About a virtual subset

Two constructed prime number subsets (called prime brother & sisters and prime cousins) lead to a third one (called isolated primes) so that all three disjoint subsets together generate the prime number set. It should be suggested how the subset of isolated primes give a new approach to expand the set theory by using virtual subsets.

💡 Research Summary

**
The paper proposes a novel partition of the set of all prime numbers P into three mutually disjoint subsets: “brothers & sisters” (B), “cousins” (C), and “isolated primes” (I). The motivation is to generalize the well‑known twin‑prime concept and to use the leftover primes to define a new kind of set, called a “virtual subset,” which the author claims could enlarge classical set theory.

Section I – Brothers & Sisters (B).
Two consecutive primes p_i and p_{i+1} are called brothers & sisters if their difference d = p_{i+1} – p_i equals a power of two, i.e. d = 2ⁿ for some non‑negative integer n. The author groups all such pairs into blocks B₁, B₂, … according to gaps that are not powers of two separating the blocks. Examples are given (B₁ = {2,3,5,7,11,13,17,19,23}, B₂ = {29,31}, B₃ = {37,41,43,47}, etc.). The paper assumes, without proof, that infinitely many such blocks exist.

Section II – Other Primes (O) and Cousins (C).
The complement of B in P is denoted O = P \ B. O is again split into blocks O₁, O₂, … (e.g. O₁ = {53}, O₂ = {157}, O₃ = {173}, …). Within O, a pair (p,q) with p > q is called a cousin if p − q = 2ⁿ. The author lists several examples (173 − 157 = 16 = 2⁴, 557 − 541 = 16, etc.) and defines C as the set of all such cousin pairs. By construction C ∩ B = ∅.

Section III – Isolated Primes (I) and Relative Primes (R).
A “relative prime” is any prime that has at least one partner at distance 2ⁿ, which simply reunites B and C. The author then defines isolated primes as those primes that belong to neither B nor C: I = P \ (B ∪ C). The existence of such primes is not ruled out; the paper attempts to exhibit candidates (e.g. 53) and discusses the difficulty of proving isolation because one must verify an infinite number of distance conditions. A “candidate” notion is introduced, requiring that no smaller prime in O is at distance 2ⁿ and that all numbers of the form p + 2^i (1 ≤ i ≤ p) are either composite or belong to B.

Section IV – Infinite‑ness Indicator ψ and Combination Function κ.
The author defines a function ψ that assigns 1 to infinite sets, 0 to finite sets, and –1 to “undefined.” By evaluating ψ on B, C, and I, twelve possible triples (ψ(B), ψ(C), ψ(I)) are listed, each denoted κ₁ … κ₁₂. He then attempts to eliminate most combinations using heuristic arguments based on the Prime Number Theorem, Dirichlet’s theorem on arithmetic progressions, the twin‑prime constant, and the bounded‑gap results of Goldston‑Pintz‑Yıldırım. The conclusion is that ψ(C)=1, ψ(B)=1, and ψ(I)<1, i.e. B and C are infinite while I is either finite or empty, but the exact status of I remains undecidable within the paper’s framework.

Section V – Virtual Subsets.
A “virtual subset” V of a non‑empty set W is defined by three conditions: (a) V ≠ W, (b) W can be written as a disjoint union of finitely many ordinary subsets U₁,…,Uₙ together with V, and (c) analogous to Gödel’s incompleteness, it is undecidable whether V is empty or non‑empty. The isolated‑prime set I is presented as a concrete example of such a virtual subset. The author suggests that virtual subsets could constitute a new category in set theory, possibly containing a finite but non‑countable number of elements.

Appendix – Connection to Wieferich Primes.
Only two Wieferich primes (p such that 2^{p‑1} ≡ 1 (mod p²)) are known: 1093 (which lies in B) and 3511 (which lies in C). The paper conjectures that any further Wieferich prime must belong to I, or else no further Wieferich primes exist if I is empty.

Critical Evaluation.
The paper introduces an interesting linguistic generalization of twin primes by allowing gaps of any power of two, but the resulting “brother & sister” sets are extremely sparse and the claim of infinitely many such blocks lacks proof. The “cousin” construction depends on the complement O, yet the paper does not establish that O contains infinitely many pairs at power‑of‑two distances; the examples are isolated and the density arguments are heuristic. The definition of isolated primes is essentially “primes that have no partner at distance 2ⁿ in either B or C,” which is a well‑posed notion, but the paper provides no method to decide membership beyond exhaustive search, and the claim that its existence is undecidable is not substantiated; Gödel’s incompleteness does not directly apply to this concrete arithmetic question.

The ψ/κ framework attempts to formalize the infinite/finite status of the three subsets, but the arguments rely heavily on unproven conjectures (e.g., Elliott–Halberstam, bounded‑gap conjecture) and on informal probabilistic reasoning. Consequently, the deductions that ψ(B)=1, ψ(C)=1, ψ(I)<1 remain speculative. Moreover, the “virtual subset” concept is essentially a restatement of a set whose emptiness is independent of a given axiomatic system; such phenomena are already known (e.g., the set of counterexamples to Goldbach’s conjecture). The paper does not demonstrate that I possesses any novel logical properties beyond those already captured by existing independence results.

In summary, while the paper offers a creative classification of primes and an attempt to link arithmetic properties with logical undecidability, its definitions are loosely formulated, many central claims are unproved, and the proposed “virtual subset” does not constitute a genuinely new object in set theory. Further rigorous development would be required to turn the ideas into substantive contributions.