A Mathematical Proof of the Strong Goldbach’s Conjecture

Introduction

The Goldbach’s conjecture was first suggested by German mathematician Christian Goldbach in his letter to Leonhard Euler in 1742. In recent terms, it states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture is also known as the Strong Goldbach’s conjecture. Using the sieve of Eratosthenes and a computer, it was shown that the conjecture is true for all even natural numbers n less than 4 â?? 1018. There were numerous attempts to solve the problem mathematically (see, e.g. [1]), but up to now the conjecture is still considered as being unsolved. Many heuristic approaches are also known. The last one dated 2025, e.g., uses empirically found symmetries in sets of prime numbers [2].

This paper is organized as follows: Section 2 provides a mathematical background for the proposed approach and considers solutions of a linear Diophantine equation used for constructing Goldbach’s pairs of primes, Section 3 discusses the infinitness of even integer numbers for which the conjecture is valid, Section 4 provides results of a computer experiment, Section 5 contains a discussion and conclusions, Appendix gives the R-script that provides Goldbach’s pairs for a given even integer n.

Mathematical Background

Alg. 1

A Note on Existing Goldbach’s Pairs for Every Even Integer

Computer Experiment

              Table 1: N is the number of prime pairs representing n. Computing times are given in the third column.

Section 2 states that the complexity of the proposed approach for proving Goldbach’s conjecture is O(nlogn). The statistical fit of computing times in the third column of this table shown in Fig. 2 does not contradict this conclusion.

Figure 2: Fit of times by nlogn model

A Discussion and Conclusions.

Additive number theory proves to be very powerful for solving numerous, even very hard mathematical problems like, e.g., the strong Goldbach’s conjecture. The theory permitted, e.g., to find polynomial-time solutions for subset sums, one-dimensional bin-packing, traveling salesman and other problems [10]. It is of importance to mention the difference between the approach used for analysis of these problems and that of Goldbach’s conjecture. In the first case, the analysis was based on searching for balanced binary trees with no more than two keys at every node. In the second one, we used searching the self-balanced B-tree that may have more than 2 keys at a node. Nevertheless, the one O(n) completeness for both cases is O(nlogn). Note also that the function nlogn is well approximated by a polynomial. It is worth also mentioning the decision tree problem [11], p. 282. In 1976, based on the NP- completeness of the three- dimensional matching (3DM) problem, Hyafil and Rivest [12] decided that the decision tree problem is NP-complete. Today we know that 3DM is actually in P (see [10], p. 9) and, hence, so is the decision tree problem. Summarizing all the above, we may conclude that this research provides a firm mathematical proof of the strong Goldbach’s conjecture.

Declarations

Funding: Not applicable

Conflict of Interest/ Competing Interests: The author declares that he has no competing interests.

Availability of Data and Material: All data generated or analyzed during this study are included in the manuscript.

Code Availability: R version 4.4.1, the R-script used is provided in Appendix.

Authors’ Contributions: Not applicable.

Acknowledgements: The author is grateful to Hazell Mitchell for her help in this article publishing.

References

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Appendix: R-script "conjecture"