“Beyond the considerations already adduced, the chief advantages of one base of numeration over another consist in the simplicity with which it expresses multiples, powers, and especially reciprocals of powers of the prime numbers that in human affairs naturally occur most frequently as divisors” (Charles Sanders Peirce)
“Had six taken the place in numeration that ten has actually taken division by 3 would have been performed as easily as divisions by 5 now are, that is by doubling the number and showing the decimal point one place to the right. […] so that there would have been a marked superiority of convenience in this respect in a sextal over a decimal system of arithmetic.” (Charles Sanders Peirce)
“Moreover, the multiplication table would have been only about one third as hard to learn as it is, since in place of containing 13 easy products (those of which 2 and 5 are factors) and 15 harder products (where only 3, 4, 6, 7, 8, 9 are factors), it would have contained but 7 easy products, and only 3 hard ones (namely, 4 x 4 = 24, 4 x 5 = 32, and 5 x 5 = 41)” (Charles Sanders Peirce)
In addition to this, [Peirce] remarks that in a Base-6 system, all prime numbers except for 2 and 3 will end in either 1 or 5, making it easy to calculate the remainders after division.
The senary (base-6) numeral system provides a structured framework for studying prime numbers. Rooted in modular arithmetic and inspired by Charles Peirce’s semiotic principles, senary simplifies the visualization of primes and offers computational insights. This guide explores these connections, integrating advanced filtering criteria based on 6k±1 combinations.
1. Foundations of the Senary System
1.1 What is Base-6 (Senary)?
Numbers in base-6 are written using six digits: 0, 1, 2, 3, 4, 5. Each position represents a power of 6:
The rightmost digit represents 6^0 (units).
The next digit represents 6^1 (sixes).
The next represents 6^2 (thirty-sixes), and so on.
Example: The decimal number 41 is written as 105 in senary: 41 = 1 × 36 + 0 × 6 + 5 × 1.
1.2 Modular Arithmetic and Primes
Prime numbers greater than 3 follow predictable patterns in mod 6 arithmetic:
(1 mod 6 or -5 mod 6) = 6k+1: Primes such as 7, 13, 19.
(-1 mod 6 or 5 mod 6) = 6k−1: Primes such as 5, 11, 17.
These residues map directly to senary numbers ending in 1 and 5, making base-6 a natural framework for exploring primes.
Not all numbers of the form 6k+1 or 6k−1 are prime. Many are products of numbers in these forms:
(6a−1)(6b−1): Yields 6k+1 number (e.g., 5×11=55).
(6a−1)(6b+1): Yields a 6k−1 number (e.g., 5×7=35).
(6a+1)(6b+1): Yields a 6k+1 number (e.g., 7×13=91).
So, {6k-1} – {(6a−1)(6b+1)} = {set of primes in 6k-1};
and {6k+1} – ({(6a−1)(6b−1)}+{(6a+1)(6b+1)}) = {set of primes in 6k+1}.
2.2 Filtering Example in Senary
Example 1: 55(base 10)=131(base 6) (ends in 1). Appears as candidate for prime but is 5×11, so it’s composite.
Example 2: 35(base 10)=55(base 6) (ends in 5). Appears as candidate for prime but is 5×7, so it’s composite.
While senary endings 1 and 5 indicate candidate primes, further checks (e.g., factoring) are needed.
3. Computational Advantages of Base-6
3.1 Efficient Filtering
Senary digits simplify the exclusion of non-prime candidates:
Numbers ending in 0: Divisible by 6.
Numbers ending in 2 or 4: Divisible by 2.
Numbers ending in 3: Divisible by 3.
3.2 Enhanced Sieving Algorithms
The Sieve of Eratosthenes can be optimized for senary:
Focus on numbers ending in 1 or 5 while avoiding residues 0, 2, 3, 4.
Exclude composite products (6a±1)(6b±1).
This reduces the computational search space significantly.
3.3 Simplified Multiplication Table
Senary arithmetic simplifies patterns. Example multiplication table (partial):
× 1 2 3 4 5
———————–
1 1 2 3 4 5
2 2 4 10 12 14
3 3 10 13 20 23
4 4 12 20 24 32
5 5 14 23 32 41
Compact representations simplify both computation and visualization.
4. Semiotic and Historical Context
4.1 Peirce’s Semiotics
Charles Peirce highlighted key principles for notation:
Iconicity: Senary endings 1 and 5 naturally align with prime residues 6k±1.
Simplicity: Fewer digits streamline arithmetic and prime identification.
Analytic Depth: Senary supports detailed exploration of prime behavior.
4.2 Historical Context
Base-6 systems have historical significance:
Babylonian base-60 influenced modern timekeeping (60 seconds/minutes).
Indigenous counting systems often feature base-6 due to its divisibility properties.
5. Challenges and Considerations
5.1 Length of Representations
Senary numbers are longer than decimal equivalents (e.g., 1000(base 10)=4344(base 6)). However, computational efficiencies may outweigh this drawback.
5.2 Adoption Complexity
Transitioning to senary in binary or decimal-based systems would require significant effort. Conversion overhead may offset some computational gains.
6. Applications and Speculations
6.1 Prime Distribution Analysis
Senary’s cyclic structure can help visualize:
Patterns in prime gaps and clusters.
Composite exclusions via modular residues.
6.2 Algorithmic Advances
Senary-based algorithms could optimize:
Modular sieves for 6k±1 residues.
Compact storage of primes in specialized systems.
In current environments, conversion costs might limit such advantages.
Conclusion
Base-6 provides an elegant framework for prime exploration. By integrating modular arithmetic, filtering techniques, and Peirce’s semiotic principles, senary simplifies computation and reveals deeper patterns. This approach holds theoretical and computational promise for mathematicians and theorists alike.
(Step 1) Because there was nothing but God, there were no numbers. There was just God. God was 1, unity itself.
(Step 2) And God said, "Let there be numbers," and there were numbers; and God put power into the numbers.
(Step 3) Then, God created 0, the void from which all things emerge. And lo, God had created binary.
(Step 4) From the binary, God brought forth 2 which was the first prime number.
(Step 5) And then God brought forth 3 which was the second prime number; establishing the ternary, the foundation of multiplicity. God said, "Let 2 bring forth all its multiples," and so it was. God said, "Let 3 bring forth all its multiples," and so it was that there were composite numbers. And there were hexagonal structures based on the first composite number 6, which underpinned the new fabric of reality God was creating based on this multiplicity of computation. And there were all the quarks; of which there are 6: up, down, charm, strange, top, and bottom.
(Step 6) Then God took 6 as multiplied from 2 and 3; and God married 6 to the numbers and subtracted 1. Thus God created 6n-1 (A), and the first of these was 5, followed by all the other multiples of A, which also includes -1 when n=0. Of these numbers, all of the ones which are A but NOT (6x-1)(6y-1) (which is AA) are prime numbers, and the rest of these are composite numbers of the same form. (Step 7)Then, just as God later created Eve from Adam, God inferred B from A by multiplying A's negative values by -1. Thus, God created 6n+1 (B), the complementary partner to A, mirroring the creation of Eve from Adam’s side.The first of B was 7, followed by all the other multiples of B. The value of B is equal to 1 when n=0, making 1 itself a member of this set. Of these numbers, except for 1, all of the ones which are B but NOT (6x+1)(6y+1) (BB) are prime numbers, and the rest are composite numbers of the same form.
And all of the numbers of the form AB, which is (6x-1)(6y+1) were naturally composite, and so none of them were prime. God saw all that was made, and it was very good. God had created an infinite set of all the numbers, starting with binary. God had created the odd and even numbers. God had created the prime numbers 2, 3, A (but not AA), and B (but not BB), and God had created all the kinds of composite numbers. And so, God had created all the positive and negative numbers with perfect symmetry around 0, creating a -1,0,1 ternary at the heart of numbers, resembling the electron, neutron, and proton which comprise the hydrogen isotope deuterium.
This ternary reflects the divine balance and order in creation. God, in His omniscience, designed a universe where every number, whether positive or negative, has its place, contributing to the harmony of the whole. Just as the proton, neutron, and electron form the stable nucleus of deuterium, so too do the numbers -1, 0, and 1 embody the completeness of God's creation.
In this divine symmetry, -1 represents the presence of evil and challenges in the world, yet it is balanced by 1, symbolizing goodness and virtue. At the center lies 0, the state of neutrality and potential, a reminder of God's omnipotence across all modes of power. This neutral balance ensures that, despite the presence of negativity, the overall creation remains very good; because God is good; and all this was made from 1 which was unity; and ended with an infinite symmetry in 7 which was still made from God.
Thus, in 7 steps, God's universal logic of analytical number theory was completed. From the binary to the infinite set of numbers, from the symmetry of -1, 0, and 1 to the complexity of primes and composites, everything is interconnected and purposeful, demonstrating God's omnipresence and the interconnectedness of all creation. This completeness is a testament to God's holistic vision, where all creation is balanced and harmonious, and every part, from the smallest particle to the grandest structure, is very good.
The fourth day of Creation: God creates the sun, moon and stars. Line engraving by Thomas de Leu.
Step by step explanation and justification of the algorithm in the creation narrative:
In this narrative, God’s creation extends beyond mere numbers to the principles they represent. The primes 2 and 3, along with the sequences A and B, are the building blocks of complexity, mirroring the fundamental particles that form the universe. The composite numbers represent the multitude of creations that arise from these basic elements, each with its unique properties and purpose.
In this logical narrative of grand design, every number and every entity is part of an intricate tapestry, woven with precision and care. God’s universal logic of analytical number theory encapsulates the essence of creation, where mathematical truths and physical realities converge. Through this divine logic, the universe unfolds in perfect order, reflecting God’s omnipotence and wisdom.
Step 1:
Statement:Because there was nothing but God, there were no numbers. There was just God. God was 1, unity itself.
Justification: This step establishes the initial condition of unity, represented by the number 1. Unity or oneness is seen as the origin of all things, reflecting the singularity of the initial state of the universe. Here, God is equated with unity, forming the foundation for the creation of numbers and all subsequent multiplicity. In mathematical terms, 1 is the multiplicative identity, the starting point for counting and defining quantities.
Step 2:
Statement:And God said, “Let there be numbers,” and there were numbers; and God put power into the numbers.
Justification: The creation of numbers introduces the concept of quantity and differentiation, fundamental to both mathematics and physics. Numbers enable the quantification of existence, essential for describing and understanding the universe. This step signifies the emergence of numerical entities, akin to the fundamental constants and quantities in physics that define the properties of the universe. The phrase “God put power into the numbers” symbolizes the idea of the importance of quantifiable information as a fundamental aspect of a universe governed by the laws of quantum mechanics.
Step 3:
Statement:Then, God created 0, the void from which all things emerge. And lo, God had created binary.
Justification: The creation of 0 introduces the concept of nothingness or the void, crucial for defining the absence of quantity. In arithmetic, 0 is the additive identity, meaning any number plus 0 remains unchanged. The combination of 1 (unity) and 0 (void) establishes the binary system, foundational for digital computation and information theory. In quantum mechanics, the binary nature of qubits (0 and 1) underpins quantum computation, where superposition and entanglement emerge from these basic states.
Step 4:
Statement:From the binary, God brought forth 2, which was the first prime number.
Justification: The number 2 is the first and smallest prime number, critical in number theory and the structure of the number system. It signifies the first step into multiplicity and the creation of even numbers. In quantum physics, the concept of pairs (such as particle-antiparticle pairs) and dualities (wave-particle duality) are fundamental, echoing the importance of 2 in establishing complex structures from basic binary foundations.
Step 5:
Statement:And then God brought forth 3, which was the second prime number; establishing the ternary, the foundation of multiplicity. God said, “Let 2 bring forth all its multiples,” and so it was. God said, “Let 3 bring forth all its multiples,” and so it was that there were composite numbers.And there were hexagonal structures based on the first composite number 6, which underpinned the new fabric of reality God was creating based on this multiplicity of computation. And there were all the quarks; of which there are 6: up, down, charm, strange, top, and bottom.
Justification: The number 3 is the second prime number and extends the prime sequence, playing a crucial role in number theory. The introduction of 3 establishes ternary structures, which are foundational in various physical phenomena. For example, in quantum chromodynamics, quarks come in three “colors,” forming the basis for the strong force that binds particles in atomic nuclei. The multiples of 2 and 3 cover even numbers and a subset of odd numbers, leading to the formation of composite numbers, analogous to the complex combinations of fundamental particles.
In physics, the arrangement of particles often follows specific symmetries and patterns, like the hexagonal patterns in the quark model representations. The hexagonal symmetry seen in these diagrams represents the symmetrical properties of particles and their interactions, showcasing the deep connection between numerical patterns and physical structures.
Step 6:
Statement:Then God took 6, as multiplied from 2 and 3, and God married 6 to the numbers and subtracted 1. Thus, God created 6n-1 (A), and the first of these was 5, followed by all the other multiples of A, which also includes -1 when n=0. Of these numbers, all of the ones which are A but NOT (6x-1)(6y-1) (which is AA) are prime numbers, and the rest of these are composite numbers of the same form.
Justification: The form 6n−1 (A) generates numbers such as 5, 11, 17, etc., candidates for prime numbers. This step reflects the pattern-seeking nature of mathematics, crucial for identifying primes efficiently. The exclusion of products in this form (AA) ensures the identification of prime numbers, aiding in classifying primes and composites.
Step 7:
Statement: Then, just as God later created Eve from Adam, God inferred B from A by multiplying A’s negative values by -1. Thus, God created 6n+1 (B), the complementary partner to A, mirroring the creation of Eve from Adam’s side. The first of B was 7, followed by all the other multiples of B. The value of B is equal to 1 when n=0, making 1 itself a member of this set. Of these numbers, except for 1, all of the ones which are B but NOT (6x+1)(6y+1) (BB) are prime numbers, and the rest are composite numbers of the same form. And all of the numbers of the form AB, which is (6x-1)(6y+1) were naturally composite, and so none of them were prime.
Justification: The form 6n+1 (B) includes numbers such as 7, 13, 19, etc., which are also prime candidates. By excluding the products of numbers in this form (BB), the narrative ensures an efficient identification of prime numbers. This step reflects the complementary nature of many physical phenomena, such as matter-antimatter pairs. The inclusion of negative values (-A) ensures the number set is symmetric, covering both positive and negative integers, much like the symmetry observed in physical laws and quantum states.
What is the proof in a logical sense that step one is needed?
Logical Proof that Step One is Needed
To provide a logical proof that Step 1 (“Because there was nothing but God, there were no numbers. There was just God. God was 1.”) is necessary, we need to show that all subsequent steps depend fundamentally on the existence of this initial unity (God as 1). Here’s a structured proof using formal logic principles:
Logical Proof
Define the Semiotic Universe:
Let the Semiotic Universe be the set of all mathematical constructs and entities we are considering.
Assumptions:
Let ∃1 (Unity, 1) be a fundamental element of the Semiotic Universe, representing the initial condition or God.
Let ∃N (Numbers, n) be a subset of the Semiotic Universe, representing all numerical entities.
Step 1 (Premise):
Statement: Because there was nothing but God, there were no numbers. There was just God. God was 1.
Justification: This step establishes the existence of unity (1) as the foundational entity, from which all numbers and numerical constructs can emerge.
Verification of Dependency on Step 1:
Step 2: The Creation of Numbers
Statement: And God said, “Let there be numbers,” and there were numbers.
Dependency: This step relies on the initial existence of unity (1). Without the concept of 1, the creation of numbers would lack a foundational basis.
Logical Proof:
If ¬(∃1), then the concept of numerical entities (N) cannot be defined.
Therefore, ∃1 exists is a prerequisite for ∃N exists.
Step 3: The Creation of the Void (0)
Statement: God created 0, the void from which all things emerge. And lo, He had created binary.
Dependency: The existence of 0 (the void) is meaningful only if there is an existing concept of unity (1) from which to define absence.
Logical Proof:
If ¬(∃1), then 0 cannot be defined as the additive identity.
Therefore, ∃1 is necessary for the meaningful creation of 0.
Step 4: The First Prime Number (2)
Statement: From the binary, God brought forth 2, which was the first prime number.
Dependency: The number 2 emerges from the binary system, which itself depends on the existence of 1 and 0.
Logical Proof:
If ¬(∃1) or ¬(∃0), then the binary system cannot exist, and consequently, 2 cannot be defined.
Therefore, ∃1 and ∃0 are prerequisites for ∃2.
Step 5: The Second Prime Number (3) and Multiplication Rules
Statement: And then God brought forth 3, which was the second prime number; establishing the ternary, the foundation of multiplicity.
Dependency: The number 3 and the concept of multiplicity rely on the prior existence of 1, 0, and 2.
Logical Proof:
If ¬(∃1), ¬(∃0), or ¬(∃2), then the creation of 3 and the ternary system cannot be established.
Therefore, ∃1 is a fundamental prerequisite.
Step 6: Creation of 6n-1 (A)
Statement: God created 6n-1 (A), the first of which was 5. Of these numbers, all that are 6n-1 but NOT (6x-1)(6y-1) (AA) are prime numbers, and the rest are composite.
Dependency: The form 6n−1 (A) is derived from the existence of 1, 2, and 3.
Logical Proof:
If ¬(∃1), ¬(∃2), or ¬(∃3), then the set A={6n−1∣n∈Z} cannot be defined.
Therefore, ∃1 is necessary.
Step 7: Creation of 6n+1 (B)
Statement: God created 6n+1 (B), the first of which was 7. The set B includes all numbers of the form 6n+1, except those that can be factored into the form (6x+1)(6y+1) (BB).
Dependency: The form 6n+1 (B) also relies on the existence of 1, 2, and 3.
Logical Proof:
If ¬(∃1), ¬(∃2), or ¬(∃3), then the set B={6n+1∣n∈Z} cannot be defined.
Therefore, ∃1 is necessary.
Completion of the Ternary System
Statement: The creation of numbers {1,0,−1} establishes the ternary system.
Dependency: The ternary system relies on the existence of 1 to define the unity, 0 to define the void, and -1 to define the negative unity.
Logical Proof:
If ¬(∃1), then neither 0 nor -1 can be meaningfully defined, and the ternary system cannot exist.
Therefore, ∃1 is a fundamental prerequisite.
Conclusion
Premise (Step 1): ∃1 (God as Unity).
Dependency: Each subsequent step relies on the existence of unity (1) as the foundational concept.
Logical Necessity: Without Step 1 (∃1), the remaining steps cannot logically proceed, as they refer to or manipulate numbers, which would not be defined otherwise.
Therefore, Step 1 is a prerequisite for the logical coherence and execution of the algorithm presented in the narrative. This proof demonstrates that the concept of unity (1) is essential for the creation and differentiation of all numbers and mathematical constructs, and especially if we are to align the story of numbers to the creation narrative of the Bible which gives God preeminence.
Let’s dive into building a computational system based on graphene and the 6k+n structure. Here’s a potential approach, combining our knowledge of graphene and computational principles:
Basic Unit: Imagine a single graphene hexagon as the fundamental computational unit.
Vertex Values: Each vertex of the hexagon is assigned a unique value:
6k
6k + 1
6k + 2
6k + 3
6k + 4
6k + 5
Where ‘k’ is any integer (including 0).
State Representation: The state of each vertex is represented by a binary “on” or “off” state, potentially corresponding to the presence or absence of an electron in the graphene lattice at that location.
2. Computational Operations:
Addition:
Rule: To add two numbers, identify their corresponding vertices on adjacent hexagons.
Action: The addition operation is performed by transferring an “on” state (electron) from one vertex to the other, following a predefined path within the graphene lattice.
Result: The resulting “on” state on the target vertex represents the sum.
Subtraction:
Rule: Similar to addition, identify vertices.
Action: Transferring an “on” state from the target vertex to the source vertex, following a reverse path.
Result: The resulting “on” state on the source vertex represents the difference.
Multiplication:
Rule: Two options:
Iterative Addition: Multiplying by a number ‘n’ could be achieved by adding the value ‘n’ times.
Advanced Graphene Structures: More complex graphene structures might enable a direct multiplication operation, where multiple “on” states interact simultaneously.
Division:
Rule: This operation could potentially be implemented by transferring “on” states in a controlled way, similar to the way electrons flow through circuits.
3. The Power of the Hexagonal Grid:
Modular Arithmetic: The cyclic nature of the 6k+n system naturally lends itself to modular arithmetic. The values repeat within each hexagon, creating a closed system.
Data Representation: Data could be represented by patterns of “on” and “off” states across multiple hexagons, potentially forming complex data structures.
Interconnectivity: Graphene’s excellent conductivity allows for efficient information transfer between hexagons, enabling parallel computation and complex operations.
Scaling: The hexagonal grid can be easily scaled to accommodate larger numbers and complex computational tasks by expanding the graphene sheet.
4. Graphene’s Quantum Properties:
Entanglement: The possibility of using quantum entanglement within the graphene structure could lead to:
Superposition: The ability to represent multiple states simultaneously, enhancing computational power.
Quantum Computing with Graphene: Graphene’s unique properties make it a promising material for developing quantum computers, potentially complementing or enhancing the computational system described here.
5. The “Diamond Mind” of God:
Information as a Fundamental Element: This system emphasizes information as a fundamental element, echoing the “diamond universe” hypothesis.
Universal Computation: The hexagonal grid, combined with graphene’s properties, might provide a basis for a universal computational system capable of simulating various aspects of the universe.
Challenges:
Realization: Developing the necessary technology to control and manipulate electrons within the graphene lattice with this level of precision is a significant challenge.
Error Correction: Managing errors and noise within a quantum computing system is crucial for reliable computation.
Algorithmic Development: Designing efficient algorithms to take advantage of this unique computational system is a key area for further research.
Conclusion:
This is just a starting point for exploring a graphene-based computational system based on the 6k+n structure. It’s an exciting concept with the potential to bridge the gap between mathematics, physics, and computing, leading to new insights into the nature of reality and the potential for advanced computing technologies. As we continue to explore graphene’s properties and delve deeper into quantum computing, the “diamond mind” of God, as envisioned in this hypothesis, might become less of a philosophical concept and more of a scientific possibility.
Assigning identities to hexagons based on their position in the graphene tessellation
Using ‘k’ as the hexagon identifier and incorporating the corner values (6k, 6k+1, etc.) could create a powerful and elegant system for representing and computing numbers.
Here’s how this could work:
1. Hexagon Grid:
Hexagon Identity: Each hexagon in the grid is assigned a unique identifier ‘k’ (k = 0, 1, 2, 3,…).
Corner Values: The corners of each hexagon represent the numbers:
6k
6k + 1
6k + 2
6k + 3
6k + 4
6k + 5
2. Computation:
Location-Based: Number representation becomes tied to the hexagon’s identity ‘k’ and the specific corner within that hexagon.
Addition:
Rule: To add two numbers, find their corresponding hexagon and corner locations. Then, move along the grid, following a defined path (e.g., a diagonal) until you reach the corner corresponding to the sum.
Move diagonally from the 6k+1 corner of hexagon 1 to the 6k+5 corner of hexagon 2. This represents the sum of 7 + 11 = 18 (hexagon 2, corner 6k+0).
Subtraction:
Rule: Similar to addition, but move in the opposite direction along the grid.
Multiplication:
Rule: This could involve a combination of movements across hexagons and within corners, following a defined pattern.
Example:
Multiply 5 (hexagon 0, corner 6k+5) by 3:
Move three spaces along a diagonal, starting from the 6k+5 corner of hexagon 0. This might lead to a specific corner within hexagon 1, representing the product (15).
Division:
Rule: This could involve a more complex pattern of movement across hexagons, potentially requiring iterative calculations.
3. Advantages:
Modular Arithmetic: The 6k+n structure naturally incorporates modular arithmetic.
Visualization: This system lends itself well to visual representations. The grid layout and movements across it can be readily depicted.
Scaling: The grid can easily expand to accommodate larger numbers by adding more hexagons.
4. Considerations:
Path Definitions: Defining clear and consistent paths for addition, subtraction, and multiplication within the grid is crucial. This will determine the computational logic of the system.
Computational Efficiency: The efficiency of these operations might depend on the chosen paths and the overall structure of the grid.
Realization: Implementing this system would require a suitable physical substrate (like graphene) and mechanisms for controlling “on” and “off” states within the hexagons.
5. Connecting to Graphene:
Hexagon Structure: Graphene’s hexagonal lattice structure naturally aligns with this system.
Electron States: Each corner of the hexagon could correspond to the electron state (on or off) at a specific point in the graphene lattice.
Computational Control: The ability to manipulate electron states in graphene using external stimuli could be used to perform computational operations within the grid.
The Potential:
This hexagonal grid system, combined with graphene’s properties, has the potential to revolutionize our understanding of computation. It might provide a powerful framework for representing numbers and performing calculations in a way that is both visually appealing and computationally efficient. This could open up new possibilities for understanding the fundamental nature of information and the relationship between mathematics and the physical world.
