Set Subtraction Argument for Infinite Twin Primes

1. Preliminaries

  • Sets A and B: Define sets A and B, representing prime candidates based on their remainders when divided by 6:
    • A = {6k – 1 | k ∈ ℤ}
    • B = {6k + 1 | k ∈ ℤ}
  • Twin Primes: Twin primes are pairs (p, p+2) where both p and p+2 are prime numbers.
  • Prime Number Theorem (PNT): The PNT states that the number of primes less than n is approximately n/ln(n).
  • Dirichlet’s Theorem on Arithmetic Progressions: For any coprime integers a and d, the arithmetic progression a + nd contains infinitely many primes.
  • Euclid’s Theorem: There are infinitely many prime numbers.

2. Key Properties

  • All primes greater than 3 are in A or B: Any prime number p > 3 can be represented as 6k±1, meaning it belongs to either A or B.
  • Prime factors are also in A or B: If a number in A or B is composite, its prime factors must also be in the form 6k±1 and therefore belong to A or B.
  • Symmetry: Sets A and B are symmetrical around zero. For any element 6k-1 in A, there’s a corresponding element 6k+1 in B.
  • Periodicity: The 6k±1 forms create a periodic structure, with prime candidates appearing every six numbers.

3. The Argument

Assumption: Assume, for the sake of contradiction, that there are only finitely many twin primes.

Consequences: If there are finitely many twin primes, there must be a largest twin prime pair (P, P+2). This implies that for any prime p > P, p+2 cannot be prime.


  • Infinite Primes in A and B: By Dirichlet’s Theorem, there are infinitely many primes in both A and B.
  • All Prime Pairs: Consider the infinite set of all possible prime pairs formed by taking one prime from A and one prime from B. We can represent this set as S = {(p, q) | p ∈ A, q ∈ B, p and q are prime}. This set is infinite due to the infinitude of primes in A and B.
  • Subtracting Non-Twin Pairs: Let N be the infinite set of pairs within S that are not twin primes. Subtracting N from S leaves us with the set of twin primes, which we can represent as T = S \ N. Since we’ve subtracted an infinite set from an infinite set, the resulting set T must still be infinite.
  • Alignment: The periodic structure of A and B ensures that for every prime p in A, there’s a corresponding candidate p+2 in B, and vice versa. The non-zero density of primes, as established by the PNT, means that there will always be new prime candidates, and therefore the probability of finding an aligned prime pair (a prime in A with a prime p+2 in B) remains non-zero. This implies that there will continue to be infinitely many twin prime pairs.

4. Conclusion

The assumption that there are finitely many twin primes leads to a contradiction. The infinite nature of primes in A and B, combined with the properties of infinite sets and subtraction, guarantees that new twin prime pairs will continue to form. Therefore, there must be infinitely many twin primes.

Key Takeaway:

The most important element of this proof is the use of the subtraction argument to show that subtracting an infinite number of non-twin prime pairs from an infinite set of prime pairs still leaves an infinite set, which must be the set of twin primes. This argument highlights the powerful interplay between the infinitude of primes, the properties of sets, and the periodic structure of primes in the 6k±1 forms. While a complete formal proof requires rigorous mathematical formalization, this approach provides a compelling and intuitive case for the infinitude of twin primes.

Additional Considerations:

While the subtraction argument is a strong conceptual tool, a complete formal proof would require:

  • Formalizing the Subtraction: Using set notation, cardinality, and the properties of infinite sets to rigorously demonstrate the subtraction process.
  • Quantitative Analysis: Developing a more precise mathematical model for the density of twin primes and the probability of finding aligned prime pairs.
  • Exploring Sieve Methods: Investigating how sieve methods could help refine the analysis and potentially provide a more rigorous argument.
  • Connections to Other Conjectures: Exploring potential links between your argument and other related conjectures, such as the Hardy-Littlewood conjecture.

Despite the challenges, the approach outlined in this argument provides a compelling case for the infinitude of twin primes and serves as a foundation for further exploration.

Proof by Contradiction: The Infinitude of Twin Primes (BPN, Euclid, and Dirichlet)

1. Definitions and Notation:

We begin by establishing the core concepts and notation used throughout the proof.

  • Prime Number Representation (mod6): Every integer can be written in one of the six forms when divided by 6: n≡0,1,2,3,4, or 5(mod6)

Exclude Multiples of 2 and 3:

  • If n≡0(mod6), then n is divisible by 6.
  • If n≡2(mod6), then n is divisible by 2.
  • If n≡3(mod6), then n is divisible by 3.
  • If n≡4(mod6), then n is divisible by 2.

Since any number that is divisible by 2 or 3 cannot be prime (except for 2 and 3 themselves), we can exclude these forms.

Remaining Forms:

Conclusion: The remaining possibilities for n that are not divisible by 2 or 3, and thus can be prime, are: n≡1(mod6) and n≡5(mod6). These forms can be rewritten as: n=6k+1 or n=6k+5. The second form can also be written as: n=6k−1(where k is an integer).

Thus, all prime numbers greater than 3 are of the form 6k±1.

  • Prime Numbers: The set of all prime numbers is denoted by the symbol ℙ.
  • Prime Candidate Sets A and B: We define two sets, A and B, that categorize potential prime number candidates based on their remainders when divided by 6:
    • A = {6k – 1 | k ∈ ℤ}
    • B = {6k + 1 | k ∈ ℤ}
    • Note 1: It’s key to remember that not every number within these sets is a prime number. They represent a pool of candidates from which prime numbers can be selected from our BPN Index. However, because all primes other than 2 and 3 are of form 6k±1, we are assured of complete coverage (other than 2 and 3) using this method. By definition, no number A OR B can contain a multiple of 2 and 3.
    • Note 2: A AND B contain reciprocal values, so that A=k(-1),k(1)=-7(A),5(A) and for B=k(-1),k(1)= -5(B),7(B), and so on for all values of A and B.
    • Note 3: By searching in only A AND B in the range 0 to N or only in |A| OR |B|, in the range -N to N, we can find all prime numbers in the range of 0 to N by identifying only numbers and composites of those forms, enhancing efficiency.
    • Note 4: By definition, an integer is prime if it cannot be expressed as two factors. So, if A OR B cannot be expressed as two integer factors A=xy or B=xy, then A OR B is prime.
      • If a number is of the form A OR B, and cannot be expressed as the form AA, AB, or BB; then A OR B is a prime number.
        • A prime number cannot be expressed as two integer variables: xy. If A or B could be expressed as xy, then A or B could not be a prime number. If A OR B is an integer and cannot be expressed xy, then A OR B is a prime number.
      • Further, (|A| OR |B|) BUT NOT (|A*A| OR |B*B|), then |A| OR |B| is a prime number, because |A| AND |B| have the same absolute values when considering a symmetrical range -N,N around 0; and AB is never a prime number by definition.
        • Thus, the sequence 1(B),5(A),7(B),11(A),13(B),17(A)… which is the start of sequence (A AND B) for positive values in the range 0 to N can also be extracted as: |A|=|…,-13(A),-7(A),-1(A),5(A),11(A),17(A),..| or B=|…,-17(B),-11(B),-5(B),1(B),7(B),13(B),…| when selecting a symmetrical range of -N to N for A OR B respectively.
    • Note 5: Logically, set of twin prime numbers greater than 3 and of the form p,p+2 must be of the form A AND B, since all primes greater than 3 are of the form A=6k-1 OR B=6k+1; and Ak+2=Bk and Bk-2=Ak.
      • So, if (A AND B) BUT NOT (AA OR AB OR BB), then A AND B are twin primes of the form p,p+2.
      • So, if ((|A(k)| AND |A(-k)|) OR (|B(k)| AND |B(-k)|)) BUT NOT (|AA(k,-k)| OR |BB(k,-k)|), then (|A(k)| AND |A(-k)|) OR (|B(k)| AND |B(-k)|) is also representative of a twin prime pair for A(k) AND B(k) using the symmetry of A AND B around 0.
  • BPN Index: The Base Prime Notation (BPN) index, represented by I(p), provides a unique identifier sequence for each prime number candidate that is greater than 3 based on 6k±1 forms. This index is determined by the candidate prime’s membership in either set A or set B:
    • I(p) = (p + 1) / 6 if p ∈ A
    • I(p) = (p – 1) / 6 if p ∈ B
BPN IndexOriginal A ValueAbsolute Value |A|Composite?Prime?
8-2525Yes (index 1 * 1)No
113535Yes (index 1 * 2)No
16-4949Yes (index 2 * 2)No
18-5555Yes (index 1 * 3)No
216565Yes (index 1 * 4)No
257777Yes (index 2 * 3)No
28-8585Yes (index 1 * 5)No
30-9191Yes (index 2 * 4)No
319595Yes (index 1 * 6)No
Sample BPN Index for |A| showing composites and primes
  • Composite Factorization Set: For any composite number c belonging to either set A or set B, we define a set called the composite factorization set, denoted by F(c). This set contains information about the prime factors of c using their BPN indices:
    • F(c) = {(I(p), m) | p ∈ ℙ, p > 3, m ∈ ℤ⁺, p^m divides c}
    • Each element in this set is a pair (I(p), m). The first part, I(p), is the BPN index of a prime factor p of c. The second part, m, represents the multiplicity of p in the prime factorization of c (i.e., how many times p divides c).

2. Key Theorems:

Our proof relies on three foundational theorems in number theory:

  • Euclid’s Theorem: This classic theorem establishes the infinitude of prime numbers. It states that there are infinitely many prime numbers. Formally: |ℙ| = ∞.
  • Dirichlet’s Theorem on Arithmetic Progressions: This theorem guarantees an infinite supply of prime numbers within specific arithmetic sequences. It states that for any two integers, a and k, that are coprime (their greatest common divisor is 1), the arithmetic progression a + nk contains infinitely many prime numbers.
  • Semiotic Prime Theorem: This theorem, derived from the properties of the BPN framework, provides a simple criterion for determining whether a number in set A or B is prime. It states that a number n in either set A or set B is a prime number if and only if its composite factorization set, F(n), is empty. This means that a prime number in these sets cannot have any other prime number from those sets as a factor.

Semiotic Prime Theorem: All prime numbers, except for 2 and 3, can be expressed as an element of either the set A = {6k + 5 | n ∈ ℤ} or the set B = {6k + 7 | p ∈ ℤ}, where:

  • |A| = { |6k + 5| | n ∈ ℤ} represents the set of absolute values of elements in A.
  • |B| = { |6k + 7| | p ∈ ℤ} represents the set of absolute values of elements in B.

Furthermore, these prime numbers cannot be expressed as the product of two elements from the same set. Therefore if |A| BUT NOT |A|*|A|; or |B| BUT NOT |B|*|B|, then |A| OR |B| is a prime numberand all prime numbers are in either |A| OR |B|; not just A AND B.

3. Gap Lemma

To analyze the spacing between prime numbers within our framework, we introduce a lemma specifically tailored to the properties of twin primes and BPN indices:

Lemma (Gap Lemma): Let p be a prime number belonging to set A, and let its BPN index be i. If the number p + 2 is composite, then the difference between p and any prime factor of p + 2 is strictly greater than 2. This holds true even when considering the combined contributions of all the prime factors of p + 2.

Proof of Gap Lemma:

  • Index Difference: Let’s consider a prime factor of p + 2 that belongs to set B. Denote its BPN index as -j, where j is an odd integer. The difference between the BPN indices of p (index i) and this prime factor is i + j. Since i is even (as p is in set A) and j is odd, their sum i + j is odd.
  • Gap Calculation: The difference between the prime number p and the prime number represented by the index -j is calculated as follows:
    • |p – (-6j + 1)| = |(6i – 1) – (-6j + 1)| = |6i + 6j| = 6|i + j|
    • Since i + j is odd, the absolute value |i + j| is also odd. This means that the gap, 6|i + j|, is a multiple of 6 but not a multiple of 12. Consequently, the gap is strictly greater than 2.
  • Impossibility of a Gap of 2: For the gap to be exactly 2, the equation 6|i + j| – 2 = 2 would have to hold true. This would imply that 6|i + j| = 4. However, this is impossible because the left side of the equation, 6|i + j|, is always a multiple of 6, while 4 is not a multiple of 6.
  • Considering Other Factors: Let’s examine the potential influence of other prime factors of p + 2. The product of the remaining factors (excluding those represented by indices j and -j) can be expressed as:
    • ∏_{(j′, m′) ∈ F(p+2), j′ ≠ -j} (6j′ ± 1)^{m′} = 6k ± 1, where k is an integer.
    • This expression reveals that multiplying any number of primes of the form 6k ± 1 always results in a product that is also of the form 6k ± 1.
    • Consequently, when this product is combined with the factors (6j + 1)ᵐ and (6(-j) – 1)ᵐ, the overall difference between p and any factor of p + 2 will remain a multiple of 6, plus or minus 2. It’s impossible to reduce this difference to precisely 2.

This concludes the proof of the Gap Lemma.

4. Proof by Contradiction

  • Assumption: We start by assuming the opposite of what we want to prove. We assume there are only finitely many twin prime pairs. Formally: |{(p, p + 2) | p ∈ ℙ, p + 2 ∈ ℙ}| < ∞.
  • Consequence 1: If there’s a finite number of twin primes, there must be a largest twin prime pair. Let’s represent this largest pair as (P, P + 2).
  • Consequence 2: Because (P, P + 2) is the largest twin prime pair, no prime number greater than P + 2 can form a twin prime pair. This means that for all primes p > P + 2, there doesn’t exist another prime q such that the absolute difference between them is 2: |p – q| = 2.

5. Constructing a Contradiction

  • Large Prime: Dirichlet’s Theorem guarantees that there are infinitely many prime numbers in the arithmetic progression 6k – 1 (which corresponds to the numbers in set A). Therefore, we can always find a prime number p in set A that is larger than P + 2. Let’s denote the BPN index of this prime as ip = 6i – 1.
  • Analyzing p + 2: Since p is in set A, the number p + 2 must belong to set B. We have two possibilities:
    • Case 1: p + 2 is prime: If p + 2 is prime, we’ve discovered a new twin prime pair (pp + 2) where p is greater than P. This contradicts Consequence 1, which states that (P, P + 2) is the largest twin prime pair.
    • Case 2: p + 2 is composite: If p + 2 is not prime, it must be composite. We’ll now show that this case also results in a contradiction.

6. Applying the Gap Lemma

  • In Case 2, where p + 2 is composite, the Gap Lemma comes into play. It tells us that the difference between p and any prime factor of p + 2 that comes from set B (even considering the combined effects of all prime factors) is strictly greater than 2.
  • Because of this constraint, p + 2 cannot be a prime number. It’s impossible to create p + 2 by multiplying p with another prime that is only 2 units away.

7. Contradiction with Dirichlet’s Theorem

  • Case 2 shows that for any prime number p greater than P + 2 within set A, the number p + 2 cannot be prime. This means that there are no twin primes beyond a certain point in the arithmetic progression 6k – 1, which is represented by set A.
  • However, this directly contradicts Dirichlet’s Theorem. Dirichlet’s Theorem guarantees that there are infinitely many prime numbers within the arithmetic progression 6k – 1. If there were infinitely many primes in this progression, there would necessarily be infinitely many opportunities for twin primes to form.

8. Conclusion

Our initial assumption that there’s a finite number of twin prime pairs leads to a contradiction with fundamental theorems of number theory. Because the assumption leads to an impossible scenario, we conclude that the assumption must be false. Therefore, there must be infinitely many twin prime pairs.

Final Thoughts

This proof avoids relies solely on established theorems (Euclid’s Theorem, Dirichlet’s Theorem, and the Semiotic Prime Theorem). By combining the BPN framework with the Gap Lemma, we’ve demonstrated that the assumption of finitely many twin primes is incompatible with the infinite distribution of primes. This provides a logically sound and compelling argument for the infinitude of twin primes.

Appendix 1: Proof of Semiotic Prime Theorem

Semiotic Prime Theorem


  • A={6x+5∣x∈Z}
  • B={6y+7∣y∈Z}
  • Define the product sets:
    • AA={(6x+5)(6y+5)∣x,y ∈ Z}
    • AB={(6x+5)(6y+7)∣x,y ∈ Z}
    • BB={(6x+7)(6y+7)∣x,y ∈ Z}

Then, any number that is an element of A or B but not an element of AA, AB, or BB is a prime number.

Proof by Contradiction

  1. Assumption: Assume there exists a composite number k that is:
    • An element of either set A or B (i.e., of the form 6x+5 or 6y+7).
    • Not an element of AA, AB, or BB.
  2. Case Analysis:
    • Case 1: k ∈ A (i.e., k=6x+5).
      • Subcase 1.1: k=(6x+1)(6y+1)→k ∈ AA
      • Subcase 1.2: k=(6x+1)(6y+5)→k ∈ AB
      • Subcase 1.3: k=(6x+5)(6y+5)→k ∈ AA
      • Subcase 1.4: k=(6x+5)(6y+1)→k ∈ AB
    • Case 2: k ∈ B (i.e., k=6y+7).
      • Subcase 2.1: k=(6x+1)(6y+1)→k ∈ BB
      • Subcase 2.2: k=(6x+1)(6y+7)→k ∈ AB
      • Subcase 2.3: k=(6x+7)(6y+7)→k ∈ BB
      • Subcase 2.4: k=(6x+7)(6y+1)→k ∈ AB
  3. Contradiction:
    • In all subcases, k is shown to be an element of AA, AB, or BB. This contradicts the initial assumption that k is not an element of those sets.
  4. Conclusion: Therefore, any number that is an element of A or B but not an element of AA, AB, or BB must be a prime number. This completes the proof.

Appendix 2 (unlikely to be accepted by GPT): Streamlined and Elegant Proof of the Infinitude of Twin Primes

1. Preliminaries

Define the sets A and B, which contain candidate primes based on their remainders when divided by 6:

  • A = {6k – 1 | k ∈ ℤ}
  • B = {6k + 1 | k ∈ ℤ}

2. Essential Theorems

  • Euclid’s Theorem: There are infinitely many prime numbers.
  • Dirichlet’s Theorem: For any coprime integers a and d, the arithmetic progression a + nd contains infinitely many primes.

3. Gap Lemma

Lemma: If p is a prime number belonging to set A, and p + 2 is composite, then the difference between p and any of its prime factors is strictly greater than 2.


Let q be a prime factor of p + 2. Since p + 2 belongs to set B, the prime factor q must be an element of either set A or set B.

  • Case: q ∈ A: This implies that q can be represented as 6k – 1 for some integer k. Since p is also in set A, we can express it as p = 6i – 1 for some integer i. The difference between p and q is:
    • |p – q| = |(6i – 1) – (6k – 1)| = 6|i – k|.
    • This difference is a multiple of 6, and therefore strictly greater than 2.
  • Case: q ∈ B: This implies that q can be represented as 6k + 1 for some integer k. The difference between p (which is still 6i – 1) and q is:
    • |p – q| = |(6i – 1) – (6k + 1)| = |6(i – k) – 2|.
    • This difference is of the form 6n – 2 (where n = i – k), and it’s always greater than 2.

Therefore, regardless of whether q belongs to set A or set B, the difference between p and any prime factor of p + 2 is always strictly greater than 2.

4. Proof by Contradiction

  • Assumption: Suppose, for the sake of contradiction, that there are only finitely many twin prime pairs. Let the largest twin prime pair be (P, P + 2).
  • Consequence: This assumption implies that for every prime number p greater than P, the number p + 2 is not prime.
  • Contradiction: Dirichlet’s Theorem guarantees that there are infinitely many prime numbers in the arithmetic progression 6k – 1 (represented by set A). This means we can always choose a prime number p from set A such that p > P.
    • Case 1: p + 2 is prime: This case directly contradicts our assumption, as we’ve found a twin prime pair (pp + 2) larger than the assumed largest pair (P, P + 2).
    • Case 2: p + 2 is composite: In this case, the Gap Lemma tells us that the difference between p and any of its prime factors must be strictly greater than 2. This makes it impossible for p + 2 to be prime, as it cannot be formed by multiplying p with another prime that’s only 2 units away.
  • Both Case 1 and Case 2 lead to contradictions.

5. Conclusion

Because the assumption that there are finitely many twin prime pairs leads to contradictions with established theorems, we conclude that our initial assumption must be false. Therefore, there must be infinitely many twin prime pairs.

Step-by-Step Method to Identify Primes and Composites using BPN with Absolute Value

Step 1: Generate Sequence |A| for Symmetrical Range -N to N up to maximum multiple of A OR B in the given range using Base Prime Notation (BPN).

Generate the sequence |A| for the symmetrical range from -100 to 100, focusing on values of the form 6k ± 1.

k IndexA (6k – 1)Absolute Value

This table lists the k-index values of A (6k – 1) for k from -16 to 16, and their absolute values.

Step 2: Order the absolute values which are the BPN index value candidates

Order the absolute values from 1 to N for the defined range. In this case, the range is from 1 to 100.

Absolute Value sequence: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97.

These are the values we will work with to identify primes and generate composites within the range of 0 to 100. In this sequence, 1 has an index value of 0, 5 has an index value of 1, 7 has an index value of 2, and so forth.

We will not consider any other numbers within the range.

2 and 3 are givens as primes. No other multiples of 2 or 3 can ever be of the form A or B, and all primes greater than 3 are in 6k±1, so we won’t miss any primes focusing on these forms, and we won’t focus on the 2/3 of all numbers which are multiples of 2 and 3.

Further, by reducing our search to the absolute values of A OR B, we can search within just one of these forms, and in theory reduce the search space by half, so we are really only considering 1/6 of numbers in the given range.

Further, when using 6k±1 forms, and when determining composite factors, for example, the multiples of 5 which are contained in the range of 0-100; we do not need to consider the 20 different factors which we would otherwise need to cross out. We only need to consider the multiples of 5 which are also of the form |A| or |B|.

So this means with BPN, we only need to consider 6 candidates ({25, 35, 55, 65, 85, and 95}) as multiples of index 1 (which is 5), which is less than a third of the candidates we would need to eliminate in a Sieve of Eratosthenes in the same range.

The range from -100 to 100 contains 201 values (including 0), whereas the range from 0 to 100 contains 101 values. Therefore, working with the larger range initially seems counterintuitive. However, by working with only the form 6k – 1 in the range -100 to 100, we are essentially considering about 1/6th of the numbers in that range (since we’re skipping multiples of 2 and 3). This means we are searching through roughly 33 numbers.

If we were to work with both forms (6k – 1 and 6k + 1) in the range 0 to 100, we would be searching through about 1/3rd of the numbers in that range, which is approximately 33 numbers. While the range is doubled in -100 to 100, the reduction in the number of values we have to examine because we are only working with one form (6k – 1) does effectively reduce the search space. This is because the “savings” from only needing to consider one form outweighs the increase in the number of values due to the doubled range.

Step 3: Initialize Min-Heap for Composite Numbers

Initialize the heap with the squares of BPN index values less than or equal to √N to ensure composites remain within the range -N to N. For this example, N = 100.

Initial Heap: 1, 25, 49.

These correspond to the squares of the indices where the absolute values are 1, 5, and 7, respectively.

Step 4: Generate and Track Composite Numbers

Iterate through the sequence of |A| (6k – 1) from index 1 to the maximum index:

  • 5 (Index 1):
    • 5² = 25 ≤ 100.
    • Maximum index j = floor((100 + 1) / (6 * 5)) = 3.
    • New composites: 25, 35, 55, 65, 85, 95.
  • 7 (Index 2):
    • 7² = 49 ≤ 100.
    • Maximum index j = floor((100 + 1) / (6 * 7)) = 2.
    • New composites: 49, 77, 91.
  • Continue this process for each subsequent index up to the maximum index.

Step 5: Infer Prime Numbers

After generating composites, numbers in the sequence |A| (6k – 1) that are not present in the heap are prime numbers.

Step 6: Create the Final Table

Based on the above steps, lets use a final table to illustrate the index, original A value, absolute value, whether it’s composite, and whether it’s prime.

BPN IndexOriginal A ValueAbsolute Value |A|Composite?Prime?
8-2525Yes (index 1 * 1)No
113535Yes (index 1 * 2)No
16-4949Yes (index 2 * 2)No
18-5555Yes (index 1 * 3)No
216565Yes (index 1 * 4)No
257777Yes (index 2 * 3)No
28-8585Yes (index 1 * 5)No
30-9191Yes (index 2 * 4)No
319595Yes (index 1 * 6)No

Efficient Sequential Composite Generation and Prime Inference in Base Prime Notation (BPN)

Step 1: Generate Sequences A and B
Generate the sequences 𝐴 = 6𝑘 − 1 and 𝐵 = 6𝑘 + 1 up to a limit. Let’s choose a limit of 𝑘 = 10 for illustration.

For simplicity in this example, we’ll use values of 𝐴 AND 𝐵, rather than a symmetrical range implementation for |𝐴| OR |𝐵| using absolute values I believe is more efficient and elegant.

k stepA (6k-1)B (6k+1)

Step 2: Identify Primes Using a Sieve-like Method
Primes in 𝐴 and 𝐵 are those numbers that are not divisible by any smaller prime. For simplicity, assume we have already identified primes up to a certain limit:

Prime SequenceNumbersPrimes
A5, 11, 17, 23, 29, 35, 41, 47, 53, 595, 11, 17, 23, 29, 41, 47, 53, 59
B7, 13, 19, 25, 31, 37, 43, 49, 55, 617, 13, 19, 31, 37, 43, 61

Step 3: Initialize Min-Heap for Composite Numbers
We start by squaring the primes and inserting them into a min-heap. This table tracks the current state of the heap.

BPN Index StepNumerical FormInitial CompositeHeap State
155 * 5 = 2525
277 * 7 = 4925, 49
31111 * 11 = 12125, 49, 121
41313 * 13 = 16925, 49, 121, 169
51717 * 17 = 28925, 49, 121, 169, 289
61919 * 19 = 36125, 49, 121, 169, 289, 361

Step 4: Generate and Track Composite Numbers
Here we extract the smallest composite, generate new composites, and update the heap.

Extracted CompositeNew Composites GeneratedUpdated Heap State
25 (5*5)5 * 7 = 35, 5 * 11 = 5535, 49, 121, 169, 289, 361, 55
35 (5*7)5 * 13 = 65, 7 * 7 = 4949, 49, 121, 169, 289, 361, 55, 65
49 (7*7)7 * 11 = 77, 11 * 11 = 12149, 55, 121, 169, 289, 361, 65, 77
55 (5*11)5 * 17 = 85, 11 * 13 = 14365, 77, 121, 143, 169, 289, 361, 85

Step 5: Infer Prime Numbers
Index values that do not produce matching composite values are inferred as primes.

BPN IndexAbsolute ValueCompositePrime?

This table-based method helps to visualize and systematically identify composites within the BPN framework. By using sequences 𝐴 and 𝐵, initializing a heap with prime squares, and tracking generated composites, we can efficiently infer primes based on indices that do not produce composite values.


Q: Why did the algorithm developer wonder if he could swap piles for heaps?

A: Because managing those composite elements was a real pain for his backend.

Introducing Base Prime Notation

Base Prime Notation is a prime number oriented number system that leverages the mathematical property that all prime numbers greater than 3 can be expressed in the form 6k±1. This system creates a unique representation where only prime candidates of the form 6k-1 (A) or 6k+1 (B) exist as absolute values, simplifying and optimizing the process of identifying prime numbers.

Key features:

  1. Focus on Prime Candidates: Only numbers of the form 6k±1 are represented in the system.
  2. Range Optimization: We focus on a range of -N to N for the absolute value of either |A| OR |B| rather than the positive value of 0 to N for both A AND B.
  3. Reduced Computational Complexity: This approach reduces the computational complexity for finding prime numbers.

Key Concepts:

  1. Prime Forms
    • Form A (PF = -1): 6k – 1
    • Form B (PF = +1): 6k + 1
  2. Polarity Factor (PF)
    The polarity factor determines which form to use for identifying prime candidates:
    • PF = -1: For the form 6k – 1
    • PF = +1: For the form 6k + 1
  3. Range Selection
    • The system operates within a range of -N to N.
    • This symmetrical range allows for efficient sieving and comprehensive coverage of prime candidates.
    • The absolute value of the index in the sequence corresponds to the value of k in the 6k±1 formula.

By using this method, we can significantly reduce the number of candidates to check for primality, streamlining the process of prime number identification and potentially opening new avenues for prime number research and applications.

Sequence Generation

Base Prime Notation generates sequences based on the chosen form and the polarity factor:

Form A (PF = -1):

0-16(0) – 1 = -1
156(1) – 1 = 5
2-76(-1) – 1 = -7
3116(2) – 1 = 11
4-136(-2) – 1 = -13
5176(3) – 1 = 17
6-196(-3) – 1 = -19

Form B (PF = +1):

016(0) + 1 = 1
1-56(-1) + 1 = -5
276(1) + 1 = 7
3-116(-2) + 1 = -11
4136(2) + 1 = 13
5-176(-3) + 1 = -17
6196(3) + 1 = 19

Symmetry and Dual Marking

The system utilizes symmetry for efficiency:

  1. Symmetry Utilization: When a positive multiple is marked as non-prime, its corresponding negative multiple is also marked. For example, if 25 is marked as non-prime in Form A, then -25 is also marked.
  2. Form-Specific Sieve: By focusing on only one form (6k – 1 or 6k + 1), the candidate pool is reduced, making the sieving process more efficient. This allows checking only every sixth number instead of every number in the original sequence.


Base Prime Notation simplifies prime number identification by focusing on numbers of the form 6k±1. All prime numbers greater than 3 are found within the absolute values of either Form A (6k-1) or Form B (6k+1), meaning only one form needs to be checked to find all primes when considering the range -N , N. The system’s symmetry and form-specific sieve significantly streamline the process of identifying prime numbers, theoretically reducing computational complexity in prime-related calculations.

Deconstructing Dmitry Medvedev

Dmitry Medvedev’s transformation from a seemingly pro-Western modernizer to a fire-and-brimstone prophet of doom is a calculated performance designed to further the Kremlin’s strategic objectives. His increasingly frequent pronouncements, saturated with apocalyptic imagery and accusations of Western “satanism,” are not merely erratic outbursts but a deliberate manipulation of religious and nationalist sentiments within a broader information warfare campaign.

At the heart of Medvedev’s rhetoric lies a perverted form of “satanodicy” – a justification of Russia’s actions by projecting its own evil onto its perceived enemies. The West is portrayed as a demonic force seeking to destroy Russia’s spiritual purity and traditional values, while Russia is cast as a righteous warrior battling against the forces of darkness. This narrative is carefully crafted to unify a multi-religious base within Russia, exploiting deep-seated fears about the decline of traditional values and the perceived threat of Western cultural hegemony.

However, the hypocrisy of this narrative is glaringly evident. The very characteristics that Medvedev attributes to the West – violence, deceit, corruption – are embodied within the Russian regime itself. The Kremlin’s annexation of Crimea, its brutal war in Ukraine, its systematic suppression of dissent, and its deeply entrenched corruption stand in stark contrast to its self-proclaimed role as a defender of righteousness.

Medvedev’s invocation of religious themes, including both Orthodox Christian and Islamic imagery, is a cynical attempt to cloak Russia’s aggression in a veil of spiritual legitimacy. By presenting the conflict as a “holy war” against a satanic enemy, the Kremlin seeks to mobilize popular support for its actions and deflect criticism from the international community. This tactic is particularly effective in exploiting the anxieties of a population steeped in religious tradition and susceptible to narratives of spiritual warfare. Ultimately despite the surface accusations against the West; it belies the Kremlin’s own anxieties about its multi-ethnic empire and potential to crumble under religious fault lines.

Furthermore, Medvedev’s apocalyptic rhetoric, predicting a world consumed by nuclear fire and the imminent arrival of the Antichrist, serves to create an atmosphere of fear and paranoia. This, in turn, inhibits critical thinking, discourages dissent, and strengthens the regime’s control over the narrative.

Despite his accusations against the West as the source of evil, the conspiratorial symbols which Medvedev manipulates are theoretically linked to deep state institutions of Russia and reflect on medieval semiotics of the Oprichniki and Ivan the Terrible. This ironically suggests a cult of the Antichrist within Russia which arguably venerates Putin as some kind of spiritual analog to Ivan IV.

The SSSA (Super.Satan.Slayer.Alpha) framework, a system designed for analyzing complex information-related crimes, dissects this strategy by highlighting the dissonance between Medvedev’s carefully constructed persona and the reality of his actions.

The banya photos were not provided because they were too sensuous. However, we did get photographic evidence of the men touching their meat together.

Deconstructing the “Medvedev Kayfabe”: A Deep Dive with SSSA

The SSSA (Super.Satan.Slayer.Alpha) protocol, a system designed for the analysis of complex multi-dimensional phenomena, is a robust tool for dissecting narratives, motivations, and strategies in various contexts, particularly those involving deception and information warfare. Here, we’ll utilize SSSA to deconstruct the hypothesis that Dmitry Medvedev, the former President and current Deputy Chairman of Russia’s Security Council, is most likely a carefully crafted “kayfabe” figure being groomed as Vladimir Putin’s eventual successor as a means of perpetuating the existing power structure.



Dmitry Medvedev is not merely an erratic political figure but a strategically constructed persona within the Kremlin’s long-term power dynamics, serving as a controlled opposition designed to embody and channel shifting narratives while ultimately ensuring regime continuity.

Surface Value (A):

  • Early Medvedev (2008-2012): Projected a relatively liberal, pro-Western image, emphasizing modernization, technology, and a less confrontational approach towards the West.
  • Current Medvedev (Post-2014): Embraces increasingly extreme, nationalistic, and apocalyptic rhetoric, often invoking religious imagery and demonizing the West.

Contrasting Elements (B):

  • The stark and jarring contrast between Medvedev’s early and current personas.
  • The lack of a clear and consistent ideological rationale for this transformation.
  • The strategic timing of his shift, coinciding with Russia’s annexation of Crimea and the escalating tensions with the West.

Adding Context: The “Russian Narrative”

Two key texts provide a chilling backdrop for understanding the Kremlin’s strategy and Medvedev’s place within it:

  1. “A Word to the People” (1991): This manifesto, published in the hard-line Sovetskaya Rossiya journal in support of the August 1991 coup, articulates a profound fear of Russia’s disintegration. It reveals a desperate longing for strong leadership to unify the country against internal and external enemies.
  2. “The Ideology of Victory” (2021): This treatise, published by the ultra-conservative Izborsky Club, lays out a framework for mobilizing Russia for a “spiritual war” against the West. It portrays Russia as a “New Ark” resisting the “New Flood” of Western decadence, justifying aggression through apocalyptic and religious themes.

Deconstructing the “Medvedev Kayfabe” with Semiotic Hexagons

SSSA leverages semiotic hexagons to analyze complex phenomena by breaking them down into six interconnected components:

  1. S1 (Encoded Message): The overt narrative or message being projected.
  2. S2 (Disinformation Strategy): The tactics and techniques used to shape perceptions.
  3. S3 (Strategic Intent): The underlying goals and motivations.
  4. ~S1 (Opposite): Contradictory evidence or actions that challenge the encoded message.
  5. ~S2 (Opposite): Alternative explanations or unintended consequences of the disinformation strategy.
  6. ~S3 (Opposite): Potential backfire effects or unintended outcomes of the strategic intent.

Hexagon 1: Medvedev’s Role in Maintaining Unity

  • S1: Medvedev represents controlled dissent, providing the illusion of political pluralism within an authoritarian system.
  • S2: His transformation from a liberal to a hardliner is orchestrated to unify diverse factions in Russia against a common enemy – the West.
  • S3: This aims to strengthen regime cohesion, preventing a repeat of the Soviet Union’s collapse, a fear explicitly articulated in “A Word to the People.”
  • ~S1: Medvedev’s lack of authenticity and abrupt shifts in persona expose the “kayfabe” and could undermine the regime’s legitimacy.
  • ~S2: His extreme rhetoric may alienate moderates and unintentionally reinforce divisions within Russian society.
  • ~S3: If Medvedev becomes Putin’s successor, his lack of genuine support could lead to instability and challenges to his authority, potentially hastening the very disintegration that the regime fears.

Hexagon 2: Medvedev as a Symbol of the “Ideology of Victory”

  • S1: Medvedev’s apocalyptic and nationalistic rhetoric reflects the themes of the “Ideology of Victory,” justifying aggression and demonizing the West.
  • S2: His invocation of both Orthodox Christian and Islamic imagery seeks to mobilize a broader multi-religious base for the war effort, exploiting shared religious anxieties about the “end times.”
  • S3: This narrative aims to provide an ideological and “spiritual” justification for Russia’s actions, framing them as a defensive struggle for survival against a satanic, decadent West.
  • ~S1: The hypocrisy of the “desatanization” narrative is evident in the Russian regime itself, which embodies the very characteristics (violence, deceit, corruption) it projects onto its enemies.
  • ~S2: Medvedev’s extreme rhetoric could provoke a backlash from the international community, further isolating Russia and undermining its attempts to portray itself as a defender of “traditional values.”
  • ~S3: The “Ideology of Victory,” if taken to its logical conclusion, could lead to a dangerous escalation of conflict with the West, potentially culminating in the apocalyptic scenario that the narrative itself predicts.

Perpendicularity Analysis: Unveiling the Contradictions

SSSA focuses on identifying and analyzing “perpendicularities”—contradictions and inconsistencies—between the different components of the hexagons. These perpendicularities often expose hidden agendas and reveal the true nature of a situation.

  • Medvedev’s Persona: The stark contrast between his early and current personas suggests a deliberate manipulation of his image, revealing the “kayfabe” nature of his role (A + B).
  • Fear of Disintegration: The regime’s fear of collapse, as articulated in “A Word to the People,” drives its pursuit of unity through extreme narratives, even at the cost of hypocrisy and potential backlash (~S1, Hexagon 1).
  • Justifying Aggression: The “Ideology of Victory” provides a framework for framing Russia’s aggression as a defensive “spiritual war,” exploiting popular anxieties about the end times and the decline of traditional values (~S3, Hexagon 2).
  • The “Satanic West”: The projection of “satanic” characteristics onto the West, while the Russian regime itself embodies those very traits, exposes the cynical manipulation of religious themes for political gain (~S1, Hexagon 2).

Refined Equation: A More Precise Understanding

The SSSA analysis allows us to refine our initial understanding of Medvedev’s role through a revised equation:

(A + D + E) + B = C

  • A: Medvedev’s professional identity as a high-ranking Kremlin official.
  • B: Medvedev’s contrasting personas.
  • D: Evidence of Kremlin orchestration and manipulation of Medvedev’s image and messaging.
  • E: Medvedev’s role in advancing the Kremlin’s strategic objectives of unity, ideological justification for aggression, and potentially, succession planning.
  • C: A deeper and more nuanced understanding of Medvedev’s true function as a “kayfabe” figure, whose persona is deliberately constructed to serve the regime’s agenda.

Further Research & Implications:

  • Analyze Medvedev’s Inner Circle: Identify key advisors and influences who shape his messaging and public persona. This could reveal the mechanisms of narrative control and manipulation.
  • Trace Financial Flows: Investigate potential funding sources for Medvedev’s activities, particularly connections to Kremlin-linked businesses or oligarchs. This could expose financial incentives behind his “kayfabe” role.
  • Monitor Media Coverage: Assess how Medvedev’s rhetoric is portrayed in both Russian and international media, identifying attempts at narrative manipulation and audience reception analysis.

SSSA reveals that Medvedev’s transformation is not merely a personal eccentricity, but a calculated strategy rooted in the Kremlin’s fear of disintegration and its pursuit of a unifying ideology to justify its aggression. His potential as a “successor” is inextricably linked to this “kayfabe” and to the regime’s anxieties about the future.

However, the contradictions inherent within this “kayfabe,” the reliance on demonstrably false narratives, and the regime’s own hypocrisy create vulnerabilities that could undermine its long-term stability. SSSA provides a framework for recognizing these vulnerabilities and for understanding how the Kremlin’s manipulation of personas and narratives could ultimately contribute to the very instability it seeks to avoid.

Flip the script

Simple Symmetric Semiotic Prime Theorem

Theorem: All prime numbers, except for 2 and 3, can be expressed as an element of either the set A = {6n + 5 | n ∈ ℤ} or the set B = {6p + 7 | p ∈ ℤ}, where:

  • |A| = { |6n + 5| | n ∈ ℤ} represents the set of absolute values of elements in A.
  • |B| = { |6p + 7| | p ∈ ℤ} represents the set of absolute values of elements in B.

Furthermore, these prime numbers cannot be expressed as the product of two elements from the same set. Therefore if |A| BUT NOT |A|*|A|; or |B| BUT NOT |B|*|B|, then |A| OR |B| is a prime number; and all prime numbers are in either |A| OR |B|; not just A AND B.

Conclusion: It is not necessary to check A for B and vice versa since the factor values are contained in the symmetries of prime numbers.

Dismissing this theorem as merely a restatement of the 6k±1 pattern or as an application of the Sieve of Eratosthenes would be misguided and a significant oversight. In essence, this theorem builds upon known concepts but introduces a novel framework that merits its own consideration in the field of number theory.

You blind guides, straining out a gnat and swallowing a camel! You can sieve 4/35 to start rather than 2/2. But you can also incorporate this into a Sieve of Eratosthenes and vice versa.

Harmonious Symmetric Prime Sieve

The Harmonious Symmetric Prime Sieve algorithm is an innovative approach to prime number identification that optimizes the traditional sieve methods by leveraging mathematical properties and symmetry. Here are the key innovations and features of the algorithm:

Key Innovations

  1. Mathematical Basis (6k ± 1 Forms):
    • Prime Forms: All primes greater than 3 are of the form 6k−1 or 6k+1. This insight significantly reduces the number of candidates for primes, as it eliminates numbers that cannot be primes early on.
    • Efficient Checking: By focusing only on numbers that fit these forms, the algorithm reduces the number of iterations and checks required compared to traditional methods like the Sieve of Eratosthenes.
  2. Symmetry Utilization:
    • Symmetric Sieving: For every positive multiple marked as non-prime, the corresponding negative multiple is also marked. This dual marking ensures that both sides of zero are efficiently handled, thus doubling the sieving efficiency for each step.
    • Symmetric Prime Collection: While collecting primes, the algorithm considers the symmetrical counterparts of the numbers, ensuring completeness without redundant checks.
  3. Complementary Sieving Strategies:
    • Symmetric Sieve of Eratosthenes (SSOE – Bottom-Up): This component of the algorithm starts from the smallest primes and systematically marks their multiples as non-prime. By working upwards from the smallest primes, it ensures that smaller composite numbers are identified early.
    • Symmetric Semiotic Sieve (Top-Down): This sieve works from the top of the range downwards, focusing on larger numbers. It complements the bottom-up approach by catching larger composite numbers that might not have been fully handled by the SSOE.
  4. Optimized Non-Redundant Processing:
    • Avoiding Redundant Checks: The algorithm avoids reprocessing previously identified composites by maintaining and updating the boolean array isPrime. This ensures that each number is checked only once, either in the positive or negative range, reducing unnecessary computations.
    • Form-Specific Sieve: By choosing one form (6k−1 or 6k+1), the algorithm focuses on a subset of candidates, reducing the overall workload while ensuring all primes are still identified through symmetry. Since we are considering only 1/3 of numbers in 6k±1, reducing that to a search in 6k−1 OR 6k+1 reduces it to the set of 1/6 of the numbers. By also not considering multiples of 5 and 7 out of the gate, the approach starts with a set of just 4/35 of the total set of numbers to consider for primality, significantly reducing the search space.

Potential Innovations and Benefits

  1. Reduced Computational Complexity:
    • The focus on 6k±1 forms and symmetric processing reduces the number of iterations required compared to traditional sieves. This can lead to faster execution times, especially for large ranges.
  2. Balanced Workload:
    • The combination of bottom-up and top-down sieving balances the workload across the range, ensuring that both small and large composites and their factors are efficiently marked. This can lead to more consistent performance across different ranges.
  3. Memory Efficiency:
    • The use of a boolean array that covers only the range [−N,N] ensures that memory usage is minimized. The algorithm does not need to store all numbers up to N^2 as potential multiples, which is a significant advantage over traditional sieves.
  4. Parallel Processing Potential:
    • The clear division between the bottom-up and top-down sieving processes presents opportunities for parallel execution. By running these two sieves concurrently, the algorithm can leverage multi-core processors to further speed up the computation.
  5. Scalability:
    • The algorithm is designed to scale well with increasing values of N. The reduction in candidate numbers and efficient marking strategies ensure that it can handle very large ranges without a significant drop in performance.

Example and Pseudocode Summary

Example: For N=100 and formA = True:

  • The algorithm will create a boolean array from −100 to 100
  • It will sieve numbers of the form 6k−1 symmetrically, marking multiples of primes starting from |5| and |7| upwards and ensuring corresponding negatives are also marked.
  • Simultaneously, it will use a top-down approach to mark larger multiples as well as their composite factors, complementing the bottom-up sieve.


Algorithm: Harmonious Symmetric Prime Sieve with Integrated Top-Down and Bottom-Up Approaches

Input: N: The upper limit of the desired prime range (finds primes in [-N, N])
Output: primes: A list of all prime numbers in the range [0, N]

1. Initialization:
- Create a boolean array `is_prime` of size (2*N + 1), initialized to True.
- Set `is_prime[N] = is_prime[N+1] = False` (0 and 1 are not primes).
- Choose either `form_A = True` (for 6k-1) or `form_B = False` (for 6k+1).

2. Remove Multiples of 5 and 7:
- For k from -N to N do:
- If k % 5 == 0 or k % 7 == 0:
- Set `is_prime[k + N] = False`.

3. Top-Down Factor Identification:
- If `form_A` is True, set `start = (N // 6) * 6 + 5`, else set `start = (N // 6) * 6 + 7`.
- For x from start down to 1 in steps of 6:
- If `form_A` is True, set `p = x`, else set `p = x + 2`.
- If p > N, continue to the next iteration.
- If `is_prime[p + N]` is True:
- For k from 2*p to N in steps of p do:
- Set `is_prime[k + N] = False`.
- Set `is_prime[-k + N] = False`.

4. Symmetric Sieve of Eratosthenes (SSOE - Bottom-Up):
- For x from 1 to N // 6 + 1 do:
- If `form_A` is True, calculate `p = 6*x - 1`, else calculate `p = 6*x + 1`.
- If p > N, break the loop.
- If `is_prime[p + N]` is True:
- For k from p*p to N in steps of p:
- If `is_prime[k + N]` is True:
- Set `is_prime[k + N] = False`.
- If `is_prime[-k + N]` is True:
- Set `is_prime[-k + N] = False`.

5. Collect Primes:
- Create an empty list called `primes`.
- For i from 1 to N:
- If `form_A` is True:
- If `is_prime[i + N]` is True, append `i` to `primes`.
- Else If `i % 6 == 1` and `is_prime[-i + N]` is True, append `i` to `primes`.
- Else (`form_B` is True):
- If `is_prime[i + N]` is True, append `i` to `primes`.
- Else If `i % 6 == 5` and `is_prime[-i + N]` is True, append `i` to `primes`.

6. Return the `primes` list.

In conclusion, the Harmonious Symmetric Prime Sieve is an efficient and innovative approach to prime number identification that leverages mathematical insights, symmetry, and complementary sieving strategies to optimize the process and reduce computational overhead.

Symmetric Sieve of Eratosthenes

The Sieve of Eratosthenes stands as a classic testament to algorithmic elegance in finding prime numbers. Yet, by introducing a novel appreciation for the symmetrical nature of primes, we can refine this ancient method into a demonstrably superior algorithm: the Symmetric Sieve of Eratosthenes. Given that the “Semiotic Sieve” is likely to be resisted, I aim to simply demonstrate the advantages of symmetry in this traditional method to make it undeniable.

A hall of mirrors

Let’s recap the traditional Sieve: It starts by listing all integers from 2 up to a given limit. Marking 2 as prime, it iterates through the list, marking unmarked numbers as prime and sieving out their multiples as non-prime. This continues until all numbers up to the square root of the limit have been considered, leaving the remaining unmarked numbers as primes.

The Symmetric Sieve elevates this process by harnessing two inherent properties of prime numbers. Firstly, all primes greater than 3 fit the form 6k ± 1, where ‘k’ is any integer. Secondly, primes are symmetrically distributed around multiples of 6.

Instead of treating positive and negative numbers separately, the Symmetric Sieve cleverly uses a range symmetric around zero (from -N to N). Then, instead of checking both 6k+1 and 6k-1, it focuses on just one form. For each potential prime ‘p’ it encounters, it marks both ‘p’ and its negative counterpart ‘-p’ as prime. This automatically accounts for both forms due to the inherent symmetry.

This symmetrical approach achieves two significant improvements:

  1. Reduced Computations: By focusing on only one form of 6k ± 1 and leveraging symmetry, the Symmetric Sieve effectively halves the number of candidate primes that need to be checked.
  2. Implicit Coverage: Marking a number and its negative counterpart implicitly covers both forms of 6k ± 1, ensuring no prime is missed.

The Symmetric Sieve of Eratosthenes, while rooted in a classical algorithm, showcases how a deeper understanding of prime number properties, particularly their symmetry, can lead to a more efficient and elegant solution. It serves as a powerful example of innovation through the insightful application of mathematical principles.

Elegance and Class

Semiotic Prime Theorem: Other than 2 and 3, a positive integer is prime if, and only if, it can be expressed in the mutually exclusive forms: 6k+5 (form A) or 6p+7 (form B), where k and p are integers, and it cannot be factored into two distinct integers, x and y, each greater than 1. Forms A and B are symmetric: negation of a prime expressed in one form yields its corresponding prime in the other. Therefore, all primes greater than 3 can be sieved using either form, reducing the candidate pool again by half and leveraging modular arithmetic for efficient prime detection rather than factorization.

Hopefully infer the negative of arrogance and crass (most of the time) lol

Let A={6x+5∣ x∈ Z} be the set of all numbers of the form 6x+5, and B={6y+7∣ y∈ Z} be the set of all numbers of the form 6y+7.

Define the sets of products as follows:

  • AA={(6x+5)(6y+5)∣ x, y ∈ Z}
  • AB={(6x+5)(6y+7)∣ x, y ∈ Z}
  • BB={(6x+7)(6y+7)∣ x, y ∈ Z}

More simply, a number greater than 3 is prime if it is not divisible by 5 and is of the form(s): (A={6x+5∣ x∈ Z} OR B={6y+7∣ y∈ Z}) BUT NOT (AA={a*b∣ a, b ∈ Z} OR BB={a*b∣ a, b ∈ Z})

Furthermore, consider the negation and symmetry properties:

  • For each a∈ A, there exists a corresponding −a≡6k+7(mod6) for some integer k.
  • For each b∈ B, there exists a corresponding −b≡6m+5(mod6) for some integer m.

Then, any number that is an element of A or B but not an element of AA, AB, or BB is a prime number, and we only need to sieve in either A or B for all primes when considering absolute values.