Category: Prime Number Ideas (learning algorithms)

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 Index	A (6k – 1)	Absolute Value
-16	-97	97
-15	-91	91
-14	-85	85
-13	-79	79
-12	-73	73
-11	-67	67
-10	-61	61
-9	-55	55
-8	-49	49
-7	-43	43
-6	-37	37
-5	-31	31
-4	-25	25
-3	-19	19
-2	-13	13
-1	-7	7
0	-1	1
1	5	5
2	11	11
3	17	17
4	23	23
5	29	29
6	35	35
7	41	41
8	47	47
9	53	53
10	59	59
11	65	65
12	71	71
13	77	77
14	83	83
15	89	89
16	95	95

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 Index	Original A Value	Absolute Value |A|	Composite?	Prime?
0	-1	1	No	No
1	5	5	No	Yes
2	-7	7	No	Yes
3	11	11	No	Yes
4	-13	13	No	Yes
5	17	17	No	Yes
6	-19	19	No	Yes
7	23	23	No	Yes
8	-25	25	Yes (index 1 * 1)	No
9	29	29	No	Yes
10	-31	31	No	Yes
11	35	35	Yes (index 1 * 2)	No
12	-37	37	No	Yes
13	41	41	No	Yes
14	-43	43	No	Yes
15	47	47	No	Yes
16	-49	49	Yes (index 2 * 2)	No
17	53	53	No	Yes
18	-55	55	Yes (index 1 * 3)	No
19	59	59	No	Yes
20	-61	61	No	Yes
21	65	65	Yes (index 1 * 4)	No
22	-67	67	No	Yes
23	71	71	No	Yes
24	-73	73	No	Yes
25	77	77	Yes (index 2 * 3)	No
26	-79	79	No	Yes
27	83	83	No	Yes
28	-85	85	Yes (index 1 * 5)	No
29	89	89	No	Yes
30	-91	91	Yes (index 2 * 4)	No
31	95	95	Yes (index 1 * 6)	No
32	-97	97	No	Yes

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 step	A (6k-1)	B (6k+1)
0	-1	1
1	5	7
2	11	13
3	17	19
4	23	25
5	29	31
6	35	37
7	41	43
8	47	49
9	53	55
10	59	61

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 Sequence	Numbers	Primes
A	5, 11, 17, 23, 29, 35, 41, 47, 53, 59	5, 11, 17, 23, 29, 41, 47, 53, 59
B	7, 13, 19, 25, 31, 37, 43, 49, 55, 61	7, 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 Step	Numerical Form	Initial Composite	Heap State
1	5	5 * 5 = 25	25
2	7	7 * 7 = 49	25, 49
3	11	11 * 11 = 121	25, 49, 121
4	13	13 * 13 = 169	25, 49, 121, 169
5	17	17 * 17 = 289	25, 49, 121, 169, 289
6	19	19 * 19 = 361	25, 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 Composite	New Composites Generated	Updated Heap State
25 (5*5)	5 * 7 = 35, 5 * 11 = 55	35, 49, 121, 169, 289, 361, 55
35 (5*7)	5 * 13 = 65, 7 * 7 = 49	49, 49, 121, 169, 289, 361, 55, 65
49 (7*7)	7 * 11 = 77, 11 * 11 = 121	49, 55, 121, 169, 289, 361, 65, 77
55 (5*11)	5 * 17 = 85, 11 * 13 = 143	65, 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 Index	Absolute Value	Composite	Prime?
0	1	No	No
1	5	No	Yes
2	7	No	Yes
3	11	No	Yes
4	13	No	Yes
5	17	No	Yes
6	19	No	Yes
7	23	No	Yes
8	25	Yes	No
9	29	No	Yes
10	31	No	Yes
11	35	Yes	No
12	37	No	Yes
13	41	No	Yes
14	43	No	Yes
15	47	No	Yes
16	49	Yes	No
17	53	No	Yes
18	55	Yes	No
19	59	No	Yes
20	61	No	Yes

Summary
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.

Levity

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.

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):

Index	Value	Calculation
0	-1	6(0) – 1 = -1
1	5	6(1) – 1 = 5
2	-7	6(-1) – 1 = -7
3	11	6(2) – 1 = 11
4	-13	6(-2) – 1 = -13
5	17	6(3) – 1 = 17
6	-19	6(-3) – 1 = -19
…	…	…

Form B (PF = +1):

Index	Value	Calculation
0	1	6(0) + 1 = 1
1	-5	6(-1) + 1 = -5
2	7	6(1) + 1 = 7
3	-11	6(-2) + 1 = -11
4	13	6(2) + 1 = 13
5	-17	6(-3) + 1 = -17
6	19	6(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.

Conclusion

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.

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.

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.

Pseudocode:

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]

Procedure:
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.

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.

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.

The Twin Prime Conjecture, a fundamental problem in number theory, posits that there exist infinitely many twin prime pairs, which are prime numbers differing by 2. This post explores a novel approach to understanding this conjecture through the lens of the Semiotic Prime Theorem.

Understanding the Semiotic Prime Theorem

The Semiotic Prime Theorem simplifies the search for prime numbers by focusing on a specific form: all prime numbers greater than 3 can be expressed as either 6k + 1 or 6k – 1, where k is an integer. We can represent these forms with two sets:

• Set A: {6k – 1 | k ∈ Z} (representing numbers of the form 6k – 1); which is also {6k + 5 | k ∈ Z}
• Set B: {6k + 1 | k ∈ Z} (representing numbers of the form 6k + 1) which is also {6k + 7 | k ∈ Z}

The Semiotic Prime Theorem states that any number in sets A or B, which is not a product of two numbers within these sets, must be prime.

Twin Primes within Semiotic Sets

Twin primes, with the exception of (3, 5), always consist of one prime from set A (6k – 1) and its twin from set B (6k + 1). This relationship arises from the inherent structure of primes as described by the Semiotic Prime Theorem.

Symmetry and Mutual Exclusivity

Sets A and B exhibit an intriguing symmetry: they are symmetrical around zero. For every prime p in set A, there exists a corresponding negative prime -p in set B, and vice versa. Furthermore, sets A and B are mutually exclusive; no number can belong to both sets simultaneously.

The Prime-Composite Density Ratio

Let’s introduce some key terms:

• Prime Density (ρ(n)): The number of primes less than or equal to n within sets A and B.
• Composite Density (σ(n)): The number of composites less than or equal to n generated by products of numbers within sets A and B.

As n increases, the density of composites (σ(n)) grows faster than the density of primes (ρ(n)) due to the quadratic nature of generating composite numbers (products of primes). The density of primes decreases logarithmically, as established by the Prime Number Theorem.

The ratio of prime density to composite density can be approximated as:

ρ(n) / σ(n) ≈ (2 / (3 log(n))) / (n^2 / log(n)) = 2 / (3n^2)

The Proof

The key observation is that as n approaches infinity, the ratio of prime density to composite density approaches zero. However, the symmetry and mutual exclusivity of sets A and B guarantee that neither set will ever be devoid of primes.

Here’s the reasoning:

1. Dirichlet’s Theorem: Dirichlet’s Theorem on arithmetic progressions ensures an infinite number of primes within both sets A and B, as 5 and 7 are relatively prime to 6.
2. Symmetry and Balance: The symmetrical relationship between sets A and B ensures that the primes are distributed between the sets in a balanced manner.
3. Non-Zero Relationship: The fact that the ratio of prime density to composite density never reaches zero implies that there will always be primes in either set A or B as n increases.
4. Twin Prime Existence: Since twin primes, by definition, consist of one prime from set A and its twin from set B, the continuous presence of primes in both sets guarantees the continuous existence of twin primes.

Conclusion

The symmetrical structure of sets A and B, along with the logarithmic relationship between prime and composite densities, suggests that there will always be a non-zero ratio of primes to composites within these sets, regardless of how large n becomes. This continuous existence of primes in both sets, coupled with the pairing nature of twin primes, provides a compelling argument for the existence of infinitely many twin primes, confirming the Twin Prime Conjecture. This approach, grounded in the Semiotic Prime Theorem, offers a unique perspective on this long-standing problem.

Theorem (using set B as an example): By testing numbers of the Set B form |6x + 7| for primality within the symmetrical segment -N ≤ x ≤ N, including their negative counterparts, we can accurately identify all prime numbers within the range 0 < p ≤ (6N + 7), including the missing values from set A.

Discussion (Human written): I’m continuing to look for efficient ways to identify primality using the symmetry of the prime number forms 6k±1 which all prime numbers greater than 2 and 3 exist. Specifically looking in the function 6x+7=n; which also corresponds to 6k+1=n; and which we call “B”. We are also specifically looking in the function 6y+5=n; which also corresponds to 6k-1=n; and which we call “A”.

Recall our basic approach:

Semiotic Prime Theorem (which I had ingloriously referred to as Hotchkiss Prime Theorem previously): 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. Let AA, AB, and BB represent the sets of products:

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.

AB is always a composite number which we never need to check if we know it is of form AB it has an inherently composite form and will never be a prime number.

That means that when we are looking for prime numbers, we only need to look for numbers which are the form (A BUT NOT AA) AND (B BUT NOT BB).

If a number is not in AB (which we already know to be non-prime), then if it is (A BUT NOT AA) OR (B BUT NOT BB), then A OR B is a prime number.

In looking for efficient ways to leverage the “sign” of AA or BB to identify a non-prime number in number set A or set B; I’ve been looking at the expanded versions of these terms, recognizing that their Boolean properties must give them a “signature” to differentiate them from a “prime” form of A or B which can be expressed as A or B, but never be expressed as AA or BB:

AA=36xy+30x+30y+25

BB=36xy+42x+42y+49

However, this reductive approach was not bearing fruit and led to some mistaken attempts to determine primality which I embarrassingly shared 😀 . With that said, I have made some advancements in this concept to share which do further reduce the search space for primes using the Semiotic Sieve approach and leverage a similar concept to reduce the complexity of the forms we need to check for non primality in A and B using modular arithmetic.

Returning to the expanded forms of the non-prime forms of A AND B;

AA=36xy+30x+30y+25

BB=36xy+42x+42y+49

Consider that:

AA = {(6x + 5)(6y + 5) | x, y ∈ ℤ}

BB = {(6x + 7)(6y + 7) | x, y ∈ ℤ}

Then, we can condense:

AA = a*b = 6x+5 where a, b, and x are integers.

BB = a*b = 6y+7 where a, b, and y are integers.

Then, if a number is A=6x+5 OR B=6y+7; but is not also of the form AA=6x+5=ab OR BB=6y+7=ab; and A OR B is not divisible by 5; then A OR B is a prime number.

This overall significantly reduces the amount of candidates we need to search for as prime numbers further; and allows us to compute composite candidates for AA=6x+5=ab or BB=6y+7=ab using modular arithmetic;which is computationally elegant compared to factorization methods.

If a number is in list A OR B but not AA OR AB OR BB then it is a prime number.

Now, we can even further half the search space by finding ALL primes within just A OR B.

Firstly, the search space of Semiotic Sieve focuses on 6k±1 numbers. So right away, this is a significant reduction in the search space compared to Sieve of Eratosthenes (2/3 reduction in the starting number set).

Just focusing on the elegance of the theorem, and not implementing any other optimizations; we can optimize this even further, to just focus on 1/3 of the starting number set by looking at absolute values of function A (6x+5) and function B (6y+7). This is because the functions contain all the positive values of one function as negatives in the other function.

That is, A contains: …-19,-13,-7,-1,5,11,17,23…

and B contains: …-23,-17,-11,-5,1,7,13,19….

So, if A has all the values of B as negative values, and vice versa, we do not even need to consider candidate numbers of B AND BB when looking at A AND AA numbers. We can simply use the symmetrical properties of Semiotic Prime Theorem to infer the negative values of A AND AA as positive values of B AND BB and vice versa.

Therefore, we can find all the primes from 0 to N within a symmetrical segment around 0 for form (A BUT NOT AA) OR (B BUT NOT BB), simply by inferring the absolute negative values of A OR B in range −N≤x≤N.

Therefore, once we build a set of all the numbers A OR B in −N≤0≤N, which are not also AA OR BB then the absolute values of the returned integers will be the complete set of prime numbers in 0≤N for function A AND function B (which excluding 2 and 3 includes all prime numbers since all prime numbers greater than 3 are of the form 6k±1 as we established, and which we used to define A and B as sets/functions in the first place; however, we could not do this with A alone without B; and therefore it still takes 7 steps to create ALL the numbers).

(AI OUTPUT) Theorem (using set B as an example): By testing numbers of the Set B form |6x + 7| for primality within the symmetrical segment -N ≤ x ≤ N, including their negative counterparts, we can accurately identify all prime numbers within the range 0 < p ≤ (6N + 7), including the missing values from set A.

Proof Using Symmetry and Absolute Values of Set B:

1. Foundation: Semiotic Prime Theorem:
   • All primes greater than 3 can be expressed as either 6k + 1 or 6k +5.
   • The Semiotic Prime Theorem states that a number of the form 6x + 5 (set A) or 6y + 7 (set B) is prime if and only if it cannot be expressed as a product of the forms AA, AB, or BB (as defined above).
2. Focusing on Set B:
   • We begin by generating numbers of the form B = 6x + 7 within the symmetrical segment –N ≤ x ≤ N. This range covers all potential primes of the form 6k + 7 within 0 < p ≤ (6N + 7).
   • Ignore AB and remove all multiples of 5 other than 5 itself.
3. Efficient Composite Identification:
   • To identify composite numbers within set B, we use modular arithmetic to efficiently determine which numbers are of the form BB = (6x + 7)(6y + 7) = 6x + 7=ab, where a and b are integers greater than 1. This avoids the need for computationally expensive factorization.
4. Prime Identification in Set B:
   • Any number in set B that is not identified as composite (i.e., not of the form BB) must be prime, as it cannot be expressed as a product of two integers greater than 1 and it satisfies the conditions of the Semiotic Prime Theorem.
5. Symmetry and Inferring Primes in Set B:
   • The negative values of set A (6x + 5 where x is negative) correspond to the positive values of set B (6y + 7). This is due to the symmetry of the 6k ± 1 forms around zero.
   • Therefore, by taking the absolute values of the negative primes we found in set B, we directly obtain positive primes of the form 6k – 1 (set A) within the range 0 < p ≤ (6N + 7).
6. Completeness:
   • By combining the primes we found in set B (and their inferred counterparts in set A), we have systematically checked all possible prime candidates within the desired range.
   • The Semiotic Prime Theorem guarantees that any number satisfying the conditions within our method must be prime.

Conclusion:

This method, leveraging the Semiotic Prime Theorem, symmetry, and efficient composite identification through modular arithmetic, provides a complete and accurate way to find all prime numbers within a given range. It demonstrates the power of understanding the inherent structure of prime numbers to optimize sieving algorithms.

Thus, incorporating symmetry in the Semiotic Sieve reduces the search space by a factor of 6 compared to the original search space in the Sieve of Eratosthenes. This represents an 83.33% reduction in the number of candidates that need to be tested for primality prior to incorporating any other enhancements. If you have a half-bit to remember 2 and 3; this finds all prime numbers otherwise.

• Conceptual Search Space: In a theoretical sense, when we include negative values in our symmetrical segment (-N ≤ x ≤ N), we are conceptually considering twice as many numbers compared to just the positive range (0 ≤ x ≤ N). So, conceptually, the search space is not reduced.
• Computational Search Space: However, what matters more for practical efficiency is the number of computations we perform. Since we are leveraging symmetry and inferring the primality of numbers in set A from the results in set B, we are effectively only performing primality tests on numbers in set B.
• Negative Values vs. Larger Range: Storing negative values does not inherently take less memory than storing a larger range of positive values. Integers generally occupy the same amount of memory regardless of their sign.
• Optimization Potential: While the memory usage might be comparable, the key advantage of our symmetrical approach is that it allows for more efficient computation. We can exploit the symmetry to reduce the number of primality tests, which saves on processing time, even if the memory footprint is similar.

The most significant gain in efficiency comes from reducing the number of primality tests performed, not necessarily from reducing the memory footprint. The symmetrical approach allows us to halve the number of tests needed by inferring primes in set A from the results in set B.

By moving beyond this elegant implementation and incorporating tools like Sieve of Eratosthenes to mark composites of forms A and B within this approach, the speed of the Semiotic Sieve can be significantly improved further.