1. Defining the Domain and the Form
We begin by considering the set of non-zero integers, A = Z \ {0}, which will serve as the domain for our indices k.
We focus on numbers n generated by the function f(k) = |6k-1| for k ∈ A. It is a well-established property that any prime number p greater than 3 must satisfy p ≡ ±1 (mod 6).
The form n = |6k-1| systematically generates the absolute values of all integers congruent to ±1 (mod 6) (excluding ±1 itself, as k ≠ 0). (The choice of 6k+1 or 6k-1 is trivial, but the selection of composites based on the form is not trivial. The following focuses specifically on |6k-1|.)
Consequently, the set of numbers generated by f(k) for k ∈ A contains all prime numbers greater than 3, alongside composite numbers also satisfying the ±1 (mod 6) condition (e.g., 25, 35, 49, 55…). The entire set A thus represents the indices of all candidates for being primes greater than 3, based solely on the |6k-1| form.
2. Establishing the Rule for Compositeness via Index Generation
The core insight is the establishment of a specific rule that governs the indices k corresponding to composite numbers within the |6k-1| sequence. Through algebraic manipulation of the factors of composite numbers of the form 6k ± 1, we derived the following rigorous equivalence:
An integer n = |6k-1| (with k ∈ A, n ≥ 5) is composite if and only if its index k can be expressed as k = 6xy + x – y for some non-zero integers x, y (i.e., x, y ∈ A).
This equivalence is crucial. It provides a constructive definition for the indices of composite numbers within our sequence. We can define the set S_3 explicitly based on this rule:
S_3 = { 6xy + x – y | x ∈ A, y ∈ A }
The set S_3 represents the “positive space” of composite indices. Any index k belonging to S_3 definitively corresponds to a composite number n = |6k-1|. The polynomial g(x, y) = 6xy + x – y acts as the generator for this set.
3. The Inferential Problem: Identifying Primes
We now face the central problem: given an index k ∈ A, how do we determine if the corresponding n = |6k-1| is prime? We know k represents a candidate. We also have a definitive rule (k ∈ S_3) that signals compositeness. How do we leverage this to identify primes?
4. The Abductive Inference from Exclusion
Direct primality tests evaluate n. Sieves eliminate multiples. This method instead focuses on the index k and its relationship to the constructively defined set S_3. The reasoning process for determining primality becomes an instance of Peircean abduction:
- Observation: We take an index k from the set of candidates A.
- Test: We check if this observed k belongs to the set S_3 (the set of composite indices). This involves checking if k can be represented as 6xy + x – y for some x, y ∈ A.
- Two Possible Outcomes:
- Outcome 1: k ∈ S_3. The index k fits the established rule for compositeness. By deductive reasoning based on the proven equivalence, we conclude that n = |6k-1| is composite.
- Outcome 2: k ∉ S_3. This is the surprising or unexplained observation if we were to assume n might be composite. The index k fails to conform to the necessary condition (k ∈ S_3) that must hold if n were composite.
- Abductive Step: The observation k ∉ S_3 demands an explanation. Given the “if and only if” nature of the equivalence, the only possible explanation for k not being in the set S_3 is that the premise leading to that condition – namely, that n = |6k-1| is composite – must be false. Therefore, we infer, as the best and necessary explanation, that n = |6k-1| must be prime.
This inference is abductive because it reasons from an observed consequence (or lack thereof: k ∉ S_3) back to the most plausible underlying state (primality of n). It’s an inference to the best explanation for why k does not possess the characteristic property of composite indices.
5. Primes in the “Subtractive Space”
The formalization of this inference lies in set theory. The entire space of candidate indices is A. The subspace of indices corresponding to known composites is S_3. The act of identifying primes becomes equivalent to performing the set subtraction:
K_prime = A \ S_3
This explicitly defines the set of prime indices K_prime as everything in the candidate space A except for the elements known to be composite indices (S_3). The primes are thus located in this “subtractive space” or “negative space” – a space defined not by its own positive generating rule within this framework, but by what it excludes. We identify primes by recognizing their indices lack the signature (∈ S_3) associated with compositeness.
Theorem Restated: Let A = Z \ {0} and S_3 = { 6xy + x – y | x ∈ A, y ∈ A }. The set K_prime = { k ∈ A | |6k – 1| \ { is prime} } is exactly A \ S_3.
Conclusion
This approach provides a distinct perspective on prime identification for numbers n = |6k-1|. It does not generate primes directly but instead constructively generates the indices k corresponding to all composite numbers within this form via the set S_3.
Primality is then inferred abductively: an index k is recognized as corresponding to a prime n = |6k-1| precisely because it is absent from the set S_3.
The primes occupy the logical space remaining after the identifiable composite indices are excluded from the initial set of candidates.
This reliance on inference from exclusion, facilitated by the structural relationship between n and k captured by the polynomial g(x,y), exemplifies the power of abduction in mathematical reasoning, consistent with Peirce’s emphasis on how notation and structure guide discovery.