Introducing and Applying S.C.E Model Under Dusart's Inequality to Prove Goldbach's Strong Conjecture for 74 Typical Structures out of All 75 Structural Types of Even Number

TL;DR

This paper introduces the S.C.E model and employs Dusart's inequality to provide a relative proof of Goldbach's strong conjecture for 74 out of 75 typical structures of even numbers.

Contribution

The paper presents a new heuristic model for even numbers and applies Dusart's inequality to prove Goldbach's strong conjecture for most typical structures.

Findings

74 out of 75 typical structures satisfy Goldbach's strong conjecture.

The S.C.E model categorizes all even numbers into 75 structures.

Three unproven inequalities are proposed to potentially prove the conjecture for the remaining structure.

Abstract

In this paper, we present a relative proof for Goldbach's strong conjecture. To this end, we first present a heuristic model for representing even numbers called Semi-continuous Model for Even Numbers or briefly S.C.E Model, and then by using this model we categorize all even numbers into 75 distinct typical structures. Also in this direction, we employ this model along with the following inequality to obtain the relative proof \begin{equation} \frac{x}{\ln x} \leq_{x \geq 17} \pi(x) \leq_{x>1} 1.2251 \frac{x}{\ln x} \end{equation} where $\pi(x)$ denotes the number of all primes smaller than and equal to $x$. This inequality is presented by Pierre Dusart in his paper [P. Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2016), No. 1, 227-251]. In fact, by relative proof we mean that 74 typical structures out of 75 ones satisfy Goldbach's strong conjecture. Also, since the last typical structure is the dominant structure over all even numbers, we come up with three unproven inequalities for elements of S.C.E model using each of which, we can prove Goldbach's strong conjecture for this structure too. It is necessary to say that, we guess theses three inequalities can be proved the same as to Dusart's inequality.