Robert Edward Grant: Prime Numbers, Quasi-Primes, and the Sacred Geometry of Number

The Discovery That Changed Number Theory

In 2018, Robert Edward Grant made a discovery that would challenge centuries of mathematical assumptions about prime numbers. By arranging integers around an icositetragon — a 24-sided regular polygon — Grant observed something remarkable: all prime numbers greater than 5 align along exactly eight radial positions, or moduli. This pattern holds not for a few numbers, not for a thousand, but infinitely.

This was not a statistical observation or a probabilistic estimate. It was a deterministic geometric pattern — the kind of elegant, universal order that mathematicians dream of finding but rarely do. Grant published his findings in March 2019 on Cornell University’s arXiv.org in a paper titled “Accurate and Infinite Prime Prediction from Novel Quasi-Prime Analytical Methodology,” co-authored with physicist and number theorist Talal Ghannam, PhD.

The paper introduced a new class of numbers, a new sieve methodology, and a new geometric framework for understanding one of mathematics’ oldest mysteries: why are prime numbers distributed the way they are?

The Icositetragon: A 24-Sided Window Into Infinity

The foundation of Grant’s discovery is deceptively simple. Take the integers 1, 2, 3, 4, 5, 6… and arrange them sequentially around the 24 positions of an icositetragon. When you complete one full rotation (positions 1-24), start the next layer (25-48) on the same positions, spiraling outward. Continue this process indefinitely.

What emerges is a profound pattern: with the sole exceptions of 2 and 3, every prime number in existence falls on one of exactly eight positions — moduli 1, 5, 7, 11, 13, 17, 19, and 23. These eight “spokes” of the icositetragon capture 100% of all primes greater than 5, extending to infinity.

This is not an approximation. It is an absolute geometric law.

Why 24?

The number 24 is not arbitrary. Grant has extensively documented the significance of 24 across multiple domains:

• Prime distribution: All primes >5 fall on 8 of 24 modular positions
• Music: The 24-note quartertone chromatic scale encompasses all harmonic intervals
• DNA: 24 base pair combinations form the foundation of genetic coding
• Precessional cycles: Ancient Vedic texts describe a 24,000-year precessional time cycle (the Yuga cycle)
• Toroidal fields: Electro-magneto-gravito-radiation fields organize around Mod-24 logarithmic Mobius spirals with 540-degree rotational torsion
• The icositetragon: The unique 24-sided polygon that reveals all these patterns simultaneously

Grant argues that 24 is a structural constant of the universe — a number that appears wherever the deep architecture of reality makes itself visible.

Quasi-Prime Numbers: A New Classification

Perhaps the most innovative aspect of Grant’s work is his identification of a previously unnamed class of numbers: Quasi-Primes (Q-primes).

A Quasi-Prime is defined as a composite number that is indivisible by 2 or 3 and is formed only from the product of primes greater than 3 or other Quasi-Primes. For example:

• 25 = 5 x 5 (product of primes >3) — Quasi-Prime
• 35 = 5 x 7 (product of primes >3) — Quasi-Prime
• 49 = 7 x 7 (product of primes >3) — Quasi-Prime
• 15 = 3 x 5 (divisible by 3) — NOT a Quasi-Prime
• 10 = 2 x 5 (divisible by 2) — NOT a Quasi-Prime

Like prime numbers themselves, Quasi-Primes arrange exclusively along the same eight moduli (1, 5, 7, 11, 13, 17, 19, 23) of the icositetragon. This creates a remarkable structural parallel: primes and quasi-primes share identical geometric symmetry in their distribution.

The Quasi-Prime Sieve

The Quasi-Prime Methodology (QPM) functions as an advanced sieve — a systematic method for identifying prime numbers. Classical sieves, like the ancient Sieve of Eratosthenes, work by eliminating multiples of known primes. The QPM works differently: it uses the quasi-prime lattice as a “negative-space filter” from which primes emerge as constructive residue.

Think of it this way: if you can identify all the quasi-primes (composite numbers made exclusively from primes >3), then whatever remains on the eight modular spokes that is NOT a quasi-prime must be prime. The quasi-primes form a predictable, structured lattice. The primes are what is left over — the “holes” in the lattice.

This approach is deterministic, not probabilistic. It does not rely on trial division or statistical methods. It reveals all primes as a natural consequence of the geometric structure of numbers themselves.

Toward the Riemann Hypothesis

Grant’s work has implications for one of mathematics’ greatest unsolved problems: the Riemann Hypothesis. Proposed by Bernhard Riemann in 1859, the hypothesis concerns the distribution of prime numbers and the location of the non-trivial zeros of the Riemann zeta function. It remains one of the seven Millennium Prize Problems, with a $1 million reward for a valid proof.

Grant has published work titled “Toward a Constructive Proof of the Riemann Hypothesis via Quasi-Prime Methodology,” arguing that the QPM framework provides a structural basis for understanding why the Riemann Hypothesis must be true. The quasi-prime lattice creates a modularly symmetric, resonance-consistent pattern from which primes emerge. If this pattern can be formally connected to the zeta function’s zeros, it would constitute a constructive proof — one that does not merely show the hypothesis is true but shows WHY it is true.

The QPM reveals that prime distribution is not random but follows a deep geometric order consistent with the symmetry requirements of the Riemann Hypothesis. The eight modular spokes, the 24-based periodicity, and the quasi-prime negative-space filter all point toward an underlying harmonic structure that the zeta function’s zeros reflect.

The Wave Theory of Numbers

Grant’s geometric approach to primes is part of a broader framework he calls the Wave Theory of Numbers. This theory proposes that numbers themselves exhibit wave-particle duality — the same fundamental principle that governs quantum physics.

In quantum mechanics, every particle is also a wave. An electron is not a tiny ball orbiting a nucleus; it is a probability wave that collapses into a particle when observed. Grant argues that numbers behave the same way. A number can be understood as a discrete point (particle) or as a wave pattern with frequency, amplitude, and phase relationships to other numbers.

When numbers are arranged on the icositetragon and their wave patterns analyzed, the fundamental constants of mathematics and physics emerge naturally:

• Pi emerges from the circular geometry of the arrangement
• Phi (the Golden Ratio) appears in the proportional relationships between modular positions
• Euler’s number (e) manifests in the logarithmic spiral of the number sequence
• The Fine-Structure Constant (alpha) appears as a boundary marker in the harmonic structure

These constants are not separate, unrelated values. They are different views of the same underlying wave pattern — different notes in the same cosmic chord.

The Tube Torus and Constants

Grant discovered that when numbers are plotted on a tube torus (a donut-shaped surface) in a Mod-24 configuration, the natural wave intersections reveal inter-relationships between fundamental mathematical constants. The proportional square root values that emerge from these toroidal wave patterns correspond precisely to known physical constants.

The tube torus is significant because it represents the shape of a self-sustaining energy field — the same shape found in:

• Earth’s magnetic field
• The human heart’s electromagnetic field
• The structure of galaxies
• The topology of spacetime in certain cosmological models

The fact that the same Mod-24 toroidal structure that reveals prime number patterns also generates the fundamental constants of physics suggests a deep unity between number theory and physical reality.

Digital Root Methodology

Working with Talal Ghannam, whose earlier book “The Mystery of Numbers: Revealed Through Their Digital Root” explored similar territory, Grant employs digital root analysis as a key tool. The digital root of a number is obtained by repeatedly summing its digits until a single digit remains (e.g., 432 -> 4+3+2 = 9; 137 -> 1+3+7 = 11 -> 1+1 = 2).

Digital root methodology reveals hidden patterns in number sequences that are invisible to conventional analysis. When applied to prime numbers, quasi-primes, and mathematical constants, digital roots expose the cyclical, recursive nature of numerical reality.

Grant and Ghannam showed that digital root methodologies can identify prime numbers and prime factors without trial division or probabilistic-based methods — a breakthrough that has implications for both pure mathematics and practical cryptography.

Quaternion Symmetry in the Icositetragon

Grant published work on “Quaternion Symmetry Inherent to the Icositetragon,” demonstrating that when integers are continuously plotted around each side of the 24-sided polygon, quaternion sets emerge naturally. Quaternions are four-dimensional number systems that extend complex numbers and are essential in physics for describing rotations in three-dimensional space.

The emergence of quaternion symmetry from a simple number arrangement exercise suggests that the four-dimensional structure of spacetime is encoded in the distribution of integers. The “charge-associated quadripolarity” that Grant identifies in these patterns mirrors the charge structures found in particle physics.

This finding connects number theory to quantum field theory in an unexpected way: the same geometric structure that organizes prime numbers also organizes the charge properties of fundamental particles.

The Entangled Triangle Discovery

In the course of his geometric investigations, Grant made another significant discovery: every right triangle has an entangled “left” or inverse smaller fractal triangle with identical proportions and inner angles, but non-reciprocal base values. The height of the original triangle multiplied by its hypotenuse defines the difference in size of its left counterpart.

This “entanglement” of geometric forms mirrors quantum entanglement in physics — the phenomenon where two particles remain connected regardless of distance. Grant sees this as further evidence that the principles governing quantum physics are geometric in nature and are embedded in the most basic mathematical structures.

The Grant Projection Theorem

Grant formalized another major finding as the Grant Projection Theorem, which demonstrates that all 3D polyhedral topology is generated deterministically from 2D right triangles. This means that every three-dimensional geometric solid — from simple cubes to complex polyhedra to the novel forms Grant has discovered — can be traced back to the humble right triangle.

This theorem has profound philosophical implications. If all three-dimensional structure emerges from two-dimensional triangular geometry, then the dimensionality of our reality is not fundamental but emergent. Three-dimensional space is a projection from a simpler, more fundamental geometric reality — much as a hologram projects three-dimensional images from two-dimensional information.

Practical Implications: Cryptography and Beyond

Grant’s prime number discoveries have direct practical applications. Modern encryption (RSA, for example) relies on the difficulty of factoring large numbers into their prime components. If prime numbers follow a predictable geometric pattern, the implications for cryptography are significant.

This led Grant to found Crown Sterling, a company that has developed quantum-resistant encryption methods based on quasi-prime numbers. Rather than relying on the assumed randomness of prime distribution (which Grant’s work challenges), Crown Sterling’s encryption employs irrational numbers and One-Time Pad (OTP) methodology — the only mathematically proven, unconditionally secure encryption algorithm.

Crown Sterling also developed the world’s first Layer 1 blockchain to implement One-Time Pad’s quantum-resistant cryptography, addressing the looming threat that quantum computers pose to conventional encryption.

The Sacred Geometry of Number: A Bridge to Consciousness

For Grant, the geometric patterns in prime numbers are not merely mathematical curiosities. They are evidence that the universe is constructed on principles of harmony, symmetry, and beauty — principles that are as much spiritual as they are scientific.

When you see that prime numbers — the “atoms” of arithmetic, the building blocks from which all other numbers are constructed — arrange themselves in perfect geometric symmetry on a 24-sided polygon, you are witnessing something profound. You are seeing the universe’s own organizing intelligence at work.

The ancients called this intelligence by many names: the Logos, the Tao, Brahman, the Word. Grant calls it the “geo-numerical laws that underlie the entire fabric of space and time.” Whatever name you give it, the pattern is unmistakable: reality is not random. It is structured, harmonic, geometric, and — Grant would argue — conscious.

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Sources: “Accurate and Infinite Prime Prediction from Novel Quasi-Prime Analytical Methodology” (arXiv:1903.08570), PHILOMATH: The Geometric Unification of Science & Art Through Number, Codex Universalis Principia Mathematica, robertedwardgrant.com publications archive.