WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 21, 2022
A Proof of Goldbach Conjecture by Mirror Prime Decomposition
Author:
Abstract: This work presents a formal proof of Goldbach conjecture based on a novel theory of Mirror-Prime
Decomposition (MPD) for arbitrary even integers. A new concept of mirror primes $$ \Bbb{P}_{μ}\subset \Bbb{P} \times \Bbb{P} $$ is introduced
as a set of pairs of primes that are symmetrically adjacent to any pivotal even number $$ n_{e} \in \Bbb{N}_{e} \subset \Bbb{N} $$ on both sides
in finite distance k bounded by 1 ≤ k ≤ (ne/2) − 2. As a counterpart of the Euclidean Fundamental Theorem of
Arithmetic for natural number factorization, the MPD theory enables arbitrary even number decomposition by
mirror primes. MPD paves a way to prove the Goldbach conjecture, i.e., where denoted by the big-R calculus
for representing recursive structures and manipulating recursive functions. An algorithm of Goldbach conjecture
testing is designed for demonstrating the formal proof of the Goldbach theorem. i.e., $$\forall 4 \leq \frac{n_{e}}{2}<\infty$$,
$$n_{e}= f(p^\frac{n_{e}}{2}_{\mathrm{μ}^{-}}, p^\frac{n_{e}}{2}_{\mathrm{μ}^{+}} ) = R_{k=1}^{(\frac{n_{e}}{2}-2)}(p^\frac{n_{e}}{2}_{\mathrm{μ}^{-}}+ p^\frac{n_{e}}{2}_{\mathrm{μ}^{+}} )$$,
where
$$(p^\frac{n_{e}}{2}_{\mathrm{μ}^{-}}= \frac{n_{e}}{2}-k, p^\frac{n_{e}}{2}_{\mathrm{μ}^{+}}+k) \in \Bbb{P}_{μ}$$
denoted by the big-R calculus for representing recursive structures and manipulating recursive functions. An
algorithm of Goldbach conjecture testing is designed for demonstrating the formal proof of the Goldbach
theorem.
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Keywords: Number theory, Goldbach conjecture, proof, mirror primes, mirror prime decomposition, recursive
sequence, numerical algorithm
Pages: 563-571
DOI: 10.37394/23206.2022.21.63