## A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers

Daniel Sankei *

Department of Mathematics, Meru University of Science and Technology, P. O. Box 972, Meru, Kenya.

Loyford Njagi

Department of Mathematics, Meru University of Science and Technology, P. O. Box 972, Meru, Kenya.

Josephine Mutembei

Department of Mathematics, Meru University of Science and Technology, P. O. Box 972, Meru, Kenya.

*Author to whom correspondence should be addressed.

### Abstract

The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes P_{i} and P_{j} such that P_{i}+P_{j }cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form E_{ij} = n_{i } + n_{j} + (n_{j }- n_{i})^{n} or all^{n} ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.

Keywords: Goldbach conjecture, goldbach partition, even numbers, odd numbers, prime numbers, natural numbers

**How to Cite**

*Asian Research Journal of Mathematics*,

*20*(4), 8–17. https://doi.org/10.9734/arjom/2024/v20i4793

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