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.
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.