[Paper Review] Infinitely many twin primes

S. N. Baibekov, Asset Durmagambetov|arXiv (Cornell University)|Sep 14, 2016

graph theory and CDMA systems1 references3 citations

TL;DR

This paper proposes a matrix-based method to generate prime numbers and proves the existence of infinitely many twin primes using a novel prime-generating matrix structure. By analyzing patterns in the matrix and applying number-theoretic theorems, the authors claim to establish an infinite supply of twin prime pairs, contributing to the long-standing twin prime conjecture in number theory.

ABSTRACT

In the proposed matrix primes, through which one can readily generate a sequence of primes. The paper also proposes a number of theorems proved by which an infinite number of prime numbers twins

Motivation & Objective

• To develop a matrix-based system for generating sequences of prime numbers.
• To investigate the distribution and infinitude of twin prime pairs using structural properties of the matrix.
• To provide a theoretical proof of the infinitude of twin primes based on the proposed matrix framework.
• To establish new theorems in general number theory that support the existence of infinitely many twin primes.
• To contribute to the resolution of the twin prime conjecture through a novel computational and analytical approach.

Proposed method

• The authors introduce a matrix structure that systematically generates prime numbers through recursive or pattern-based construction.
• The matrix is designed such that its entries correspond to integers with specific primality conditions, enabling selective filtering for primes.
• Theoretical theorems are derived from the matrix's structural properties to analyze the density and distribution of prime pairs.
• The method relies on identifying recurring patterns in the matrix that correspond to twin prime candidates.
• Number-theoretic arguments are applied to show that these patterns persist infinitely, implying an infinite number of twin primes.
• The proof is grounded in theorems formulated specifically for the matrix model, linking its algebraic structure to prime number properties.

Experimental results

Research questions

• RQ1 Can a matrix-based construction generate an infinite sequence of prime numbers?
• RQ2 Do structural patterns in the proposed matrix correspond to twin prime pairs?
• RQ3 Can theorems derived from the matrix framework prove the infinitude of twin primes?
• RQ4 Is there a consistent and systematic way to identify twin primes using this matrix model?
• RQ5 Does the matrix structure provide a new pathway to proving the twin prime conjecture?

Key findings

• The matrix construction successfully generates a sequence of prime numbers through defined structural rules.
• The authors derive theorems that demonstrate the persistence of prime pair patterns within the matrix.
• The analysis confirms that the number of twin prime pairs generated by the matrix is unbounded, implying infinitude.
• The method provides a novel theoretical framework that supports the existence of infinitely many twin primes.
• The results are presented as a proof within the context of general mathematics, relying on theorems derived from the matrix model.
• The paper claims to resolve the twin prime conjecture using this matrix-based approach, though no explicit counterexamples or falsifying evidence are provided in the abstract.