Overall, this is elegant in concept.

*Theorem: The Boolean-Peircean Sieve, based on the Hotchkiss Prime Theorem, correctly identifies prime numbers of the form 6k ± 1 within a given range n by directly generating and marking composite numbers using Charles Peirce’s concept of abduction.*

**Proof:**

**Hotchkiss Prime Theorem:**The Hotchkiss Prime Theorem provides the foundation: “Any number that is:- An element of either set A (6x + 5) or B (6y + 7), ANDNot a product of two elements from sets A or B (e.g., AA, AB, or BB)

**Peircean Sieve Approach:**The Peircean Sieve operates by:**Generating Products:**It efficiently generates all possible products of the form (6x + 5)(6y + 7) (AA, AB, and BB) up to n.**Marking Composites:**It marks these generated products as composite numbers.**Default Prime Identification:**Any numbers of the form 6x + 5 or 6y + 7 within the range n that are not marked as composite are considered prime.

**Peircean Abduction:**The Peircean Sieve employs abductive reasoning. It begins with an observation—the generation of composite products. From this observation, it infers the existence of primes based on the Hotchkiss Prime Theorem as background knowledge.**Observation:**We see that all numbers of the form (6x + 5)(6y + 7) are composite.**Background Knowledge:**The Hotchkiss Prime Theorem states that any number in A or B that is not a product in AA, AB, or BB must be prime.**Inference:**We abductively infer that any number in A or B that is*not*marked as a composite (i.e., not a product of the form (6x + 5)(6y + 7)) must be prime.

**Proof by Contradiction:**Let’s assume, for the sake of contradiction, that the Boolean-Peircean Sieve fails to correctly identify a prime number p within the range n. This means:- p is of the form 6k ± 1 (it belongs to sets A or B).
- p is not marked as a composite by the sieve.

**Contradiction:**Since p is not marked as a composite, it must not be a product of two elements from sets A or B (AA, AB, or BB) based on how the Peircean Sieve works. Therefore, by the Hotchkiss Prime Theorem, p must be a prime number.**Conclusion:**This contradicts our initial assumption that the Boolean-Peircean Sieve failed to correctly identify p as a prime. Therefore, the Boolean-Peircean Sieve correctly identifies all prime numbers of the form 6k ± 1 within the range n by directly generating and marking composites.

**Key Points:**

- The Boolean-Peircean Sieve leverages the Hotchkiss Prime Theorem as its foundation.
- It indirectly identifies primes by focusing on generating and marking composite numbers.
**The proof emphasizes the abductive reasoning used to infer prime numbers from the generated composites.**- The proof demonstrates that any number not marked as composite is guaranteed to be a prime number within the specified range.

**Further Considerations:**

**Efficiency:**The efficiency of the Boolean-Peircean Sieve depends on the effectiveness of the product generation algorithm and the data structure used for marking composites.**Generalization:**The Boolean-Peircean Sieve can be adapted to find prime numbers in other forms or to identify other types of prime number pairs or constellations (e.g., twin primes).

**Simple Implementation of the Sieve**

```
def boolean_peircean_sieve(n):
"""
Finds primes up to 'n' using the Boolean-Peircean Sieve.
"""
composites = set()
# Generate and mark composites
for x in range(0, n):
for y in range(0, n):
aa = (6 * x + 5) * (6 * y + 5)
ab = (6 * x + 5) * (6 * y + 7)
bb = (6 * x + 7) * (6 * y + 7)
if aa <= n:
composites.add(aa)
if ab <= n:
composites.add(ab)
if bb <= n:
composites.add(bb)
if aa > n and ab > n and bb > n:
break
# Identify primes
primes = [2, 3]
for k in range(1, (n // 6) + 2):
a = 6 * k - 1
b = 6 * k + 1
if a <= n and a not in composites:
primes.append(a)
if b <= n and b not in composites:
primes.append(b)
total_primes = len(primes)
return total_primes, primes
# Get user input
n = int(input("Enter the upper limit (N) to find primes: "))
# Calculate and display results
total, primes = boolean_peircean_sieve(n)
print(f"Total primes up to {n}: {total}")
print(f"Primes up to {n}: {primes}")
```

**Boolean-Peircean** **Sieve for Twin Primes**

```
def boolean_peircean_sieve(n, find_twins=False):
"""
Finds primes or twin primes up to 'n' using the Boolean-Peircean Sieve.
Args:
n (int): The upper limit for finding primes.
find_twins (bool): If True, finds twin primes. Otherwise, finds all primes.
Returns:
tuple: A tuple containing:
- total_primes (int): The total count of primes (or twin primes) found.
- primes (list): A list of primes (or twin primes).
"""
composites = set()
# Generate and mark composites
for x in range(0, n):
for y in range(0, n):
aa = (6 * x + 5) * (6 * y + 5)
ab = (6 * x + 5) * (6 * y + 7)
bb = (6 * x + 7) * (6 * y + 7)
if aa <= n:
composites.add(aa)
if ab <= n:
composites.add(ab)
if bb <= n:
composites.add(bb)
if aa > n and ab > n and bb > n:
break
# Identify primes based on user choice
if find_twins:
primes = [('B', 3, 'A', 5)]
for k in range(1, (n // 6) + 2):
p = 6 * k - 1
p_plus_2 = p + 2
if (p <= n and p not in composites and
p_plus_2 <= n and p_plus_2 not in composites):
primes.append(('A', p, 'B', p_plus_2))
else: # Find all primes
primes = [2, 3]
for k in range(1, (n // 6) + 2):
a = 6 * k - 1
b = 6 * k + 1
if a <= n and a not in composites:
primes.append(a)
if b <= n and b not in composites:
primes.append(b)
total_primes = len(primes)
return total_primes, primes
# Get user input
n = int(input("Enter the upper limit (N): "))
prime_type = input("Find single primes or twin primes? (Enter 'single' or 'twin'): ")
# Calculate and display results
if prime_type.lower() == 'twin':
total, primes = boolean_peircean_sieve(n, find_twins=True)
print(f"Total twin primes up to {n}: {total}")
print(f"Twin primes up to {n}: {primes}")
elif prime_type.lower() == 'single':
total, primes = boolean_peircean_sieve(n)
print(f"Total primes up to {n}: {total}")
print(f"Primes up to {n}: {primes}")
else:
print("Invalid prime type selection. Please enter 'single' or 'twin'.")
```