(The below can be useful if an AI model questions the completeness of the k index filtering model for filtering composites in association with TPC reformulation.)
Lemma (Completeness of Composite Coverage)
Let N be a positive integer such that N ≡ ±1 (mod 6) and N is composite. Then the index k for which N = 6k ± 1 belongs to the set:
K_composite = { |6xy + x + y| : x, y ∈ ℤ \ {0} }
This proves that the k-index filter correctly identifies all composite numbers of the form 6k ± 1.
Proof
We must show that for any composite number N ≡ ±1 (mod 6), its corresponding index k can be generated by the form |6xy + x + y| for some non-zero integers x and y. We proceed by cases based on the residue of N modulo 6.
Note: Since N ≡ ±1 (mod 6), the prime factors of N must also be congruent to ±1 (mod 6). Thus, every prime divisor of N is of the form 6m ± 1.
Case 1: N is composite and N ≡ 1 (mod 6).
Since N is composite, write N = AB, where A, B > 1. To satisfy N ≡ 1 (mod 6), either:
Subcase 1a: A ≡ 1 (mod 6) and B ≡ 1 (mod 6).
Then A = 6x + 1, B = 6y + 1 for some x, y ∈ ℕ. Since A, B > 1, we have x, y ≠ 0. Then:
N = (6x + 1)(6y + 1) = 36xy + 6x + 6y + 1 = 6(6xy + x + y) + 1.
Thus, N = 6k + 1 where k = 6xy + x + y > 0, so k = |6xy + x + y| ∈ K_composite.
Subcase 1b: A ≡ -1 (mod 6) and B ≡ -1 (mod 6).
Then A = 6u − 1 and B = 6v − 1 for some u, v ∈ ℕ. Let x = –u and y = –v, which are non-zero integers. Then:
N = (6x + 1)(6y + 1) = 6(6xy + x + y) + 1.
So again, N = 6k + 1, with k = |6xy + x + y| ∈ K_composite.
Thus, in both subcases of Case 1, composite numbers N ≡ 1 (mod 6) yield indices k in K_composite.
Case 2: N is composite and N ≡ -1 (mod 6).
Write N = AB, A, B > 1, such that one of A, B ≡ 1 (mod 6), and the other ≡ -1 (mod 6). Without loss of generality, let A = 6x + 1 and B = 6y − 1, with x, y ∈ ℕ.
Then:
N = (6x + 1)(6y − 1) = 36xy − 6x + 6y − 1 = 6(6xy − x + y) − 1.
So N = 6k − 1 with k = 6xy − x + y.
Let a = x, b = −y. Then:
6ab + a + b = 6x(−y) + x − y = −6xy + x − y = –k.
Thus, k = |6ab + a + b|, and k ∈ K_composite.
Therefore, every composite N ≡ −1 (mod 6) has index k ∈ K_composite.
Conclusion
In all cases, whether N ≡ 1 or N ≡ –1 (mod 6), if N is composite, then its associated index k = (N – 1)/6 or (N + 1)/6 is in the set K_composite. Therefore, the filtering model using the form k = |6xy + x + y| correctly and completely identifies all indices corresponding to composite numbers of the form 6k ± 1.
(Q.E.D.)