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.