Proof of the Goldbach's Strong Conjecture by Using Semi-Continuous Model of Even Numbers
Aref Zadehgol Mohammadi, Mohsen Kolahdouz
TL;DR
The paper introduces a Semi-Continuous Model for Even Numbers (S.C.E Model) that encodes each even $E$ via a quadruple $(a_E,b_E,c_E,d_E)$ representing additive interactions of odd numbers summing to $E$. By linking this model with Dusart's prime-counting bounds, it derives Teeter inequalities that bound combinations of the quadruple components and yield tight bounds on $d_E-a_E$ and related sums. Using these bounds, the authors prove a contradiction-based result showing $d_E>0$ for all $E\\geq 22864$, thereby establishing Goldbach's strong conjecture for all even numbers beyond this threshold (with smaller cases verifiable by inspection). If accepted, this provides an explicit analytic proof of the strong conjecture for the tail of even integers, while situating the result within a combinatorial-additive framework tied to prime distribution estimates.
Abstract
In this paper, we present an explicit and analytic proof for still unproven Goldbach's strong conjecture. To derive this proof, we first define a heuristic model for representing even numbers called Semi-continuous Model for Even Numbers or briefly S.C.E Model, and then we employ this model along with using the inequality \begin{center} $\dfrac{x}{\ln x}\leq_{x\geq 17}π(x)\leq_{x>1}1.2251\dfrac{x}{\ln x}$, \end{center} where $π(x)$ denotes the number of all primes smaller than and equal to $x$, and is presented by Pierre Dusart in his paper [P. Dusart, \textit{Explicit estimates of some functions over primes}, Ramanujan J. \textbf{45} (2016), No. 1, 227--251]. On the one hand, this proof is given for all even numbers $E\geq22864$. On the other hand, since the assertion of Goldbach's strong conjecture is easy to verify for all even numbers $4\leq E<22864$, we turn this conjecture into a theorem.