WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
A Proof of the Twin Prime Conjecture in the Ƥ x Ƥ Space
Author:
Abstract: This work presents a formal proof of the twin prime conjecture based on a novel mathematical
structure of mirror primes $$\Bbb{P}_{μ}\subset \Bbb{P} \times \Bbb{P}$$ in the 2-dimensional space of primes. $$\Bbb{P}_{μ}$$ is an infinite recursive
set of pairs of symmetric primes adjacent to any pivotal even number $$\frac{n_e}{2}\geq 4 \in \Bbb{N}_{e}\subset \Bbb{N}$$ in finite distances $$1\leq k \leq ( \frac{n_e}{2})-2$$. In the framework of $$\Bbb{P}_{μ}$$, the set of twin primes $$\Bbb{P}_{t}$$ is deduced to a subset of the mirror primes $$\Bbb{P}_{t}\subset \Bbb{P}_{μ}$$ where the half interval $$k\equiv1$$ Therefore, the equivalence among sets of $$\Bbb{P}_{t}$$, $$\Bbb{P}_{μ}$$, $$\Bbb{P}$$ and $$\Bbb{N}$$ is established, i.e., $$\lim_{n \to \infty} |\Bbb{P}_{t}| = \lim_{n \to \infty} |\Bbb{P}_{μ}| = \lim_{n \to \infty} |\Bbb{P}| = \lim_{n \to \infty} |\Bbb{N}| = \infty $$ based on Cantor’s principle of infinite countability, such that the twin prime conjecture holds. Experiments based on an Algorithm of Twin-Prime
Sieve (ATPS) are designed to demonstrate and visualize the proven twin prime theorem.
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Keywords: Number theory, twin prime conjecture, mirror primes, recursive sequence, algebraic number
theory, and algorithm
Pages: 585-593
DOI: 10.37394/23206.2022.21.66