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

**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.

**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.

**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.

**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

**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.

**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.

**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.

**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.

**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.