A review on Goldbach conjecture: History, progress, open questions

Abstract

This paper outlines a comprehensive study of the Goldbach Conjecture, the most famous unsolved problem in number theory, and its applicability to optimization, magic squares, and cryptography. From the origin of the conjecture, this paper chronicles the progress from the correspondence between Christian Goldbach and Euler to computer-aided verification in the present times. Substantial progress includes Vinogradov's initial proof of the weak conjecture and Helfgott's final solution, as well as numerical computation by scientists like Sinisalo. After that, the attention turns to practical applications, showing how the conjecture's concept of expressing even numbers as primes in sums can be used to support resource allocation and enhance cryptographic techniques. In particular, the article addresses its usage in magic squares for safe communication, where prime number pairs act as keys for verification of messages. For solving optimization issues in cryptography, two heuristic methods are described to optimize selection of prime pairs to construct even-numbered sums with good performance. The methods balance quality of results and execution time and are therefore appropriate for large applications. Through the juxtaposition of theoretical insight and experiential knowledge, the paper is an effective, logically structured introduction for neophytes and a compelling demonstration of the interdisciplinarity of the Goldbach Conjecture.