pub.en.md

Distribution Patterns of Prime Numbers and a Proof of the Infinitude of Twin Primes

Author: Gene Kong

Abstract: This paper builds upon the Eratosthenes sieve to propose a periodic model of prime number distribution based on modular structures. By constructing residue sets after sieving with primes such as 2, 3, and 5, we reveal the periodic distribution characteristics of "sieve remainders" and demonstrate that such properties persist in subsequent sieving processes.
Furthermore, using proof by contradiction, we analyze the possibility of the infinitude of twin primes. We argue that the irremovable nature of candidate twin prime characteristics provides theoretical support for the infinite existence of twin primes.

Keywords: Prime distribution; Twin prime conjecture; Sieve method; Modular structure; Property preservation; Sieve remainders; Candidate twin primes

1. Introduction

The twin prime conjecture (i.e., whether there exist infinitely many prime pairs of the form $(p, p+2)$ ) remains one of the central unsolved problems in number theory. A groundbreaking advance was achieved by Yitang Zhang (2013) [@zhang2014bounded], who employed an innovative GPY sieve method to prove the existence of infinitely many prime pairs with gaps smaller than 70 million. This result is widely regarded as a landmark achievement in 21st-century number theory.

Subsequently, Castryck et al. (2014) [@castryck2014new] focused on the intersection of computational number theory and sieve theory. They proposed a novel sieve strategy rooted in algebraic number theory and computational optimization, leveraging tools such as local rings and modular forms to reanalyze the distribution properties of sieve remainders.

Building upon these foundations, Maynard (2015) [@maynard2015small] independently developed an enhanced multidimensional sieve framework. By optimizing weight functions, he significantly reduced the minimal gap between prime pairs to 600. Later, the Polymath Project synthesized the methodologies of Maynard and Castryck et al., further refining the bound to 246. These collective efforts underscore the formidable potential of sieve methods in studying prime distribution.

As a pivotal tool for investigating prime distribution, the refinement and extension of sieve methods have long been a focal point in number theory. For instance, the Eratosthenes sieve—the most classical prime-sieving algorithm—constructs prime sets by iteratively eliminating composite numbers. In recent years, sieve optimization techniques incorporating modular structures have garnered substantial attention, not only improving algorithmic efficiency but also offering fresh perspectives for deciphering the patterns underlying prime distribution.

The study of sieve remainders holds critical importance in sieve optimization. Sieve remainders, defined as numbers not eliminated during sieving, exhibit distribution characteristics that directly influence the performance and applicability of sieve methods. A deeper understanding of their periodicity and distribution patterns not only facilitates the design of more efficient algorithms but also provides theoretical insights into longstanding problems such as the twin prime conjecture.

This paper proposes a modular structure-based model for prime distribution, extending the principles of the Eratosthenes sieve. By analyzing the periodic distribution properties of sieve remainders, we aim to establish preservation rules for candidate twin prime characteristics and further explore the possibility of the infinitude of twin primes. Our research contributes novel theoretical support to the twin prime conjecture and offers innovative ideas for the optimization of sieve methodologies.

2. Definitions and Notations

Definition 2.1 (Sieve Residues): For a given sequence of primes $p_1, p_2, \ldots, p_k$ , natural numbers not eliminated by this sequence are termed sieve residues, denoted as $\mathcal{S}(p_1, p_2, \ldots, p_k)$ .
Definition 2.2 (Candidate Twin Primes): If adjacent elements $(q, q+2)$ exist in the sieve residue set, they are termed candidate twin primes.

Note: Sieve residues may be genuine primes or unsieved composites, but their distribution follows periodic patterns dictated by the sieve method.

3. Analysis of Prime Sieving Patterns

3.1 Initial Sieving Stages

• Prime 2: After eliminating all even numbers, the sieve residues form the odd number set ${2m+1 \mid m \in \mathbb{Z}}$ (see Appendix Figure 1).
• Prime 3: Further eliminating multiples of 3, the residues become ${6m \pm 1 \mid m \in \mathbb{Z}}$ (see Appendix Figure 2). At this stage, the probability of candidate twin primes with a gap of 2 is $\frac{1}{6}$.

3.2 Advanced Sieving Stages

After introducing a prime $p_k$, the sieve residue set can be expressed as:

$$ \mathcal{S}(2,3,\ldots,p_k) = \bigcup_{i} \begin{Bmatrix} \left( \prod_{j=1}^k p_j \cdot m \pm a_i \right) \mid m \in \mathbb{Z} \end{Bmatrix}, $$

where $a_i$ are offsets coprime to $\prod_{j=1}^k p_j$. For example, introducing 5 yields the residue set ${30m \pm 1, \pm 7, \pm 11, \pm 13}$ (see Appendix Figure 3).

The sieve residue set exhibits a cyclic period of $\prod_{j=1}^{k} p_j$ after introducing $p_k$, where $p_1 \ldots p_j$ denotes the consecutive prime sequence from 2 to the $k$-th prime. Each new prime extends the period while preserving residue patterns.

Theorem 3.1 (Preservation of Candidate Twin Prime Properties): For any prime sequence $p_1, p_2, \ldots, p_k$, the distribution properties of sieve residues generated by the sieve method satisfy the following when introducing a new prime $p_{k+1}$ ($p_{k+1} > 3$):

• Retention of ±1 Candidate Pairs: Candidate twin prime pairs of the form $\prod_{j=1}^k p_j \cdot m \pm 1$ are retained unless $\prod_{j=1}^k p_j \cdot m \equiv \pm 1 \pmod{p_{k+1}}$ (specific values depend on modular arithmetic). The next candidate pair $\prod_{j=1}^{k+1} p_j \cdot m \pm 1$ will definitely be preserved;
• Proportion Retained Per Cycle: A fraction $\frac{p_{k+1}-2}{p_{k+1}}$ of incremental candidate twin prime pairs remain unaffected by the new prime;
• Incremental Count Per Cycle: Within the period $\prod_{j=1}^{k+1} p_j$, at least $p_{k+1}-2$ incremental candidate twin prime pairs are preserved.

Proof: Consider candidate twin prime pairs $\prod_{j=1}^k p_j \cdot m \pm 1$. Since $p_{k+1}$ is coprime to $\prod_{j=1}^k p_j$, by modular arithmetic:

$$ \prod_{j=1}^k p_j \cdot m \pm 1 \not\equiv 0 \pmod{p_{k+1}} \quad \text{unless} \quad m \equiv \pm \left( \prod_{j=1}^k p_j \right)^{-1} \pmod{p_{k+1}}. $$

Thus, only when $m$ satisfies the above congruence are the candidate pairs eliminated. The remaining $\frac{p_{k+1}-2}{p_{k+1}}$ proportion of pairs persists, corresponding to $p_{k+1}-2$ preserved incremental pairs within the extended period $\prod_{j=1}^{k+1} p_j$.

4. Proof by Contradiction for the Infinitude of Twin Primes

Assume there exists a largest twin prime pair $(P, P+2)$. Then, for all $n > P$, the sieve residue set $\mathcal{S}(2,3,\ldots,p_k)$ must not contain any candidate twin prime pairs $(q, q+2)$ that are not covered by actual primes. However, Theorem 3.1 ensures that such candidate pairs persist with a positive proportion in every sieving stage and their count within each period $\prod_{j=1}^{k} p_j$ is no less than $p_k - 2$. This contradicts the assumption of a "largest twin prime pair."

Conclusion 4.1: Within the cyclic period $\prod_{j=1}^{k} p_j$, the incremental preservation of candidate twin prime pairs is always no less than $p_k - 2$. Therefore, twin primes are infinite.

5. Discussion and Future Directions

1. Extensions: Investigating the infinitude of prime pairs with generalized gaps (e.g., 6, 12) or combining other models to explore additional preservation laws for candidate twin primes.

6. Conclusion

By constructing a periodic model of prime sieving, this paper demonstrates the persistent existence of candidate twin primes and infers the infinitude of twin primes via proof by contradiction.

Appendix

Figures illustrate sieved primes (brown), periodic sieve residue sets (boxed), and unsieved composites (dark green).

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Figure 1: Residue distribution after sieving multiples of 2

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Figure 2: Residue distribution after sieving multiples of 2 and 3

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Figure 3: Residue distribution after sieving multiples of 2, 3, and 5

Additional combinations may be analyzed using the tools provided in this document.

Code Availability Statement

The manuscript, experimental tools, and data are hosted on GitHub:

• Repository: https://github.com/GeneKong/primes
• Version: v1.0.0 (Release Date: March 2025)
• DOI: 10.5281/zenodo.15071343

References