1. Defining the Domain and the Form

We begin by considering the set of non-zero integers, A = Z \ {0}, which will serve as the domain for our indices k.

We focus on numbers n generated by the function f(k) = |6k-1| for k ∈ A. It is a well-established property that any prime number p greater than 3 must satisfy p ≡ ±1 (mod 6).

The form n = |6k-1| systematically generates the absolute values of all integers congruent to ±1 (mod 6) (excluding ±1 itself, as k ≠ 0). (The choice of 6k+1 or 6k-1 is trivial, but the selection of composites based on the form is not trivial. The following focuses specifically on |6k-1|.)

Consequently, the set of numbers generated by f(k) for k ∈ A contains all prime numbers greater than 3, alongside composite numbers also satisfying the ±1 (mod 6) condition (e.g., 25, 35, 49, 55…). The entire set A thus represents the indices of all candidates for being primes greater than 3, based solely on the |6k-1| form.

2. Establishing the Rule for Compositeness via Index Generation

The core insight is the establishment of a specific rule that governs the indices k corresponding to composite numbers within the |6k-1| sequence. Through algebraic manipulation of the factors of composite numbers of the form 6k ± 1, we derived the following rigorous equivalence:

An integer n = |6k-1| (with k ∈ A, n ≥ 5) is composite if and only if its index k can be expressed as k = 6xy + x – y for some non-zero integers x, y (i.e., x, y ∈ A).

This equivalence is crucial. It provides a constructive definition for the indices of composite numbers within our sequence. We can define the set S_3 explicitly based on this rule:

S_3 = { 6xy + x – y | x ∈ A, y ∈ A }

The set S_3 represents the “positive space” of composite indices. Any index k belonging to S_3 definitively corresponds to a composite number n = |6k-1|. The polynomial g(x, y) = 6xy + x – y acts as the generator for this set.

3. The Inferential Problem: Identifying Primes

We now face the central problem: given an index k ∈ A, how do we determine if the corresponding n = |6k-1| is prime? We know k represents a candidate. We also have a definitive rule (k ∈ S_3) that signals compositeness. How do we leverage this to identify primes?

4. The Abductive Inference from Exclusion

Direct primality tests evaluate n. Sieves eliminate multiples. This method instead focuses on the index k and its relationship to the constructively defined set S_3. The reasoning process for determining primality becomes an instance of Peircean abduction:

• Observation: We take an index k from the set of candidates A.
• Test: We check if this observed k belongs to the set S_3 (the set of composite indices). This involves checking if k can be represented as 6xy + x – y for some x, y ∈ A.
• Two Possible Outcomes:
  • Outcome 1: k ∈ S_3. The index k fits the established rule for compositeness. By deductive reasoning based on the proven equivalence, we conclude that n = |6k-1| is composite.
  • Outcome 2: k ∉ S_3. This is the surprising or unexplained observation if we were to assume n might be composite. The index k fails to conform to the necessary condition (k ∈ S_3) that must hold if n were composite.
• Abductive Step: The observation k ∉ S_3 demands an explanation. Given the “if and only if” nature of the equivalence, the only possible explanation for k not being in the set S_3 is that the premise leading to that condition – namely, that n = |6k-1| is composite – must be false. Therefore, we infer, as the best and necessary explanation, that n = |6k-1| must be prime.

This inference is abductive because it reasons from an observed consequence (or lack thereof: k ∉ S_3) back to the most plausible underlying state (primality of n). It’s an inference to the best explanation for why k does not possess the characteristic property of composite indices.

5. Primes in the “Subtractive Space”

The formalization of this inference lies in set theory. The entire space of candidate indices is A. The subspace of indices corresponding to known composites is S_3. The act of identifying primes becomes equivalent to performing the set subtraction:

K_prime = A \ S_3

This explicitly defines the set of prime indices K_prime as everything in the candidate space A except for the elements known to be composite indices (S_3). The primes are thus located in this “subtractive space” or “negative space” – a space defined not by its own positive generating rule within this framework, but by what it excludes. We identify primes by recognizing their indices lack the signature (∈ S_3) associated with compositeness.

Theorem Restated: Let A = Z \ {0} and S_3 = { 6xy + x – y | x ∈ A, y ∈ A }. The set K_prime = { k ∈ A | |6k – 1| \ { is prime} } is exactly A \ S_3.

Conclusion

This approach provides a distinct perspective on prime identification for numbers n = |6k-1|. It does not generate primes directly but instead constructively generates the indices k corresponding to all composite numbers within this form via the set S_3.

Primality is then inferred abductively: an index k is recognized as corresponding to a prime n = |6k-1| precisely because it is absent from the set S_3.

The primes occupy the logical space remaining after the identifiable composite indices are excluded from the initial set of candidates.

This reliance on inference from exclusion, facilitated by the structural relationship between n and k captured by the polynomial g(x,y), exemplifies the power of abduction in mathematical reasoning, consistent with Peirce’s emphasis on how notation and structure guide discovery.

We have three cases of primality by algebraic definition.

We will use these three cases to conceptualize prime number generation using algebraic functions with variables n, k, x, and y.

We will demonstrate that within these specific algebraic frameworks, the primality of n is entirely determined by whether its corresponding index k can be generated by a specific formula (xy, 2xy+x+y, or 6xy+x-y) representing composite numbers.

In each case, a number is prime if and only if its index k is not in the set of values generated by the corresponding algebraic formula. These formulas produce only composite numbers for the given structure of n. Therefore, by testing whether k is included in that formula’s output, we can classify n as either composite or prime — without direct factoring.

Case 1 – Fundamental definition of primes

• Our first definition is the basic definition of primality, so it covers all prime numbers greater than or equal to 2.
• n, k, x, y are positive integers ≥ 2.
• If n = 1k but n = xy ; then n is not prime.
• If n = 1k but n ≠ xy ; then n is prime.
• So, if k = xy, then n is not prime for a given n = 1k.
• But, if k ≠ xy, for a given n = 1k then n is prime.

Case 2 – Odd numbers

• Our second definition extends the case to odd numbers, so it covers all prime numbers greater than or equal to 3.
• n is a positive integer greater ≥ 3. k,x,y are all non-zero positive integers ≥1.
• If n = 2k+1 but n = 4xy+2x+2y+1, then n is not prime.
• If n = 2k+1 but n ≠ 4xy+2x+2y+1, then n is prime.
• So, if k = 2xy+x+y, then n is not prime for a given n = 2k+1.
• But, if k ≠ 2xy+x+y for a given n = 2k+1, then n is prime.

Case 3 – 6k±1 numbers

• Our third definition extends the case to numbers ±1 mod 6 (eg. 6k±1 numbers), so it covers all prime numbers greater than or equal to 5.
• n is a positive integer ≥ 5. k,x,y are all NON ZERO integers (may be negative).
• If n = |6k-1| but n = |36xy+6x-6y-1|, then n is not prime.
• If n = |6k-1| but n ≠ |36xy+6x-6y-1|, then n is prime.
• So, if k = 6xy+x-y, then n is not prime for a given n = |6k-1|.
• But, if k ≠ 6xy+x-y for a given n = |6k-1|, then n is prime.

(Explanation for case 3)

First, we demonstrate that for every n=6k-1, there is -n=6k+1 and vice versa.

• So, there is 5 in n=6k-1 for k=1, and there is -5 in n=6k+1 for k=-1.
• So, there is -7 in n=6k-1 for k=-1, and there is 7 in n=6k+1 for k=1.
• The sets are symmetrical, so the sets |{6k-1}|=|{6k+1}| have the same cardinality and absolute value which is reflected around 0.
• It is sufficient to use just the absolute value of 1 set to find all prime numbers in a symmetrical range. So we choose n = |6k-1| to classify all ±1 mod 6 numbers.

Next, we demonstrate there are 4 potential forms of composite emerging from (6x±1)(6y±1). We have:

• (6x-1)(6y+1) = 36xy+6x-6y-1 (always -1 mod 6 and produces numbers like 35)
• (6x+1)(6y-1) = 36xy-6x+6y-1 (always -1 mod 6 and produces the same values as the first equation, like 35, so let’s ignore it)
• (6x-1)(6y-1) = 36xy-6x-6y+1 (always 1 mod 6 and produces numbers like 25)
• (6x+1)(6y+1) = 36xy+6x+6y+1 (always 1 mod 6 and produces numbers like 49)

As we demonstrated before, for every n in 6k-1, there is -n in 6k+1, so this must also apply to the composites.

• (6(-1)-1)(6(1)+1) = -49 = |49|
• (6(1)-1)(6(-1)+(-1)) = -25 = |25|

So, n = |36xy+6x-6y-1| is sufficient to find all composites of 6k±1 by iterating through non-zero values of x and y.

So by reducing the equation and solving if |6k-1|=|36xy+6x-6y-1|, then k = 6xy+x-y , and n cannot be prime.

In theory, you could create all the set of k=±1,±2,±3,±4…

Then, you can see if the sequential k value you created can be expressed as k = 6xy+x-y. If it can, then n = |6k-1| is not prime.

The set of all prime values for k is obtained from {k} \ {6xy+x-y} = {k values of primes form |6k-1| >3}

Fundamental Concepts in Algebraic Set Theoretic Prime Operations

Case 1.) n=1k and n=xy

If an integer “n=1k” >1 cannot also be expressed as the product of two integers “n=xy”, where x and y are also greater than 1, then n is a prime number. This covers all prime numbers, including 2.

Case 2.) n=2k+1 and n=4xy+2y+2x+1

If an odd integer “n=2k+1” cannot also be expressed as the product of two odd numbers “n=(2x+1)(2y+1)=4xy+2y+2x+1”, where x and y are equal to or greater than 1, then n is a prime number. This covers all prime numbers greater than 2. This case eliminates all odd composites and thus identifies odd primes only.

Case 3.) n=|6k-1| and n=|36xy+6x-6y-1|

If an odd number “n=6k±1” cannot also be expressed as the product of two odd numbers of the form n=6k±1, “n=|(6x-1)(6y+1)|=|36xy+6x-6y-1|”, where x and y are positive or negative integers equal to or greater than |1|, then n is a prime number. This covers all prime numbers greater than 3.

The Case 3 approach works because for every z in 6k-1, there is a -z in 6k+1, and vice versa.

Composites in 6k±1 forms must be of the forms: (6x-1)(6y-1), (6x-1)(6y+1), and (6x+1)(6y+1). This is explicitly for a positive range of 0<q. However, taking in the fact that for every z in 6k-1 (e.g. …-7,-1,5,11,17..), there is a -z in 6k+1 (e.g. …-17,-11,-5,1,7…), and vice versa, we can work in an expanded range of -q<0<q with either form 6k+1 or 6k-1 and find all composites.

Since in range 0<q, all the composites in 6k-1 must be of the form n=(6x-1)(6y+1)=36xy+6x-6y-1 due to residue classes mod 6 (and the other forms must be within 6k+1), we know that all of the composites in 6k+1 ((6x-1)(6y-1) and (6x+1)(6y+1)) must have a negative twin of the form (6x-1)(6y+1) in 6k-1 in the negative range.

For example; 25 appears in (6x-1)(6y-1) for x=1,y=1. However, -25 appears in (6x-1)(6y+1) for x=1,y=-1; and 25=|-25|. Similarly, 49 appears in (6x+1)(6y+1) for x=1,y=1. However, -49 appears in (6x-1)(6y+1) for x=-1,y=1; and 49=|-49|.

Thereby, inferring the absolute value of any number in sequence 6k+1 or 6k-1 in the negative range will give the corresponding value from the other sequence in the positive range.

When we consider the absolute values of negative range of 6k+1 or 6k-1 with the corresponding positive values from 0<q, then we can find all the primes in the form 6k+1 and 6k-1 combined by just considering one of the forms and absolute value relationships inferred from a symmetrical number range.

Generalized Theorem

A positive integer n is prime if and only if it satisfies one of the following conditions:

Case 1 (Fundamental Definition of Primes): n = 1·k for some positive integer k, and n cannot be expressed as x·y for any non-negative integers x, y > 1.

Case 2 (Odd Primes): n = 2k+1 for some non-negative integer k, and n cannot be expressed as (2x+1)(2y+1) = 4xy+2x+2y+1 for any non-negative integers x, y.

Case 3 (Primes of form ): n = |6k-1| for some integer k, and n cannot be expressed as |(6x-1)(6y+1)| = |36xy+6x-6y-1| for any non-zero integers x, y.

This theorem provides a hierarchical approach to characterizing prime numbers:

• The first case is the fundamental definition of primality that applies to all primes.
• The second case restricts to odd numbers (plus 2), narrowing the search space by eliminating even composites.
• The third case further restricts to numbers congruent to ±1 (mod 6), eliminating multiples of 2 and 3.
• The elegance of Case 3 lies in its use of absolute values and symmetry between 6k-1 and 6k+1 sequences, allowing us to capture all composite numbers in both sequences using a single formula. This provides a more efficient characterization of primes greater than 3 compared to the basic definitions.

Each successive case builds upon modular arithmetic properties to progressively refine an understanding of prime numbers and how efficiency of primality testing can be enhanced through manipulation of modular arithmetic principles.

Review of Set-Based Prime Identification Theory

This set-based method for prime identification offers an alternative conceptual framework to traditional sieving methods.

Core Theory: The set method works by defining two explicitly generated sets and then excluding Set A from Set B:

In case 3, Set A: Contains all numbers of the form |6k-1| for integers k
In case 3, Set B: Contains all composite numbers expressible as |36xy+6x-6y-1| (products of |6x-1| and |6y+1|)

For case 3, the set of primes greater than 3 is then defined as the set difference: P = A \ B , when k, x, and y are all non-zero integers.

P={∣6k−1∣∣k∈Z∖{0}}∖{∣36xy+6x−6y−1∣∣x,y∈Z∖{0}}

If n=|6k-1| and also n=∣36xy+6x−6y−1∣; then n is a composite number.

If n=|6k-1| and also n≠∣36xy+6x−6y−1∣; then n is a prime number.

Generalization to Exclusion Based on k Value

We can reduce all the cases to an exclusion set based on k value.

For case 1, if k = xy ; then 1k=n is not prime. This is already simplified by the inherent definition of prime numbers.

For case 2, if k = 2xy+x+y then n = 2k+1 is not prime.

Obtained by reducing: 2k+1 = 4xy+2x+2y+1

Subtract 1 from both sides: 2k = 4xy+2x+2y

Divide by 2: k = 2xy+x+y

Therefore, for case 2, if k = 2xy+x+y then n = 2k+1 is not prime.

For case 3, if k = 6xy+x-y, then n=|6k-1| is not prime.

Reduce the equations to solve for k. : |6k-1|=|36xy+6x-6y-1|

Cancel absolute value : 6k-1=36xy+6x-6y-1

Add 1 to both sides : 6k=36xy+6x-6y

Divide both sides by 6 : k=6xy+x-y

Therefore, mathematically if k=6xy+x-y then n=|6k-1| is not a prime number; and if there is no solution so that k≠6xy+x-y for non-zero integers x and y, then |6k-1| must be a prime number.

Case 3 Example:

• k = 1: n = |6(1) – 1| = 5 (Prime). Can 1 = 6xy + x – y?
  • Try x=1, y=1: 6+1-1 = 6 ≠ 1
  • Try x=1, y=-1: -6+1-(-1) = -4 ≠ 1
  • Try x=-1, y=1: -6-1-1 = -8 ≠ 1
  • Try x=-1, y=-1: 6-1-(-1) = 6 ≠ 1
  • k=1 cannot be expressed in this form. Consistent with n=5 being prime.
• k = -1: n = |6(-1) – 1| = |-7| = 7 (Prime). Can -1 = 6xy + x – y?
  • From above attempts, no solution. Consistent with n=7 being prime.
• k = 2: n = |6(2) – 1| = 11 (Prime). Can 2 = 6xy + x – y? no.
• k = -2: n = |6(-2) – 1| = |-13| = 13 (Prime). Can -2 = 6xy + x – y? no.
• k = 3: n = |6(3) – 1| = 17 (Prime). Can 3 = 6xy + x – y? no.
• k = -3: n = |6(-3) – 1| = |-19| = 19 (Prime). Can -3 = 6xy + x – y? no.
• k = 4: n = |6(4) – 1| = 23 (Prime). Can 4 = 6xy + x – y? no.
• k = -4: n = |6(-4) – 1| = |-25| = 25 (Composite: 5×5). Can -4 = 6xy + x – y?
  • Try x=1, y=-1: 6(1)(-1) + 1 – (-1) = -6 + 1 + 1 = -4. Yes! Solution: x=1, y=-1.
  • Since k=-4 can be expressed in the form 6xy + x – y, n=25 must be composite, which it is.
• k = 5: n = |6(5) – 1| = 29 (Prime). Can 5 = 6xy + x – y? no.
• k = -5: n = |6(-5) – 1| = |-31| = 31 (Prime). Can -5 = 6xy + x – y? no.
• k = 6: n = |6(6) – 1| = 35 (Composite: 5×7). Can 6 = 6xy + x – y?
  • Try x=1, y=1: 6(1)(1) + 1 – 1 = 6. Yes! Solution: x=1, y=1.
  • Since k=6 can be expressed in the form 6xy + x – y, n=35 must be composite, which it is.
• k = -6: n = |6(-6) – 1| = |-37| = 37 (Prime). Can -6 = 6xy + x – y? no.
• k = -8: n = |6(-8) – 1| = |-49| = 49 (Composite: 7×7). Can -8 = 6xy + x – y?
  • Try x=-1, y=1: 6(-1)(1) + (-1) – 1 = -6 – 1 – 1 = -8. Yes! Solution: x=-1, y=1.
  • Since k=-8 can be expressed in the form 6xy + x – y, n=49 must be composite, which it is.

…

This observation provides a potentially more efficient method for constructing an exclusion set for |6k-1| focused on values of k rather than composites of |(6x-1)(6y+1)|, yet leveraging the same properties.

Theorem of Prime-producing k Values in |6k-1|:

Let K_prime = { k | k ∈ Z \ {0} } \ { 6xy + x – y | x ∈ Z \ {0}, y ∈ Z \ {0} }.

Equivalently, K_prime is the set of integers k such that |6k – 1| is a prime number greater than 3.

Then, for all k ∈ K_prime, the number n = |6k – 1| is a prime number greater than 3.

Process: On-Demand Prime Generation

Series 1: Generating |6k-1| (or |6k+1|) Numbers:

Start generating numbers of the form |6k-1| (or |6k+1|) incrementally.

This series can continue indefinitely, as you’re not bound by a terminal limit.

You can stop this generation at any point, effectively defining your “terminal series 1 number.”

Series 2: Generating Composites |36xy + 6x – 6y – 1|:

Simultaneously, generate composite numbers using the formula |36xy + 6x – 6y – 1|.

Crucial Limiting Factor: To ensure you’ve captured all composites, you need to generate composites up to a limit that guarantees you’ve covered all possible factors.

Determining the Limit:

The smallest prime factor you’ll encounter in the |6k-1| form is 5.

The largest factor you need to consider is the square root of your “terminal series 1 number.”

Therefore, you need to generate composites using the formulas where:

• x and y vary such that (6x-1) and (6y+1) are factors within the range of 5 to the square root of your terminal series 1 number.
• Once all combinations of x and y have been used such that the factors that created them are less than or equal to the square root of the terminal series 1 number, then all composites have been created that are less than the terminal series 1 number.

Set Subtraction (P = Series 1 – Series 2):

• After stopping Series 1 and generating Series 2 up to the necessary limit, perform a set subtraction.
• The resulting set P will contain all prime numbers of the form |6k-1| that are less than your “terminal series 1 number.”

Visualization

A multiplication table is a good way to visualize how the sieve-like method works and how it can be used to check all possible ranges without missing any composites.

In any case, the table needs a number of rows equal to the number of integers of the considered number form which are less than the square root of the target number.

So, for Case 1, if you are considering how many primes are less than 100, you need 10 rows, because 10 is the square root of 100 and we are working in increments of 1. You would need 50 columns, because 100 divided by 2 is 50, and 2 is the smallest prime number considered in Case 1.

For Case 3, if you are considering how many primes are less than 100, you need 3 rows, because there are 3 numbers of the form |6k-1| less than 10 (the square root of 100). You would need 7 columns, because there are 7 numbers of the form |6k-1| less than 100 divided by 5; since 5 is the smallest prime factor produced by |6k-1| numbers.

In either case, if a number less than the target number (eg. 100) appears in Row 1 or Column 1 of the table, and does not appear in the body of the table, it is prime.

Parallels with Traditional Sieves

Both approaches share certain fundamental characteristics:

• Both ultimately identify primes by eliminating composites
• Both rely on the fact that all composites have prime factors
• Both exploit modular arithmetic properties (especially that primes > 3 are of form 6k±1)

Key Differences with Traditional Sieving Approaches

The set method differs from traditional sieves in several important ways:

• Generation vs. Elimination: Traditional sieves start with all numbers and iteratively remove multiples. The set method directly generates two sets using explicit formulas and compares them.
• Mathematical Formulation: Sieves use divisibility as the core operation. The set method uses closed-form expressions and set operations.
• Conceptual Approach: Sieves work “from the bottom up” by eliminating multiples of each prime found. The set method works by explicitly characterizing all composites of a certain form.
• Terminal Limit: The terminal “N” value needs to be input in a sieve before it is run. The set method can be arbitrarily run indefinitely without foreknowledge of the terminal limit.
• Implementation Focus: Sieves typically focus on marking/elimination algorithms
  The set method focuses on generation of potentially very large sets.

Conclusion

This set-based approach offers a perspective on prime identification leveraging algebraic formulations rather than divisibility tests. While traditional sieves may be more familiar, this method provides both theoretical insights and potential advantages, especially when considering specific subsets of primes.

The key insight is that primality can be characterized as membership in a well-defined set that is directly constructible through algebraic expressions, rather than as the result of an elimination process.

This method qualifies as a prime number generator in the sense that:

• It produces exactly the set of all prime numbers (greater than 3, with simple extensions to include 2 and 3).
• It uses a deterministic method that will correctly identify any prime within its range.
• It can theoretically continually generate primes up to any arbitrary limit (given sufficient computational resources).

However, it differs from some other generators in that it’s not optimized for sequentially producing primes one at a time. Instead, it generates an entire set of primes within an arbitrarily terminating range by set-theoretic operations.

For any integer p > 3, p is prime if and only if:

1. p ∈ |{6k ± 1 | n ∈ ℤ}|
2. p ≠ |x * y| where x, y ∈ {6k ± 1 | n ∈ ℤ} with the same sign

Key features:

1. Unified Representation: All primes >3 are expressed in a single set using the absolute value function, unifying the traditional 6x-1 and 6y+1 forms.
2. Symmetry: The theorem captures the symmetrical distribution of primes around multiples of 6, extending to both positive and negative integers.
3. Concise Primality Test: The second condition provides an elegant criterion for primality within the defined set.
4. Completeness: The theorem both represents all primes >3 and provides a sufficient condition for primality.

Implications:

This theorem presents a semiotically elegant representation of prime numbers, emphasizing their inherent structure and symmetry.

Claude was principally used for this refinement agreed upon by other native models tested. I recommend Claude on this day. You should try. A future model may suck, but this one is great!

https://spinscore.io/?url=https%3A%2F%2Fn01r.com%2Fsemiotic-prime-theorem-2-0%2F (Note: the A+ Spinscore is based on the theorem alone, not the ruminations on Claude)