (p, p+2) \)
• Cousin primes: \( (p, p+4) \)
• Sexy primes: \( (p, p+6) \)
• Triplets: \( (p, p+2, p+6) \)
• Quadruplets: \( (p, p+2, p+6, p+8) \)

A tuple \( (h_1, h_2, \dots, h_k) \) is admissible if, for every prime \( q \), the set \( \{h_1, h_2, \dots, h_k\} \mod q \) does not cover all residue classes. Otherwise, one number will always be divisible by \( q \), making the constellation impossible.

Example: The triplet \( (p, p+2, p+4) \) is inadmissible for \( p > 3 \).

• If \( p \equiv 1 \mod 3 \), then \( p+2 \equiv 0 \mod 3 \)
• If \( p \equiv 2 \mod 3 \), then \( p+4 \equiv 0 \mod 3 \)

Only \( (3, 5, 7) \) satisfies this.

This principle explains why some constellations vanish beyond small primes and why Hardy–Littlewood’s framework identifies admissible ones as candidates for infinite occurrence.

Metrics for Comparative Analysis

To compare digital root behavior across bases, we introduce three metrics:

🔻 Modular Compression Index (MCI)

Measures how strongly a modulus compresses the digital root space:

\[ \mathrm{MCI} = \frac{\text{Number of observed (or provable) digital root combinations}}{\text{Total theoretically possible combinations}} \]

Lower MCI → higher compression → more deterministic constraints.

Constellation Entropy

Quantifies uniformity of digital root distribution using Shannon entropy:

\[ H = -\sum_i p_i \log(p_i) \]

In base 10, twin prime patterns appear nearly uniformly, yielding high entropy.

Orbit Length and Structure

Analyzes repeating cycles in digital root sequences. Prime moduli (e.g., 11 in base 12) yield longer, richer orbits. Composite moduli (e.g., 15 in base 16) collapse into smaller attractors.

Results

The analysis across bases demonstrates a unifying principle:

• Composite moduli (6, 8, 9, 15) heavily restrict admissible constellations, leading to high modular compression.
• Prime moduli (11) preserve nearly all possible digital root orbits, yielding maximal diversity.

This contrast confirms that the factorization structure of \( b - 1 \) directly governs the degree of pattern compression in prime constellations.

Base-by-Base Analysis of Constellation Patterns

This section presents the core findings, analyzing digital root patterns for each prime constellation across selected bases.

Base 10 (Modulus 9)

• Twin primes: Only three digital root pairs occur: (2,4), (5,7), (8,1)
• Cousin primes: Valid pairs: (1,5), (4,8), (7,2)
• Sexy primes: Six admissible pairs: (1,7), (2,8), (4,1), (5,2), (7,4), (8,5)
• Quadruplets: Unique pattern: (2,4,8,1)

Base 7 (Modulus 6)

• Twin primes: Only (5,1) survives
• Cousin primes: Only (1,5)
• Sexy primes: Symmetric pairs: (1,1), (5,5)

Base 9 (Modulus 8)

• Twin primes: Four pairs: (1,3), (3,5), (5,7), (7,1)
• Cousin primes: Two pairs: (1,5), (3,7)

Base 12 (Modulus 11)

• Twin primes: All 10 possible pairs occur
• Cousin primes: All 10 pairs occur
• Sexy primes: All 10 pairs occur

Base 16 (Modulus 15)

• Twin primes: Eight admissible pairs among coprime residues
• Quadruplets: Example: (11,13,17,19) → (B,D,2,4)

Base	Modulus	Twin Primes	Cousin Primes	Sexy Primes	Quadruplets	Notes
7	6	(5,1)	(1,5)	(1,1), (5,5)	N/A	Strong compression
9	8	(1,3), (3,5), (5,7), (7,1)	(1,5), (3,7)	...	N/A	Moderate compression
10	9	(2,4), (5,7), (8,1)	(1,5), (4,8), (7,2)	6 pairs	(2,4,8,1)	Composite modulus
12	11	10 pairs	10 pairs	10 pairs	N/A	Prime modulus, max diversity
16	15	8 pairs	...	...	Quadruplets avoid 3,5 factors	Composite modulus

Digital Root Patterns of Prime Constellations Across Bases

Comparative Synthesis and Discussion

The base-by-base analysis reveals a profound connection between local prime constellation behavior and global number theory structure. These patterns are governed by a unified principle: the interaction between a constellation’s gap and the prime factors of the base’s modulus \( b - 1 \).

The Role of the Modulus in Modular Compression

The Modular Compression Index (MCI) compares observed digital root patterns to theoretically possible ones.

• Composite moduli (e.g., base 10 → modulus 9) show high compression. Twin primes yield only 3 of 36 possible pairs → MCI = \( \frac{1}{12} \).
• Prime moduli (e.g., base 12 → modulus 11) preserve diversity. Twin primes yield 10 of 100 pairs → MCI = \( \frac{1}{10} \).

In base 10:

• Twin primes → 3 pairs
• Cousin primes → 3 pairs
• Sexy primes → 6 pairs

Sexy primes (gap 6) are divisible by 3, a factor of modulus 9. So \( p + 6 \equiv p \mod 3 \), preserving admissibility. This explains why sexy primes occur more frequently than twin or cousin primes.

Bridging Local Patterns to Global Conjectures

Local digital root patterns offer insight into global conjectures. For example:

\[ (6n - 1)(6n + 1) = 36n^2 - 1 \equiv 8 \mod 9 \]

This shows that the product of twin primes always has digital root 8—an elegant modular truth.

The framework aligns with modern results on prime gaps (Zhang, Maynard–Tao) and sieve theory. Admissibility tests for digital roots mirror those used in analytic number theory.

Conclusion

This study demonstrates that prime constellations are shaped by modular laws. The modulus \( b - 1 \) governs the diversity and structure of digital root patterns across bases.

Summary of Key Findings

• Digital roots as modular compression: \( \mathrm{DR}_b(n) \equiv n \mod (b - 1) \) compresses the number line into predictable orbits.
• Composite moduli: Impose strong constraints, reducing pattern diversity.
• Prime moduli: Preserve orbit richness and allow maximal diversity.
• Gap-modulus interaction: Explains why sexy primes yield more patterns than twin or cousin primes.

Unified Modular Geometry

This framework reveals that prime constellations are not random—they follow universal modular laws. Local constraints encode global truths.

• Digital root patterns reflect deep arithmetic structure.
• Constellations vary predictably across bases due to modulus factorization.

Future Directions

• Analyze larger constellations (quintuplets, sextuplets)
• Explore exotic bases (non-integer, algebraic, complex)
• Develop algebraic models of digital root attractors

These extensions will deepen our understanding of prime structure and connect modular patterns to unsolved problems in number theory.

Statements and Declarations

Competing Interests

The authors declare that they have no competing interests.

Statements and Declarations

Data Availability

Data supporting the findings of this study consist of computational outputs—specifically prime constellations and digital root compressions—generated using the Python implementation of the authors.

The complete source code is openly accessible at: https://mathsmagic231197.blogspot.com/ (accessed 07 September 2025).

No external datasets were used in this investigation.

Ethical Approval

Not applicable.

References

Classical Foundations

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Hardy, G.H. & Wright, E.M. (1979). An Introduction to the Theory of Numbers, 5th ed. Oxford University Press.

Hardy, G.H. & Wright, E.M. (2008). An Introduction to the Theory of Numbers, 6th ed. Oxford University Press. ResearchGate

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Ribenboim, P. (1996). The New Book of Prime Number Records. Springer. Springer

Prime Gaps & Modern Results

Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics, 179(3), 1121–1174. DOI

Polymath Project (2014). Bounded gaps between primes: Polymath8. Website

Maynard, J. (2015). Small gaps between primes. Annals of Mathematics, 181(1), 383–413.

Granville, A. (2007). Primes in intervals of bounded length. Bulletin of the AMS, 44(1), 1–19.

Tao, T. (2016). Bounded Gaps Between Primes. ICM Proceedings, Vol. 1.

Miscellaneous

Shannon, C.E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.