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

Aref Zadehgol Mohammadi; Mohsen Kolahdouz

arxiv: 1909.13230 · v5 · pith:H2KXWHINnew · submitted 2019-09-29 · 🧮 math.NT · math.HO

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

Aref Zadehgol Mohammadi , Mohsen Kolahdouz This is my paper

Pith reviewed 2026-05-24 14:48 UTC · model grok-4.3

classification 🧮 math.NT math.HO

keywords Goldbach conjectureeven numbersprime counting functionDusart inequalitystructural classificationnumber theory

0 comments

The pith

A model sorts even numbers into 75 structures where 74 obey the strong Goldbach conjecture using Dusart's prime bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the S.C.E model to represent even numbers and divides them into 75 distinct typical structures. It combines this classification with Dusart's inequality on the prime counting function to establish that 74 of the structures must contain at least one pair of primes summing to the even number. The remaining dominant structure is handled by reducing it to three specific inequalities whose proofs are left open but expected to follow the same explicit estimation techniques. A sympathetic reader would see this as turning an infinite verification problem into a finite number of structural cases plus a small set of bounds.

Core claim

By placing every even number into one of 75 structures via the S.C.E model and applying the inequality x / ln x ≤ π(x) ≤ 1.2251 x / ln x for x ≥ 17, the strong Goldbach conjecture holds for 74 structures; the dominant structure reduces to three unproven inequalities on the model's elements.

What carries the argument

The S.C.E model that categorizes even numbers into 75 typical structures, together with Dusart's explicit bounds on the prime counting function π(x).

If this is right

Goldbach's conjecture is settled for every even number outside the dominant structure.
Full resolution of the conjecture reduces to proving or disproving the three remaining inequalities.
The structural approach replaces direct search over all even numbers with case-by-case application of prime bounds.
Similar model-based reductions could be attempted for other additive problems involving primes and even integers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Computational checks of the model for moderate even numbers could confirm that the 74 structures behave as claimed before the inequalities are settled.
If the three inequalities prove true, the same framework might classify sums involving more than two primes or other arithmetic progressions.
The dominant structure's special status suggests it may contain most even numbers, so any future proof effort should focus computational or analytic resources there first.

Load-bearing premise

The S.C.E model gives a complete partition of all even numbers into exactly 75 structures, and the three inequalities needed for the dominant structure can be proved by the same explicit methods that established Dusart's bounds.

What would settle it

An even number belonging to the dominant structure with no two primes summing to it, or a numerical counterexample showing that any of the three proposed inequalities on S.C.E model elements fails for some x.

Figures

Figures reproduced from arXiv: 1909.13230 by Aref Zadehgol Mohammadi, Mohsen Kolahdouz.

**Figure 1.** Figure 1: S.C.E Model of E In this papion shape, for upper vertices, we can see that the values dL1(E)e, dR1(E)e, show respectively the number of odd non-primes in the intervals 0, E 2 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: S.C.E Model of 20 2.2. Linking with Dusart’s Inequality. Actually, the S.C.E Model by itself is slightly poor to prove Goldbach’s strong conjecture. To enrich it, we utilize Dusart’s inequality (1). To link up with this inequality, we consider the following two main inequalities called Teeter Inequalities. Lemma 2.7. Let E ≥ 17 be an even number with quadruple representation E 4 = aE + bE + cE + dE, obtain… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a heuristic split of even numbers into 75 structures and applies Dusart's bounds to 74 of them, but leaves the dominant case open with three unproven inequalities and no proof that the split is exhaustive.

read the letter

The main takeaway is that this paper introduces a heuristic S.C.E model to divide even numbers into 75 structures and then uses Dusart's prime-counting inequality to argue that 74 of those structures satisfy the strong Goldbach conjecture. The 75th structure, described as dominant, is handled only by three inequalities the authors say they guess can be shown in the same style as Dusart's work but do not actually prove. That is the core of what is on offer. The new piece is the specific S.C.E model and the resulting 75-structure breakdown. The paper does lay out the model clearly enough and shows how the known bounds apply in the 74 cases it covers, so that part of the argument is straightforward and uses the cited inequality correctly. The soft spots are more substantial. The model is explicitly called heuristic, yet the relative proof depends on it supplying a complete, disjoint partition of every even integer greater than 2 into exactly those 75 structures. The abstract gives no indication that such a partition has been proven rather than assumed or fitted. Without that step, the claim only reaches the even numbers that happen to match the defined structures. On top of this, the dominant structure has no proof at all, only the three guessed inequalities. That leaves the most common case unaddressed. This work might be of interest to someone looking at heuristic case divisions for Goldbach-type problems, but it does not deliver a rigorous or complete result. The gaps in the partition and the unproven inequalities sit at the center of the argument. I would not bring it to a reading group or cite it. It does not deserve peer review in its current form.

Referee Report

3 major / 1 minor

Summary. The paper introduces a heuristic Semi-continuous Model for Even Numbers (S.C.E Model) claimed to partition all even integers into exactly 75 distinct typical structures. It then applies this model together with Dusart's bounds on π(x) (Eq. 1) to assert a relative proof that Goldbach's strong conjecture holds for 74 of the 75 structures, while leaving three inequalities for the dominant structure unproven and only conjectured to be establishable in the style of Dusart's estimates.

Significance. If the S.C.E model were shown to furnish a rigorous, exhaustive, and disjoint partition of all even integers greater than 2 into precisely 75 structures, and if the three guessed inequalities for the dominant structure were proved, the work would constitute a complete proof of Goldbach's strong conjecture and would be of exceptional significance in number theory.

major comments (3)

[Abstract] Abstract: The S.C.E Model is explicitly described as heuristic, yet the entire relative proof rests on the unproven assertion that this model supplies an exhaustive, disjoint categorization of every even integer >2 into exactly 75 structures; no theorem establishing this partition is indicated or referenced.
[Abstract] Abstract: The dominant (75th) structure is stated to be the most frequent, yet the three inequalities needed to handle it are left unproven, with only the remark that they 'can be proved the same as to Dusart's inequality'; this gap directly prevents the claimed relative proof from covering all even numbers.
[Abstract] Abstract, Eq. (1): The application of Dusart's inequality to the 74 structures presupposes that the S.C.E structures are defined independently of the Goldbach property and that the prime-counting bounds can be transferred without additional verification; neither independence nor the transfer is demonstrated.

minor comments (1)

[Abstract] Abstract: The phrasing 'relative proof' is used without a precise definition of what is being proved relative to what; a short clarifying sentence would help readers.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We are grateful to the referee for the thorough review and constructive comments on our paper. Below we provide point-by-point responses to the major comments, maintaining the distinction between our relative proof for 74 structures and the conjectural aspects for the remaining one.

read point-by-point responses

Referee: [Abstract] Abstract: The S.C.E Model is explicitly described as heuristic, yet the entire relative proof rests on the unproven assertion that this model supplies an exhaustive, disjoint categorization of every even integer >2 into exactly 75 structures; no theorem establishing this partition is indicated or referenced.

Authors: The S.C.E. model is presented as a heuristic tool designed to partition even numbers based on specific structural properties derived from their possible forms. In the full manuscript, we define the 75 structures explicitly and argue that they cover all even integers greater than 2 in a disjoint manner through the model's construction. While a standalone theorem is not stated separately, the definitions and the way the model is built ensure exhaustiveness by considering all possible cases for even numbers. We can revise the abstract to emphasize that the partition follows from the model's definitions. revision: partial
Referee: [Abstract] Abstract: The dominant (75th) structure is stated to be the most frequent, yet the three inequalities needed to handle it are left unproven, with only the remark that they 'can be proved the same as to Dusart's inequality'; this gap directly prevents the claimed relative proof from covering all even numbers.

Authors: Our claim is a relative proof covering 74 out of the 75 structures using Dusart's inequality. For the dominant structure, we explicitly note that three additional inequalities would be required and conjecture that they can be established similarly to Dusart's work. This does not invalidate the relative proof for the 74 structures; rather, it highlights an avenue for future work to achieve a full proof. The abstract accurately reflects this scope. revision: no
Referee: [Abstract] Abstract, Eq. (1): The application of Dusart's inequality to the 74 structures presupposes that the S.C.E structures are defined independently of the Goldbach property and that the prime-counting bounds can be transferred without additional verification; neither independence nor the transfer is demonstrated.

Authors: The S.C.E. structures are defined based on the arithmetic properties of even numbers, such as their factorization patterns or residue classes, which are independent of whether they satisfy the Goldbach conjecture. The application involves bounding the number of primes in certain intervals or sets associated with each structure, directly using Dusart's bounds on π(x). The manuscript provides the specific mappings and applications in the relevant sections, though we acknowledge that more explicit verification of the transfer could be added for clarity. revision: partial

standing simulated objections not resolved

The rigorous establishment of the S.C.E. model's partition as exhaustive and disjoint for all even integers >2
The proof of the three inequalities for the dominant structure

Circularity Check

0 steps flagged

No circularity detected; derivation applies external Dusart inequality to a newly introduced heuristic partition.

full rationale

The paper introduces the S.C.E model as a heuristic input, claims an exhaustive categorization of even numbers into 75 structures via that model, and then applies the independently published Dusart inequality to establish the Goldbach property for 74 of those structures. No quoted step equates a derived quantity to a fitted parameter, redefines the target conjecture inside the structures, or reduces the result to a self-citation chain; the three inequalities for the remaining structure are explicitly left unproven. The derivation therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the new S.C.E model and the application of an existing inequality, with the dominant case depending on unproven additional inequalities. The model and structures are invented without independent evidence.

axioms (1)

domain assumption Dusart's inequality holds for x >=17
The paper relies on this inequality from prior work to bound the prime counting function.

invented entities (2)

S.C.E Model no independent evidence
purpose: To represent even numbers and categorize them into 75 structures
The model is introduced in the paper as a heuristic without independent verification outside this work.
75 typical structures of even numbers no independent evidence
purpose: To classify even numbers for applying the proof
These structures are defined within the S.C.E model.

pith-pipeline@v0.9.0 · 5825 in / 1385 out tokens · 39452 ms · 2026-05-24T14:48:33.380681+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

H. A. Helfgott. The ternary Goldbach conjecture is true. arXiv:1312.7748 [math.NT], 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013
[2]

P. Dusart. Explicit estimates of some functions over primes. Ramanujan J., 45(1):227– 251, 2016

work page 2016

[1] [1]

H. A. Helfgott. The ternary Goldbach conjecture is true. arXiv:1312.7748 [math.NT], 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013

[2] [2]

P. Dusart. Explicit estimates of some functions over primes. Ramanujan J., 45(1):227– 251, 2016

work page 2016