Theorem: Boolean Conditions for the Appearance of Twin Primes according to the Hotchkiss Prime Theorem
Introduction: The Hotchkiss Prime Theorem offers insights into the distribution of prime numbers within sets A and B, defined as {6k + 5 | k ∈ ℤ} and {6k + 7 | k ∈ ℤ} respectively. It delineates the relationship between primes and composite numbers within these sets. Expanding upon this theorem, we establish the necessary Boolean architectural conditions under which twin primes emerge without exceptions.
Statement: Let A = {6k + 5 | k ∈ ℤ} and B = {6k + 7 | k ∈ ℤ} denote the sets defined by the Hotchkiss Prime Theorem. These sets encompass all primes greater than 3 (excluding 2 and 3), as well as composite numbers formed by the products of elements within sets A and B.
Theorem: Twin primes manifest under the following Boolean conditions:
- Primes other than 2 and 3 are distributed within sets A and B as defined by the Hotchkiss Prime Theorem.
- Twin primes occur when primes from both sets A and B coincide, without being products of elements within sets AA, AB, or BB.
Formalization:
- Primes Distributed in Sets A and B:
- Let P(A) represent the presence of primes in set A, and P(B) represent the presence of primes in set B.
- Primes greater than 3 (excluding 2 and 3) are distributed within sets A and B: P(A) OR P(B).
- Primes do not emerge as products of elements within sets AA, AB, or BB: NOT (AA OR AB OR BB).
- Architectural Conditions for Twin Primes:
- Twin primes occur when primes from sets A and B coincide: P(A) AND P(B).
- Twin primes do not emerge as products of elements within sets AA, AB, or BB: NOT (AA OR AB OR BB).
Conclusion: The Hotchkiss Prime Theorem provides the architectural framework within which twin primes manifest. By establishing that all primes (excluding 2 and 3) are distributed within sets A and B, this theorem elucidates the necessary conditions for the appearance of twin primes. Understanding these architectural conditions enhances our comprehension of the distribution and occurrence of twin primes within prime number theory.