**Theorem:** *All prime numbers, except for 2 and 3, can be expressed as an element of either the set A = {6n + 5 | n ∈ ℤ} or the set B = {6p + 7 | p ∈ ℤ}, where:*

*|A| = { |6n + 5| | n ∈ ℤ} represents the set of absolute values of elements in A.**|B| = { |6p + 7| | p ∈ ℤ} represents the set of absolute values of elements in B.*

*Furthermore, these prime numbers cannot be expressed as the product of two elements from the same set. Therefore if |A| BUT NOT |A|*|A|; or |B| BUT NOT |B|*|B|, then |A| OR |B| is a prime number*; *and all prime numbers are in either |A| OR |B|; not just A AND B.*

**Conclusion:** **It is not necessary to check A for B and vice versa since the factor values are contained in the symmetries of prime numbers. **

Dismissing this theorem as merely a restatement of the 6k±1 pattern or as an application of the Sieve of Eratosthenes would be misguided and a significant oversight. In essence, this theorem builds upon known concepts but introduces a novel framework that merits its own consideration in the field of number theory.