Hedi Zarkouna

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Independent Researcher
Supervisors: Ridha HAJALI and Nejib BOULAHIA
Phone: +216 25287473
Address: Ariana, Tunisia

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Deterministic Proof of the Composition of Fermat Numbers (n ≥ 5): Arithmetic Index Theory, Absolute Space, and Symmetry Breaking

Engineering of Inifinity, 2026

Since the 17 th century, factoring Fermat numbers (F_n = 2^(2^n) + 1) has been hindered by algori... more Since the 17 th century, factoring Fermat numbers (F_n = 2^(2^n) + 1) has been hindered by algorithmic complexity. This paper provides a structural resolution to the problem through the Arithmetic Index Theory (AIT). By modeling the matrix of these numbers via a topology called Absolute Space, we unveil the internal recursivity governing their multipliers. We demonstrate that primality is merely a transient state of spatial vacuity (F_0 to F_4). Starting at F_5 , the matrix undergoes a symmetry breaking forcing the emergence of a Fractal Jump (Diophantine Clasp) to balance the entangled energy. Ultimately, this paper establishes a deterministic proof by contradiction of infinite non-primality: the hypothesis of a return to a prime state for n ≥ 5 causes a triple collapse of the conservation laws of the Absolute Space. This entropic irreversibility definitively proves that all Fermat numbers beyond the temporal initialization point are strictly composite.

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Arithmetic Index Theory (AIT): Unification and Deterministic Resolution of Major Linear and Quadratic Conjectures

The Engineering of Infinity, 2026

This article synthesizes the fundamental results of the Arithmetic Index Theory (AIT) applied to ... more This article synthesizes the fundamental results of the Arithmetic Index Theory (AIT) applied to the resolution of open problems in higher arithmetic. By substituting the probabilistic paradigm of classical analytic sieves with a deterministic vector matrix space, the AIT definitively dissolves the parity problem. This article exposes the structural foundations of the Absolute Space through its alternating index generating function and its collision matrix, then formalizes the universal proof protocol based on the lower bounding of the primality reservoir, telescoping by Euler's Gamma function, and the algebraic determination of the minimum ceiling in the worst case. Two major geometric families are demonstrated here by extension: the linear family (Goldbach, Twin Primes, Polignac, Sophie Germain) and the quadratic family (Oppermann, Legendre, Brocard, Landau n² + 1).

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Deterministic Proof of Landau's Conjecture via the Theory of Arithmetic Indices and Absolute Space

The Engineering of Infinity, 2026

This article presents a deterministic proof of Landau's conjecture (the 4th problem of 1912). By ... more This article presents a deterministic proof of Landau's conjecture (the 4th problem of 1912). By projecting the quadratic form m² + 1 into the Absolute Space generated by the function f (n) = 3n + [ (3 - (-1)^n)/2 ], we demonstrate an exhaustive topological coverage of valid candidates. Through the application of a strict lower bound on matrix destruction (the spatial Eulerian product) and analytic telescoping via Euler's Gamma function, we establish a minimal survival threshold S min (X) = 0.8308 • [ R(X)^(1/3) ]. This deterministic threshold crosses unity as early as X = 17 and strictly diverges to infinity, guaranteeing the infinitude of prime numbers of the form m² + 1 without resorting to probabilistic models or the Prime Number Theorem.

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Arithmetic Index Theory - Primes Order

Tne Engineering of Infinity, 2026

This supplemental note details a fascinating structural property of the Absolute Space, defined b... more This supplemental note details a fascinating structural property of the Absolute Space, defined by the generating function f (n). We demonstrate that the product of two adjacent elements of this space consistently maps to another exact element within the same space, indexable in a deterministic manner through the relation f (f (n(n + 2))) = f (n) • f (n + 1). This algebraic closure formally generates pairs of twin and cousin prime candidates.

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Arithmetic Index Theory and Prime Numbers

The Engineering of Infinity, 2026

This article presents a deterministic re-foundation of the analysis of the distribution of prime ... more This article presents a deterministic re-foundation of the analysis of the distribution of prime numbers and their twins, breaking with the conventional probabilistic approaches inherited from Gauss and Riemann. The core of this study lies in the definition of "Absolute Space," a rigid topological structure generated by the unified indicial function f (n) = 3n + [3-(-1)^n]/2

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Proof of the Infinitude of Twin Primes via Index Theory and Absolute Space

The Engineering of Infinity, 2026

The analytical investigation of prime number distribution has long been dominated by probabilisti... more The analytical investigation of prime number distribution has long been dominated by probabilistic approaches or screening methods. This article proposes an epistemological rupture by demonstrating the twin prime conjecture through Arithmetic Index Theory. By substituting numerical value with spatial index within the Absolute Space, we prove that twin couples are governed by a free sub-index n ′. The proof relies on an asymptotic crossing: the linear expansion of the sub-index reservoir in O(N) strictly dominates the polylogarithmic rarefaction in O(1/ ln 2 N) imposed by the composition equation.

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The Engineering of Infinity: A Methodological Framework for Structural Determinism in Goldbach's Conjecture

The Engineering of Infinity, 2026

The Goldbach Conjecture has long been characterized as the "Mount Everest" of pure mathematics, a... more The Goldbach Conjecture has long been characterized as the "Mount Everest" of pure mathematics, a peak that has resisted ascent for nearly three centuries. The persistent failure of classical approaches stems from a fundamental reliance on probabilistic heuristics-treating prime distribution as a series of stochastic fluctuations rather than a structural consequence. This methodology advocates for a radical paradigm shift toward Arithmetic Index Theory. By transitioning from the observation of "arithmetic accidents" to the mapping of a "geometric grid," the conjecture is repositioned as a question of structural necessity. The introduction of the "Absolute Space" (S A) provides a topological necessity that renders previous probabilistic methods obsolete. By deriving a topological safety floor, Z * d (2N) ≈ 0.2286 • N ^ (2/3) , we prove that the combinatorial reservoir of solutions grows strictly faster than the destructive power of prime sieving, reducing the existence of Goldbach partitions to an O(1) structural certainty.

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The Index Theory of Prime Numbers: Algebraization of the "Wheel of 30", Deterministic Primality Testing, and Factorization via Index Method

Index Method for Primes & Factorization, 2026

Since antiquity, the detection of prime numbers and the factorization process have relied on the ... more Since antiquity, the detection of prime numbers and the factorization process have relied on the empirical paradigm of Euclidean division, historically leading either to computational dead ends or to the reliance on probabilistic heuristics [1]. This paper-serving as the second part of the new book : The Engineering of Infinity-introduces the Index Theory, a fundamental paradigm shift that substitutes traditional value-based arithmetic, which has long defined the unpredictable nature of prime distribution [2], with an absolute spatial geometry. The foundational breakthrough of this work is the demonstration that the classical "Wheel of 30" has been algebraized and transformed into a continuous space. This is achieved through the combination of a generating function and the pre-filter S 5 [n]. Consequently, arithmetic chaos is replaced by strict geometric determinism, governed by the unique analytical formulation of three fundamental expressions. Furthermore, deploying six (6) deterministic algorithms (Exclusion, Linear, and Quadratic paths), The Index Method completely bypasses traditional division. It unifies lightning-fast primality testing (offering a novel spatial alternative to modern deterministic polynomial-time bounds [3]) with immediate factor extraction during a spatial breakdown (Natural Factorization Theorem), opening a new era for structural analysis and cryptanalysis.

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