Generalized k-Index Filtering for Compositeness (Enhanced) - noir

AI podcast version (Deep Dive)

This document outlines an algebraic framework for identifying composite numbers within arithmetic progressions by parameterizing their indices.

1. General Framework

Let a, b, c, d be fixed integers. Consider an arithmetic progression:
N = ak + b

Suppose the index k can be expressed in the form:
k = axy + cx + dy
for some integers x, y.

We seek to find conditions under which N factors based on this structure of k.
Let’s substitute the expression for k into N:
N = a(axy + cx + dy) + b
N = a²xy + acx + ady + b

Now, consider the potential factorization:
P = (ax + c)(ay + d) = a²xy + adx + acy + cd

For N to be equal to this product P, we must have:
a²xy + acx + ady + b = a²xy + adx + acy + cd
acx + ady + b = adx + acy + cd
b – cd = adx – acx + acy – ady
b – cd = a(d-c)x – a(d-c)y
b – cd = a(d-c)(x-y)

Multiplying by -1, we get the identity:
cd – b = a(c-d)(x-y)

➤ Implication:
If the parameters a, b, c, d and the variables x, y satisfy the identity cd – b = a(c-d)(x-y), then the number N = ak + b (where k = axy + cx + dy) admits the factorization:
N = (ax+c)(ay+d)

If both factors |ax+c| > 1 and |ay+d| > 1, then N is composite.

Note on the Identity: The identity cd – b = a(c-d)(x-y) implies a constraint. If a(c-d) \neq 0, then x-y = (cd-b) / (a(c-d)), meaning x-y must be a specific constant. This restricts the independence of x and y. The framework is most powerful when this constraint is trivially satisfied or leads to a useful simplification, as seen below.

2. Simplified Case: c = d

A significant simplification occurs when c = d.
The identity cd – b = a(c-d)(x-y) becomes:
c² – b = a(c-c)(x-y)
c² – b = a(0)(x-y)
c² – b = 0
This implies b = c².

In this scenario:

The condition on x,y vanishes, allowing x and y to be independent integers.
The index expression for k becomes:
k = axy + cx + cy = axy + c(x+y)
The arithmetic progression value N = ak+b becomes:
N = ak + c²
The factorization becomes:
N = (ax+c)(ay+c)

This simplified form guarantees compositeness for N = ak+c² whenever k = axy+c(x+y), provided |ax+c| > 1 and |ay+c| > 1. This is a robust way to generate composites.

3. Applications to Specific Cases

Let k_{orig} be the original index in a progression N = A k_{orig} + B. We will use k to denote the parameterized form axy+c(x+y) which, if k_{orig}=k, implies compositeness.

Case 1: Filtering for Universal form N = k+1 (also expressible as N = n² + 1)

We can write this as N = K + 1, where K = n². So, the arithmetic progression parameters are A=1, B=1.
We use the simplified case: c=d. We require B=c². Since B=1, we have c²=1. Let c=1.
The corresponding a in the k = axy+c(x+y) formula is A=1.
The k-parameterization that generates composites is:
k = (1)xy + 1(x+y) = xy+x+y
If n² (our K) can be expressed as xy+x+y, then N = n²+1 is composite:
N = (1)(xy+x+y) + 1² = (x+1)(y+1)
Compositeness is ensured if |x+1|>1 and |y+1|>1. For N>0, we typically take x,y \ge 1.

Case 2: Filtering for an adapted form, e.g., N = 2k+1

Consider numbers N = 2K+1. This fits ak_{orig}+b with a=2 and b=1.
Using the simplified case: c=d. We need b=c². Since b=1, we choose c=1.
The a in k_{param} = axy+c(x+y) is a=2.
The k-parameterization is:
k_{param} = 2xy + 1(x+y) = 2xy+x+y.
If an index K can be expressed as k_{param} = 2xy+x+y, then N = 2K+1 is composite:
N = 2k_{param} + 1² = 2(2xy+x+y) + 1 = 4xy+2x+2y+1 = (2x+1)(2y+1).
For compositeness with factors greater than 1, if N > 0, we typically require x, y \ge 1 (so 2x+1 \ge 3, 2y+1 \ge 3).
This shows that any odd number 2K+1 where K = ( (2x+1)(2y+1)-1 )/2 for x,y \ge 1 is composite. This directly relates to the K-Filtering Model’s Case 2 (n=2k’+1, where k’=K).

Case 3: Filtering for numbers N = 6k±1 (using symmetry of {|6k-1|}={|6k+1|} in symmetrical ranges around 0)

This case involves numbers that can be prime (if >3). We aim to identify composite numbers within this family using a single k-parameterization.

Let a=6 (from the 6k_{orig} part).
Using the simplified case b=c², let c=1. So, b=1.
This naturally addresses the 6k_{orig}+1 form.
The k-parameterization is:
k_{param} = 6xy + 1(x+y) = 6xy+x+y
The corresponding number from the simplified framework is N_{framework} = ak_{param} + c² = 6k_{param} + 1.
This yields the factorization N_{framework} = (6x+1)(6y+1).

To ensure non-trivial factors (6x+1) and (6y+1), we must have 6x+1 \neq \pm 1 and 6y+1 \neq \pm 1. This means x \neq 0, -1/3 and y \neq 0, -1/3. Since x, y are integers, we require:

x, y \in \mathbb{Z} \setminus \{0\}.

Now, let’s connect this k_{param} to filtering composites in 6k_{orig} \pm 1:

Filtering 6k_{orig}+1 numbers:
If k_{orig} = k_{param} = 6xy+x+y, then
N = 6k_{orig}+1 = 6(6xy+x+y)+1 = (6x+1)(6y+1).
This N is composite.
- If x,y > 0, then k_{param} > 0. N = (6x+1)(6y+1) is a positive composite \equiv 1 \pmod 6.
  (e.g., x=1,y=1 \implies k_{param}=8, N=6(8)+1=49=(7)(7)).
- If x,y < 0 (let x’=-x, y’=-y with x’,y’>0), then k_{param} = 6x’y’ – x’ – y’.
  N = (6x+1)(6y+1) = (-6x’+1)(-6y’+1) = (6x’-1)(6y’-1).
  This N is also a positive composite \equiv 1 \pmod 6. k_{param} can be positive (e.g., x=-1,y=-1 \implies k_{param}=4, N=6(4)+1=25=(-5)(-5)) or negative or zero (but we excluded x,y=0).
Filtering 6k_{orig}-1 numbers (using negative k_{param} values):
The request is to use negative values of k_{param} = 6xy+x+y to identify composites of the form 6k_{orig}-1.
Let k_{param} < 0. Set k_{param} = -K_0 where K_0 > 0 (so K_0 = -(6xy+x+y)).
The number to test for compositeness is N = 6K_0-1.
From the framework, we know 6k_{param}+1 = (6x+1)(6y+1).
Substituting k_{param} = -K_0:
6(-K_0)+1 = (6x+1)(6y+1)
1-6K_0 = (6x+1)(6y+1)
Since K_0 > 0, 1-6K_0 must be negative. This implies that one of (6x+1) and (6y+1) is positive and the other is negative (which happens if x and y have opposite signs and are non-zero).
Then, N = 6K_0-1 = -(1-6K_0) = -((6x+1)(6y+1)) = |(6x+1)(6y+1)|.
This N is positive and \equiv -1 \pmod 6 if (6x+1)(6y+1) is \equiv 1 \pmod 6 and negative. (Actually, |(6x+1)(6y+1)| will be \equiv \pm 1 \pmod 6).
Since (6x+1)(6y+1) is \equiv 1 \pmod 6 when x,y are such that factors are \equiv \pm 1 \pmod 6, its negative is \equiv -1 \pmod 6.
So, N = |(6x+1)(6y+1)| is composite and \equiv -1 \pmod 6.
The k_{orig} for this number is K_0 = -k_{param} = -(6xy+x+y).
(e.g., x=1,y=-2 \implies k_{param} = -13. K_0 = 13. N=6(13)-1=77$. From the formula|(6(1)+1)(6(-2)+1)| = |(7)(-11)| = |-77| = 77`).

Thus, for k_{param} = 6xy+x+y (with x,y \in \mathbb{Z}\setminus\{0\}):

If k_{orig} = k_{param} and k_{param} > 0$, then6k_{orig}+1` is composite.
If k_{orig} = -k_{param} and k_{param} < 0 (so -k_{param}>0), then 6k_{orig}-1 is composite.

This means all k_{orig} values derived from |6xy+x+y| (for x,y \in \mathbb{Z}\setminus\{0\}), when used appropriately for 6k+1 or 6k-1 forms, yield composites.

4. Discussion: K-Filtering, Prime Distribution, and Number Theoretic Conjectures

The K-Filtering Model, by defining primality through index set exclusion, provides a unique lens through which to view the distribution of prime numbers and connect with established number theoretic concepts like the Prime Number Theorem (PNT) and conjectures regarding prime constellations.

4.1 Alignment with the Prime Number Theorem (PNT)

The PNT states that the number of primes less than or equal to N, denoted π(N), is asymptotically N/ln(N). This implies that the “probability” of a randomly chosen integer N being prime is approximately 1/ln(N). The K-Filtering model reflects this probabilistic insight through the density of the prime-generating index set S_p = U_k \ S_c.
K-Filtering Case 1 (n=k+1, S_c from k_c=xy+x+y):
The universe of indices U_k corresponds to integers k=n-1. The density of k \in S_p (meaning k \notin S_c) should align with the density of primes among n. Thus, the “probability” that a given k is in S_p is approximately 1/ln(k+1) \approx 1/ln(n), consistent with the PNT for all integers n \ge 2.
K-Filtering Case 2 (n=2k+1, S_c from k_c=2xy+x+y):
Here, n is restricted to odd numbers. The index k is (n-1)/2. The PNT implies that the density of primes among odd numbers n is roughly 2/ln(n).
For a given index k, the corresponding n = 2k+1. The “probability” that this n is prime is \approx 2/ln(n) = 2/ln(2k+1).
Therefore, the “probability” that an index k from this case is in S_p (i.e., k \notin S_c) is \approx 2/ln(2k+1). The K-Filtering model for odd numbers correctly reflects this higher density of prime-generating indices within its specific domain.
K-Filtering Case 3 (n=|6k’+1|, S_c from k’_c=6xy+x-y or k’_c=6xy+x+y related forms):
This case focuses on numbers n \equiv \pm 1 \pmod 6, which include all primes greater than 3. These numbers constitute roughly 1/3 of all integers. The PNT, when applied to this subset, implies that the density of primes within this subset is approximately three times the overall density, i.e., 3/ln(n).
Let k’ be the unique index for n=|6k’+1| as defined in the model. The set S_c is generated by the corresponding k’_c Diophantine equation (e.g., 6xy-x-y, 6xy+x-y, or 6xy+x+y as used here for sign clarity).
An index k’ is in S_p if k’ \notin S_c. The density of these prime-generating indices k’ within U_{k’} (non-zero integers) would reflect the 3/ln(n) behavior for the corresponding n=|6k’+1|.
The K-Filtering model, therefore, doesn’t just provide a deterministic test but also offers a structural basis for understanding why prime densities vary across different arithmetic subsequences of integers, consistent with the PNT.
Case 3 (using the framework 6k_{orig} \pm 1):
This framework focuses on numbers n \equiv \pm 1 \pmod 6, which include all primes greater than 3. These numbers constitute roughly 2/6 = 1/3 of all integers. The PNT, when applied to this subset, implies that the density of primes within this subset is approximately three times the overall density, i.e., 3/ln(n).
Let’s consider the k-parameterization k_{param} = 6xy+x+y (as discussed in the response prior to this full document draft, where c=1 was chosen for the ak+c^2 form, leading to N=6k_{param}+1=(6x+1)(6y+1)).
- If k_{orig} = k_{param} > 0, then N=6k_{orig}+1 is composite.
- If k_{orig} = -k_{param} where k_{param} < 0 (so k_{orig} > 0), then N=6k_{orig}-1 is composite (since 6k_{param}+1 = (6x+1)(6y+1) implies 6(-k_{orig})+1 is composite, so 1-6k_{orig} is composite, meaning 6k_{orig}-1 is also composite if one ensures appropriate factors).
  The set S_c for this 6k_{orig} \pm 1 scenario would be formed by values |k_{param}| = |6xy+x+y| (where x,y \in \mathbb{Z}\setminus\{0\}). An index k_{orig} is in S_p if k_{orig} \notin S_c. The density of these prime-generating indices k_{orig} within the universe of all positive integers (representing magnitudes for 6k\pm1 forms) would reflect the 3/ln(n) behavior for the corresponding n.
  The K-Filtering model, therefore, doesn’t just provide a deterministic test but also offers a structural basis for understanding why prime densities vary across different arithmetic subsequences of integers, consistent with the PNT.

4.2 The Algebraic Foundation of the Prime Number Theorem

The K-Filtering framework reveals that the Prime Number Theorem (PNT) is not merely a statistical observation about the distribution of primes, but a direct consequence of the algebraic structure underlying integer factorization. Specifically, this model shows that compositeness can be expressed through deterministic Diophantine forms, while primality arises precisely where such representations fail.

In Case 1 of the framework, we consider numbers of the form n = k + 1. Using the parameterization k = xy + x + y, we obtain n = (x + 1)(y + 1), which is guaranteed to be composite whenever x and y are positive integers. Thus, for every composite number n > 1, there exists at least one pair (x, y) such that n can be expressed as (x + 1)(y + 1). Equivalently, the index k = n – 1 lies in the set Sc of indices expressible as xy + x + y.

In contrast, primes correspond to those values of k = p – 1 that cannot be written in this form. That is, primes arise precisely at the points where the algebraic structure of bilinear parameterizations fails to cover the integers. The set Sp of prime-generating indices is therefore the complement of Sc within the index universe Uk.

The density of Sp — that is, the proportion of integers up to N that do not fall within Sc — aligns with the classical asymptotic density of primes given by the Prime Number Theorem. Specifically, as N becomes large, the size of Sp grows approximately like N divided by ln(N). This means that the familiar 1 / ln(N) density of primes is not an empirical artifact, but a structural outcome of the algebraic limitations of composite-generating forms.

What traditional analytic number theory explains through limits, integrals, and complex functions, the K-Filtering model reveals through constructive algebra. The “randomness” of primes is not truly random; it is the structural certainty that some integers — indeed, infinitely many — cannot be expressed as the product of two linear factors of the form (ax + c)(ay + c). These integers are prime because no bilinear structure captures them.

In this light, the Prime Number Theorem becomes inevitable. It holds because bilinear parameterizations, while dense, are incomplete. Their failure to fully populate the integers creates the residual set where primes must exist. The K-Filtering approach thus transforms the PNT from a probabilistic statement into an algebraic inevitability — a direct reflection of the constraints inherent in the structure of integer factorization.

4.3 Insights into Twin Primes (Derived from Case 3)

Twin primes (p, p+2), for p > 3, are always of the form (6k-1, 6k+1) for some integer k \ge 1.
Using the Case 3 definition (n=|6k+1| where k for n=6k+1 and -k for n=6k-1; S_c generated by k_c=6xy+x+y):

For p = 6k+1 to be prime, its index K_1 = m must not be in S_c (i.e., m \notin S_c).
For p+2 = 6k-1 to be prime, its index K_2 = -m must not be in S_c (i.e., -m \notin S_c).

Thus, a pair (6k-1, 6k+1) constitutes a twin prime pair if and only if both m \notin S_c AND -m \notin S_c.

The Hardy-Littlewood conjecture for twin primes suggests that π_2(N) \sim 2C_2 \int_2^N dt/(\ln t)^2, where C_2 is the twin prime constant. This implies that the “probability” of two numbers n and n+2 (within the appropriate form) both being prime is roughly (C’/\ln n)^2.
In the K-Filtering model:

The “probability” of m \notin S_c is roughly c_0/\ln(6m).
The “probability” of -m \notin S_c is also roughly c_0/\ln(6m).

If these events were independent (which they are not, but this is a common heuristic starting point), the probability of both occurring would be (c_0/\ln(6k))^2. The K-Filtering framework, by defining S_c through the Diophantine equation k_c=6xy+x+y (or alternatively k_c=6xy-x-y or k_c=6xy+x-y), provides a concrete mechanism for why these “probabilities” are correlated. The conditions k = 6xy+x+y and -k = 6x’y’+x’+y’ are linked if, for example, specific choices of (x,y) for k imply constraints on (x’,y’) for -k, or vice-versa. The structure of S_c generated by k_c=6xy+x-y implicitly contains the sieve-theoretic interactions that lead to the twin prime constant C_2 appearing in the conjecture. Further analysis of the properties of the set S_c and how it populates around m and -m could offer structural insights into the distribution of twin primes.

The K-Filtering k recasts the problem into one of understanding simultaneous exclusion from a deterministically generated set S_c. The density and structure of S_c (and its complement S_p) are directly tied to the observed distribution of primes and prime constellations.

5. Conclusion

This generalized model provides a unified, algebraic sieve for identifying composite numbers within arithmetic progressions. At its core, it uses structured parameterizations of the index variable k to determine when a number N = ak + b can be guaranteed to factor.

The method follows these basic steps:

Start with a desired arithmetic form, such as N = ak_orig + b.
Choose a k-structure based on the framework, ideally the simplified form k = axy + c(x + y) when b = c².
If the original index k_orig can be expressed as this k_param, then N is composite, since it factors as (ax + c)(ay + c).
To ensure nontrivial compositeness, both factors must be greater than 1 in absolute value. This is typically achieved by restricting x and y to nonzero (often positive) integers.

The model:

Provides a general algebraic structure for detecting composite numbers.
Simplifies significantly when c = d, eliminating constraints and producing a clean form that always yields composites.
Covers classical expressions such as n² + 1 and 2k + 1 directly.
Adapts to prime-dense forms like 6k ± 1 using a single parameterization (k_param = 6xy + x + y), with sign and symmetry considerations used to handle both +1 and -1 cases.

Beyond its constructive use, the model reveals something deeper: an algebraic basis for the distribution of prime numbers. The Prime Number Theorem (PNT), often treated as an analytic result, is here shown to follow naturally from the structure of integer factorization. The set of composite-producing indices is large and structured, but incomplete. The integers that escape this structure — those that cannot be expressed in the k_param form — are precisely the ones that remain prime.

This reframes the PNT as a structural consequence of arithmetic itself: primes exist where algebraic compositeness fails to reach. Their apparent randomness is, in truth, the deterministic result of the limitations of bilinear representation. The classic 1 / ln(N) density of primes arises because the failure rate of composite forms follows that same asymptotic behavior.

The model also offers a fresh perspective on twin primes. A twin prime pair (such as 6k – 1 and 6k + 1) corresponds to the simultaneous absence of both k and -k from the composite-generating set. This reflects a structured, symmetric sieve that naturally accounts for known prime constellations — without invoking probabilistic assumptions.

Finally, the model opens the door to efficient computation. By precomputing the composite-generating set S_c for a given range of indices, primality tests for corresponding values of N reduce to a simple membership check. Since S_c can be generated in a highly-parallelizable way and stored in advance, lookups become constant-time operations after set generation. This means that primality detection over a fixed domain can, in principle, achieve O(1) time complexity after (distributed) linear-time setup — a compelling advantage for both theoretical and computational applications.

In the end, the k-filtering model does more than classify composites. It offers a rigorous algebraic explanation for why primes appear where they do. It transforms primality from a property we test for, into a structural phenomenon — the inevitable result of what compositeness cannot explain.