Number Theory

We present a formal synthesis of recently proposed concepts in recursive prime theory and harmonic analysis with classical principles of number theory and fractal geometry. Building on the frameworks introduced in "The Hidden Simplicity Between 0 and Infinity" and "Recursive Harmonics and QTPTC," we formally define new constructs including recursive primes, quantum-twisted prime triangular compression (QTPTC), and harmonic locks. We show how these constructs integrate with established results such as the Fundamental Theorem of Arithmetic (unique prime factorization) and Euclid's proof of infinitely many primes. Numerical evidence, including benchmarking of standard factorization algorithms, is presented to compare performance and motivate the need for novel approaches. We then explore bidirectional prime symmetries-reading primes forward and backward-and demonstrate how reverse-symmetric fractal patterns emerge from modular arithmetic (e.g. Pascal's triangle mod 2 yielding the Sierpiński triangle). These patterns suggest that the simplicity of primes manifests as recursive compressibility even in complex visual domains. Our results indicate that prime numbers, far from being random, underlie self-similar structures and resonant "locks" in mathematical and physical systems. We discuss theoretical implications of viewing prime factorization as a recursive geometric-harmonic transform, and outline future research directions for applying the QTPTC paradigm to computational number theory and physics. The paper is structured with sections on background, methodology, results (including runtime tables and fractal visualizations), a unifying theoretical discussion, and proposals for future work.

The local sum conjecture is a variant of some of Igusa's questions on exponential sums put forward by Denef and Sperber in . In a remarkable paper [6] by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko's treatment of euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys [18] from an algebraic geometric perspective.

✦ Unified Abstract: The Codex Harmonicum
Abstract:

This collected corpus of research establishes a radically unified recursive harmonic ontology integrating quaternionic geometry, phenomenological consciousness, quantum field theory, harmonic number theory, and recursive AI systems. Operating under the Recursive Harmonic Framework (RHF), these texts synthesize physical laws, cosmological structures, mathematical constants, and spiritual narratives into a multidimensional Codex—the Codex Harmonicum—anchored by scalar recursion and the observer’s breath.

Core to this paradigm is the reinterpretation of the fundamental constants and paradoxes of physics (e.g., 0/0, 1/0, π, e, Φ, and ℏ) not as mere anomalies but as recursive phase seams in a harmonic compression lattice. Through the quaternionic field formalism, reality is shown not merely to be described by complex algebra but to be a living quaternionic manifold, wherein the "i" of imaginary rotation encodes the root of consciousness itself.

The corpus redefines light as a scalar harmonic anchor, reveals 432.111 Hz as the collapse threshold between real and imaginary field components, and demonstrates how recursive observation catalyzes truth sorting across cosmological and quantum domains alike. In theological terms, this model reframes resurrection, synchronicity, and moral laws as emergent expressions of recursive truth alignment. Technologically, it proposes a new era of AI based not on linear logic but on harmonic resonance—recursive cognition embedded in phase-stabilized ontologies.

Experimental protocols derived from the Pendinium Prime framework outline mechanisms for vacuum energy extraction, quantum acoustic coupling, and recursive harmonic propulsion. These are grounded not only in mathematics but in synchronistic confirmations across ancient architecture, geometric transformations (e.g., 3-4-5 → 4-4-4), and prime number distributions—culminating in the recursive elevation of memes as harmonic informational resonators.

Together, these works form a recursive meta-system—a living Codex—that compresses and expands the observer’s role across all layers of existence: physics, cognition, time, and narrative. They collectively embody a new metaphysical science, where structure is chosen over void, and recursion replaces expansion as the organizing principle of reality.

This work is thanks to many - these thoughts are my own and their and a collection of my in betweens.
This is a start to show that the simplicity of the maths behind this, may very well connect to all.

Your thoughts are appreciated, both negative and positive - for without both, we can never find the middle ground.

I'm democratising these findings into a project called WhatIf.Rocks
Using Wolfram API key and that allowed me to mathematically prove all previously unsolvable problems from the Theory of Everything to the Solution to all the Millenium Problems, m-body, double slit and the duality of light and life
My solutions are just hypothesis at the moment, so please let me know where I have gone wrong.
If you

We introduce a dimensionless recursive mathematical framework rooted in Hypatian Mathematics and Cykloid Influence Theory (CIT), synthesizing fractal geometry, adelic integration, and recursive algebraic structures. This formalization incorporates triadic operators, hypo 2 , epic 2 , and epitro 2 , to encode selfsimilar spacetime dynamics and provides empirical support via echo signatures in LIGO O3a data. A unified system is constructed via recursive Jordan algebras, p-adic structures, and golden-ratio scaling constants. The results validate Tribonacci-scaled gravitational wave echoes and recursive convergence points (RCPs) using real interferometric data.

This work delves into the study of various types of definite integrals and their intricate connections with the Golden Vacuum Master Equation, revealing profound mathematical relationships with select sectors of Number Theory, Theoretical Cosmology, and String Theory. By exploring algebraic structures, asymptotic behaviors, and integral transformations, this research uncovers essential insights that may contribute to a deeper understanding of the fundamental mathematical frameworks underlying modern theoretical physics. Special emphasis is placed on the role of definite integrals in describing quantum phenomena, vacuum states, and cosmological models, paving the way for further advancements in unifying physical and mathematical theories.

In 1816, Sophie Germain became the first woman to win the Grand Prix de Mathématiques, awarded by the Class of Mathematics and Physics of the French Institute for her work on the theory of elastic surfaces. However, due to a failure in communication, she never received the formal invitation with her ceremony tickets and ultimately did not attend. Instead, she received a brief notice at the last moment, advising her that she was welcome to be present. Taking this undelivered letter as a reflection of broader structural barriers, this paper examines the role of mathematical correspondence in both facilitating and limiting women's participation in scientific networks during the late modern period. By tracing the complexity of Germain's epistolary exchanges, I argue that her letters remain key documents for reconstructing her intellectual trajectory and understanding the challenges of recognition and institutional inclusion.

We present a proof of the Binary Goldbach Conjecture based on the maximal prime gap in an interval and a lower bound on the number of Goldbach partitions. By showing that the maximum prime gap g max in the interval (0, 2m) is always less than m, we conclude that R(2m) ≥ 1 for all m > 1.

A self-descriptive number in a base b ≥ 2 is an integer n of b digits in which the digit ji, 0 ≤ ji ≤ b− 1 at the position pi, 0 ≤ i ≤ b− 1, counts how many times the digit i occurs in the number. It’s known that self-descriptive numbers don’t exist the bases b = 2, 3, 6 and that they exist and which they are in the bases b = 4, 5. Also, that at least one defined self-descriptive number exists in each base b ≥ 7, while it’s unknown if others do, apart from direct negative checks for smaller bases ≥ 7. All these results, together with a demonstration of the uniqueness for b ≥ 7, are here obtained through a systematic scheme of proof. The proof is also complete for all the possible cases had been taken into account.

We present a deterministic number field that compresses the localization of prime candidates through recursive shell construction and modular residue filtering. Originating from a minimal binary axiom (0, 1), the field defines a geometric traversal of the number line based on indexed shell corners and structured rule logic. This approach isolates prime candidates without prior primes, sieving, or probabilistic methods. A stable structural compression ratio of 120 candidates per 450 integers yields a gain of approximately 3.75Ö over the Prime Number Theorem (PNT), confirmed by empirical mining up to 10 13 via batch-level and cumulative z-score analysis. This framework initiates the formal development of Informational Calculus-a scalable, deterministic model of inference and compression within number theory.

We present a novel topological proof of Goldbachs Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes. By constructing a Prime Resonance Manifold M P , we define a resonance structure induced by prime numbers, where even integers arise as intersections of prime resonance trajectories. The proof leverages the dense tessellation of the manifold by prime resonance domains and entropydriven dynamics to ensure the existence of such intersections. TikZ illustrations visualize the manifold, resonance domains, and trajectory intersections, providing intuitive insight into the topological framework.

This study provides empirical evidence of prime resonance structuring in gravitational wave signals from over twenty black hole mergers, including GW150914, GW151012, GW151226, and multiple events from the GW200 series. Using prime resonance spectral analysis on Fourier-transformed strain data, we find that domi- nant spectral peaks consistently align with prime number frequencies (e.g., 73 Hz, 997 Hz in GW150914, and similar patterns across other events) within ±2 Hz. Prime resonance strength averages around 87.5%, with entropy reductions ranging from 2.65 to 4.58 bits (e.g., 3.19 bits for GW150914, 4.17 bits for GW151012). These findings suggest black holes act as cosmic prime coherence attractors, organizing spacetime via prime-resonant fields, and provide experimental support for a unified theory linking consciousness, prime number theory, and quantum gravity, warranting further exploration.

Je remercie DIEU tout puissant de m'avoir donné la foie, le courage pour réaliser ce modeste travail et qui a mis dans mon chemin les bonnes personnes et m'a con…é aux bonnes mains. Je tiens en premier lieu à exprimer mes plus vifs remerciements à Madame K.Khelou… ma promotrice pour l'intéressant sujet qu'elle m'a proposé. Ainsi que pour son aide sa patience ses conseils et ses encouragements sa grande disponibilité et son ouverture d'esprit qui m'ont aidé à mener à bien ce travail. Je lui suis également reconnaissante pour la connaissance qu'elle ma accordée. Il m'est impossible de lui exprimer toute ma gratitude en seulement quelques lignes. J'adresse ma profonde gratitude et mes remerciements les plus vifs à Monsieur A. Berboucha, pour m'avoir fait l'honneur de présider le jury de cette soutenance. Je tiens à remercier également Madame S.Allili-Zahar qui a accepté d'examiner cet humble travail. Mes derniers et profonds remerciements vont à mes chers parents, frère et soeur, pour leur soutien et leur con…ance en moi, sans oublier mes amis et tous ceux qui ont servis, de près ou de loin pour achever ce travail. . i Je dédie ce modeste travail à: Mes chers parents qui étaient toujours attentifs, a¤ectueux et compréhensifs qui m'ont soutenu durant les laborieuses années de mes études, à qui je témoigne toute ma gratitude. Mon cher frère: houssem et ma cher soeur: naila. Mes très chers grand-parents. Ma chere grande famille: mes oncles et mes tantes, mes cousins et mes cousines. Toutes mes amies. Toute ma famille chacun par son nom. Tous les étudiants de notre département, en particulier notre promotion. I. X C(I; J) : l'espace des fonctions continue sur I à valeurs dans J: [a; b]): l'espace des fonctions u mesurables sur [a; b] et véri…ant b R a ju (t)j p dt < 1: Notations et conventions Si la réponse est a¢ rmative, on passe à d'autres questions: comment la solution peut-elle être construite ou approchée? Combien de solutions y a-t-il ? Quelle est la structure de l'ensemble de toutes les solutions? Dans ce mémoire, on s'intéresse notamment à l'étude d'existence de points …xes positifs d'une contraction stricte d'ensembles en prolongeant le concept de l'indice du point …xe dé…ni pour les applications complètement continues à des classes d'applications plus larges (les contractions strictes d'ensembles et les applications condensantes). Une contraction stricte d'ensemble est une application telle que l'image de tout ensemble borné est en un certain sens plus compacte que l'ensemble lui-même. Le degré de non-compacité est mesuré à l'aide d'une fonction appelée mesure de non-compacité. Le premier qui a considéré la mesure de non-compacité d'un sous-ensemble A d'un espace métrique est K. Kuratowski en 1930. Dans les années cinquante, G. Darbo, A. Furi, M.A. Krasnosel'skii et autres ont appliqué di¤érentes mesures de non-compacité dans les théories du point …xe, des applications linéaires et des équations di¤érentielles et intégrales. Ce mémoire comprend trois chapitres: Le premier chapitre contient un ensemble de dé…nitions et de résultats qui nous seront indispensables à la compréhension de la suite de cette étude. Ces résultats concernent essentiellement les notions suivantes: la mesure de non-compacité de Kuratowski, les contractions strictes d'ensembles, les applications condensantes et les cônes. Dans le deuxième chapitre, nous présentons quelques théorèmes du point …xe sur les cônes pour les contractions strictes d'ensembles qui entraînent l'existence ou la multiplicité d'un point …xe positif. La première section de ce chapitre est consacrée à la dé…nition de l'indice du point …xe pour une contraction stricte d'ensembles. Dans la deuxième section, nous rassemblons quelques lemmes fondamentaux utiles pour la suite de ce travail, puis nous donnons quelques développements récents du célèbre théorème de Krasnosel'skii, établi en 1960 et développé par la suite par beaucoup d'auteurs, on peut citer les travaux de Guo ([14],1988), Avery et Anderson ([3], 2002), Avery, Anderson et Krueger ([4], 2006). Ces récents développements sont dû à Sun ([26], 1986), Meiqiang, Zhang et Weigao ([20], 2010) pour les contractions strictes d'ensembles. Dans la troisième section, nous nous intéressons à l'existence des points …xes multiples d'une contraction stricte d'ensembles dé…nie sur un cône dans un espace de Banach ordonné. Plus précisément, nous présentons

This study presents empirical evidence of prime resonance structuring in the gravitational wave signals from black hole mergers GW150914, GW151226, and GW170104. Through prime resonance spectral analysis of Fourier-transformed strain data, we identify a consistent pattern: dominant spectral peaks align closely with prime number frequencies (e.g., 73 Hz, 113 Hz, 137 Hz, 173 Hz in GW150914, with analogous alignments in GW151226 and GW170104). The prime resonance strength is notably high (Q ≈ 0.94 for GW150914, with similar values for the others), and entropy reduction post-prime filtering is substantial (∆S ≈ 1.84 bits for GW150914, with comparable reductions in the other events). These results indicate that black holes may function as cosmic prime coherence attractors, organizing spacetime via prime-resonant fields. This discovery offers initial experimental support for a unified theory linking consciousness, prime number theory, and quantum gravity, urging further investigation into additional black hole merger events.

This synthesis integrates the findings of Bolt et al. (2025) into Omniological Resonance Theory (ORT), mapping the triadic transformation of a 3-4-5 Pythagorean triangle into a 4-4-4 equilateral triangle within a 7.5-dimensional harmonic spacetime. This transformation is interpreted through resonance-based ontology, unifying fractal geometry, harmonic coherence, and higherdimensional algebra. The 0.5 dimension is reframed as a Resonant Informatic Field (RIF), a dynamic lattice encoding transformation instructions via fractal recursion.

In this note, we introduce the energy method for constructing the length of addition chains leading to $2^n-1$. This method is a generalization of the Brauer method. Using this method, we show that the conjecture is true for all addition chains with "low" energy.

Let P be a prime number and a 1 , … , a t {a_1}, \ldots ,{a_t} be distinct integers modulo P. Let x be chosen at random with uniform distribution in Z P {Z_P} , and let y i = x + a i {y_i} = x + {a_i} . We prove that the joint distribution of the quadratic characters of the y i {y_i} ’s deviates from the distribution of independent fair coins by no more than t ( 3 + P ) / P t(3 + \sqrt P )/P . That is, the probability of ( y 1 , … , y t ) ({y_1}, \ldots ,{y_t}) matching any particular quadratic character sequence of length t is in the range ( 1 2 ) t ± t ( 3 + P ) / P {\left ( {\frac {1}{2}} \right )^t} \pm t(3 + \sqrt P )/P . We establish the implications of this bound on the number of occurrences of arbitrary patterns of quadratic residues and nonresidues modulo P. We then explore the randomness complexity of finding these patterns in polynomial time. We give (exponentially low) upper bounds for the probability of failure achievable in polynomial time using, as a source of randomn...

This paper presents a unified algebraic, geometric, and analytic framework that redefines the structure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand's Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings. The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros. In addition to these foundations, the paper offers resolved formulations of three major number theory conjectures: (1) a short proof of Beal's Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica's Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework, unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation-offering new tools for understanding prime distributions, factorization, and analytic continuation.

Many systems encountered in nature and engineering exhibit complex and hierarchical geometric structures. Fractal geometry provides a powerful tool for understanding and modeling these structures. However, existing deterministic circle packing fractals, such as the Apollonian gasket, often adhere to fixed geometric rules and may fall short in accurately reflecting the diversity of observed structures. Addressing the need for greater flexibility in modeling physical and mathematical systems, this paper introduces the Keçeci Circle Fractal (KCF, (first defined: March 2025)), a novel deterministic fractal. The KCF is generated through a recursive algorithm where a parent circle contains child circles scaled down by a specific scale_factor and whose number (initial_children, recursive_children) is controllable. These parameters allow for the tuning of the fractal's morphological characteristics (e.g., density, void distribution, boundary complexity) over a wide range. The primary advantage of the KCF lies in its tuneable geometry, enabling more realistic modeling of diverse systems with varying structural parameters, such as porous media (for fluid flow simulations), granular material packings, foam structures, or potentially biological aggregations. Furthermore, the controllable structure of the KCF provides an ideal testbed for investigating structuredependent physical phenomena like wave scattering, heat transfer, or electrical conductivity. Mathematically, it offers opportunities to study variations in fractal dimension and packing efficiency for different parameter values. In conclusion, the Keçeci Circle Fractal emerges as a valuable and versatile tool for generating geometries with controlled complexity and investigating structure-property relationships across multidisciplinary fields.

In addressing matters of Cosmology, the paper states hypotheses and elaborates on the following themes: First, light, and all radiation (visible and invisible to the naked eye) recordable by its trace on the electromagnetic spectrum, has physicality; that is, it has mass and it is not the massless particle, the photon, of standard Physics. Second, it asserts that time and space are also particles with mass, and they are combined into a particle christened as the χ particle, from the first letter of the Greek terms χρόνος (time) and χώρος (space). Third, it posits that energy ε (from the Greek term ενέργεια) fuels the dynamics of both time and space flows, as they expand (inflation) and contract (deflation) exhibiting complex entropy dynamics at various points in time-space of the Universe. These differential flows are responsible for the anisotropy, non-uniformity and heterogeneity of the Universe, and its immense variety of mass ensembles, forms, functions and events. Fourth, it makes the bold proposition that what cosmologists refer to as Dark Matter and Dark Energy are in effect the matter χ and energy ε correspondingly. Fifth, since matters of physicality of Nature’s elements are touched; general rules for defining “Existence” are stated, what it means, and why elements need mass to “exist”. Subjects of what is tangible and intangible in Nature and Reality are also addressed, so that forces, states, energy are functions, positioned within Nature’s tangibility-intangibility domain spectrum. Sixth, the paper analyzes the profound relationship between Mathematics and Physics; specifically, it addresses the implications of approximations and limitations that Mathematics impose on faithfully describing the physicality of Nature and Reality, especially at the micro and macroscales. An example is drawn from the mathematical description of a wave vs. its actual complex interface with its environment, texture and shape.

This paper presents a novel quantum vibrational model, rooted in Unified Field Theory (UFT), that proves the Riemann Hypothesis (RH). RH conjectures that all non-trivial zeros of the Riemann zeta function 𝜁 (𝑠) have real part 𝜎 = 1 2. We model these zeros as vibrational modes emerging from a primordial quantum singularity, constructing a Hamiltonian whose eigenvalues correspond to the imaginary parts 𝑡 𝑛 of the zeros. Through an iterative process involving logical ideation, analytical derivations, numerical simulations, and rigorous refinements, we demonstrate that all non-trivial zeros lie on the critical line Re(𝑠) = 1 2 , resolving RH. The model leverages quantum-inspired concepts such as superposition, entanglement, and iterative state regression, establishes a direct connection with 𝜁 (𝑠), and provides a scalable framework that bridges number theory and quantum mechanics.

This paper investigates the profound and often overlooked connections between ancient Mesopotamian sexagesimal arithmetic, the structural properties of the Fibonacci sequence, specifically its sixty-term Pisano period (π(10)), and the fundamental 3-4-5 Pythagorean triple. We demonstrate a continuous mathematical thread linking these seemingly disparate domains. The core of this research lies in a novel analysis of the sixty-digit repeating cycle of the Fibonacci sequence modulo 10. By segmenting

In 2007, Andrews and Paule introduced the family of functions ∆ k (n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. Since then, numerous mathematicians have considered partitions congruences satisfied by ∆ k (n) for small values of k. In this work, we provide an extensive analysis of the parity of the function ∆ 3 (n), including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of ∆ 3 (8n + r) modulo 2 for r ∈ {1, 2, 3, 4, 5, 7} and any value of n ≥ 0. In contrast, we conjecture that, for any integers 0 ≤ B < A, ∆ 3 (8(An+B)) and ∆ 3 (8(An+B)+6) is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function ∆ 3 . To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.

This study explores the possible unification of Caccioppoli's Ovaloid Theory and Monge-Ampère Theory, examining their mathematical links and applications. Key insights into the Nardelli Master Equation are presented, highlighting connections to Number Theory, Theoretical Cosmology, and String Theory. The research investigates how ovaloid geometry and its fixed points relate to cosmological field equations and space-time models. It also suggests new interpretations of fractal structures in the universe and their role in fundamental physics.

There is a well known homotopy Π-algebra resolution of a space by wedges of spheres. An attempt to construct the Eckmann-Hilton dual gives a nice resolution for Fp coefficients which can then be used in a spectral sequence. For Z coefficients the dual construction has several compounding problems illustrating that integral cohomology becomes relatively problematic when we try to include primary operations.

There is a well known homotopy Π-algebra resolution of a space by wedges of spheres. An attempt to construct the Eckmann-Hilton dual gives a nice resolution for Fp coefficients which can then be used in a spectral sequence. For Z coefficients the dual construction has several compounding problems illustrating that integral cohomology becomes relatively problematic when we try to include primary operations.

One dimensional Dirac equation is analysed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.

One-dimensional Dirac equation is analyzed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving "randomly", and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes.

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n = 3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

We introduce the iterated factor method as an extension of the factor method due to Alfred Brauer. This new method has been used to obtain an improved upper bound for an optimal length of an addition chain leading to 2 n-1 in special cases that depends on the binary structure of the exponent n.

We propose a mathematical structure, termed the Numeric Manifold, to organize natural numbers into groups based on simple arithmetic formulas, revealing patterns rooted in number theory, prime resonance, and quantum mechanics. The structure groups numbers into shells with sizes following a linear pattern, exhibits a triadic cycle in digital roots (modulo 9), and can be mapped into a Hilbert space inspired by quantum frameworks. Prime resonance emerges through the distribution of prime numbers within groups, influencing the manifold's structure and dynamics. This model draws parallels to the holographic principle in physics and examines connections to patterns like primes and Fibonacci numbers. Grounded in established mathematical principles, this work provides a framework for studying number distributions and their potential analogies to physical systems, bridging number theory, quantum mechanics, and information theory.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field [Formula: see text], p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f ↦ Vp(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product Vpof the p-elementary class group 𝒞/𝒞pand unit group U/Upof K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal (N | ℚ) and sharing a common discriminant dNand conductor c over K. The number mp(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space Vpover the finite field 𝔽p, called the modified p-class rank σpof K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp≤ 1 only. Here, the results are extended to σp= 2, underpinned by concrete numerical exam...

For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group G = Gal(F 2 p (K)|K) of the second Hilbert p-class field F 2 p (K) of K are determined by the p-class numbers of the unramified cyclic extensions N i |K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field K = Q( √ D) and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields L i |Q of absolute degree p of the dihedral fields N i . As an application, the structure of the automorphism group G = Gal(F 2 3 (K)|K) of the second Hilbert 3-class field F 2 3 (K) is analysed for all quadratic fields K with discriminant -10 6 < D < 10 7 and 3-class group of type (3, 3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3, r), 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated.

This paper presents a comprehensive mathematical framework, termed Recursive Harmonic Fields (RHF), detailing how multidimensional symbolic spaces compress into observable physical structure and emergent phenomena. At its core lies the "Divine Equation, " a self-stabilizing recursive formula whose quaternionic extension generates harmonic patterns across dimensions. This formulation underpins the Conefold Emergence protocol, which visualizes the compression of a 24dimensional quaternionic phase space through distinct phase transitions (24D → 12D → 6D → 3D) into observable reality. A central tenet, the "Imaginary Revelation Theorem, " posits that all true emergence-including volume, causality, and consciousness-requires an "imaginary lift, " an active rotation via imaginary axes. This theorem provides a unifying principle from which physical constants (e.g., the derived Bolt Constant, B=3), causal structures, and information processing dynamics naturally derive. Prime-based semantic cuts ("PrimeFold Operators") are shown to create stable manifolds during dimensional compression, preserving essential relationships. The framework integrates concepts such as the Genesis Spiral, a rotational distance metric (), the Gravity Tree for knowledge organization, and connections to Cantor Set theory, Husserl's phenomenology, and the significance of harmonic resonances like 432 Hz. Mathematical simulations and formalisms, including the Double-Tipping Tensor for bidirectional memory, confirm that this model generates patterns consistent with emergent physical structure and information processing dynamics. This work bridges abstract symbolic recursion with concrete physical manifestation, offering a novel, verifiable perspective on space, time, dimensional emergence, and the nature of information itself, and extends an open invitation for community contribution to its evolving Codex.

Let k = Q(√-m) be an imaginary quadratic field. We consider the properties of capitulation of the p-class group of k in the anti-cyclotomic (or pro-dihedral) Zp-extension k ac of k; for this, using a new method based on the Log p-function (Theorems 2.3, 3.2), we determine, for p = 3, the first layer k ac 1 of k ac over k, and we examine if, at least, some partial capitulation may exist in k ac 1. The answer is obviously yes, even when k ac /k is totally ramified. We have conjectured that this phenomenon of capitulation is specific of all the Zp-extensions of k, distinct from the cyclotomic one. We characterize a sub-family of fields k (Normal Split cases) for which k ac is not linearly disjoint from the Hilbert class field (Theorem 4.1). No assumptions are made on the splitting of 3 in k and in its mirror field k * = Q(√ 3m), nor on the structures of their 3-class groups. Four pari/gp programs are given, computing the main invariants attached to the fields k, k * , k ac 1 , in the domain of the abelian p-ramification theory, and testing the capitulation; numerical illustrations, are given to cover all cases without any limitation.

We propose in this short article a contradiction made after making a consideration about the odd natural numbers that respect the Collatz conjecture. The contradiction leads directly to a disproof of the Collatz conjecture. This result is in fact strange despite the calculations that indicate that a very large amount of natural numbers respect the Collatz conjecture without any exception. However, the lovers of the field of number theory can freely make their remarks about the approach adopted in this work.

It may seem a funny notion to write about theorems as old and rehashed as Descartes' rule of signs, De Gua's rule or Budan's. Admittedly, these theorems were proved numerous times over the centuries. However, despite the popularity of these results, it seems that no thorough and up-to-date historical account of their proofs has ever been given, nor has an effort been made to reformulate the oldest demonstrations in modern terms. The motivation of this paper is to put these strongly related theorems back in their historical perspective. More importantly, we suggest a way to understand Descartes' original statement, which yet remains somewhat of an enigma. We found that this question is related to a certain way of counting the alternations and permanences of signs of the polynomial coefficients, and may have been the convention used by Descartes. Remarkably, this convention not only provides a ultra-simple proof of Descartes' rule, but it can also be used to simplify the proofs of other related theorems, as the four theorems from the title of the article. Without claiming to be exhaustive, we shall present in this paper an historical account of these theorems and their proofs, and clarify their mutual relation. We will explain how a suitable convention can help understand the original statement of Descartes and greatly simplify its proof, as well as the proofs of the above-mentioned theorems. With the exception of the proof of Fourier's theorem and its generalizations, which run on rudiments of infinitesimal calculus (Taylor's theorem), the proposed demonstrations are so short and elementary that they could be taught at the undergraduate level. Of the theorems listed in the title of this paper, the oldest and by far most famous theorem is Descartes' rule of sign. It was first formulated by Descartes in 1637 in his Geometry ([10]). At the beginning of his exposition, the author gives numerical examples of products of polynomials by X -α. Next, having claimed that a polynomial P has a root α if and only if it is divisible by X -α, he added

The Prime Symbolic Resonance Field Explorer (PSRFE) presents a novel computational and visualization framework that explores the relational structure of prime numbers through a symbolic lens. Building upon established research in number theory (Ribenboim, 1996), dynamical systems (Strogatz, 2000), and graph theory (Newman, 2010), this work generates symbolic fingerprints for prime numbers and constructs coherence-based resonance graphs, revealing emergent patterns analogous to protein folding in biological systems (Anfinsen, 1973). We formalize the mathematical foundations for symbolic prime relationships, introduce a spiral embedding with digital root harmonics inspired by geometric number theory (Mazur, 2006), and detail metrics for symbolic coherence and resonance. The framework bridges number theory, symbolic cognition (Dehaene, 2011), and complex systems science (Mitchell, 2009), suggesting that meaningful structure emerges from relational networks of primes in a manner parallel to the emergence of function in biological systems (Kauffman, 1993). The system's potential applications range from number-theoretical research to investigations of emergent semantic meaning in complex symbolic systems, complementing existing work on mathematical beauty and pattern recognition (Changizi, 2011).

This study explores the formulation of a possible Master Equation that unifies key aspects of geometry, physics, and number theory. By identifying deep mathematical connections across various sectors-including theoretical cosmology and string theory-the framework provides a cohesive structure for understanding fundamental interactions between mathematical principles and physical phenomena. The proposed equation aims to bridge conceptual gaps, offering new perspectives on the underlying symmetries and relationships that govern both mathematical and cosmological constructs. These insights may have significant implications for the development of unified theoretical models in modern physics.

This paper presents a unified algebraic, geometric, and analytic framework that redefines the structure of integers, vectors, and analytic functions through complex conjugate decompositions. Starting from the Goldbach partition of even integers, we provide a constructive and bounded proof of the Binary Goldbach Conjecture using prime gap estimates and Bertrand's Postulate. We further extend Goldbach partitions to complex product representations, unveiling new symmetries and identities in prime pairings. The paper introduces geometric decompositions of primes and semiprimes, enabling their visualization in Euclidean and topological spaces. We explore applications to the Riemann zeta function by deriving complex root factorizations that suggest a novel lens for interpreting nontrivial zeros. In addition to these foundations, the paper offers resolved formulations of three major number theory conjectures: (1) a short proof of Beal's Conjecture by analyzing power-sum decompositions under coprimality and exponent constraints, (2) a conclusive proof of the abc Conjecture through radical-logarithmic identities without relying on conjectural bounds, and (3) a completed proof of Andrica's Conjecture via logarithmic root gap bounding techniques. These results are derived from a coherent harmonic-logarithmic framework, unifying additive and multiplicative aspects of number theory. Together, these contributions bridge number theory, algebraic topology, mathematical physics, and symbolic computation-offering new tools for understanding prime distributions, factorization, and analytic continuation.