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[13] ai.viXra.org:2602.0060 [pdf] submitted on 2026-02-13 20:27:17
Authors: Michel Monfette
Comments: 14 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We study the modular dynamics of prime numbers through the families SG(k) = {p ∈ P : kp + 1 ∈ P}. We present extensive computational evidence (up to 100 million Sophie Germain primes) fora modular dynamical structure in SG primes modulo 30. Two fundamental theorems establishthe triangular residue class 11,23,29 and gap constraints. Eight conjectures emerge, includinga novel "least-gap principle" and a harmonic attractor at 60. SG(k) families exhibit distinctphases in entropy/detailed-balance plane, with period-9 resonance and asymmetric sexy orbits.A third grammatical dynamic (G3) classifies all anomalies into three energy regimes. These pat-terns suggest an underlying arithmetic grammar and self-organizing behavior in Sophie Germainprimes.
Category: General Mathematics
[12] ai.viXra.org:2601.0098 [pdf] submitted on 2026-01-24 23:08:32
Authors: Al Rahimi
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper introduces the Syracuse 2-Adic Canopy, a topological framework for the Collatz Conjecture that maps the 3n+1 dynamics onto an infinite directed forest of 2-adic equivalence classes. By quotienting the set of natural numbers by powers of 2, we reveal a rigid modular architecture governed by a generalgeometric-modular predecessor formula. We demonstrate that the system functions as a dissipative map with a negative Lyapunov exponent, ensuring that the complexity of any initial state is monotonically reduced until it is captured by the global attractor at {1}.
Category: General Mathematics
[11] ai.viXra.org:2512.0031 [pdf] submitted on 2025-12-07 20:05:52
Authors: Eero Koskela
Comments: 5 Pages. (Note by ai.viXra.org Admin: Full and real author name is required on the article)
We present an elementary proof that the Diophantine equation a^n + b^n = c^n has nonon-trivial positive integer solutions for any integer n ≥ 3. The proof is based on a novelreformulation using binomial coefficients and demonstrates that the sum of weighted binomial coefficients cannot satisfy the structural requirements imposed by Fermat’s equation.This approach is independent of previous proofs and relies only on basic properties of binomial coefficients, power functions, and convexity. The method provides a unified elementary proof for all exponents simultaneously.
Category: General Mathematics
[10] ai.viXra.org:2510.0077 [pdf] submitted on 2025-10-31 16:09:09
Authors: Linia Hammache
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite all listed scientific references)
This paper introduces a novel class of hybrid sequences combining geometric and arithmeticprogression mechanisms modulated by periodic trigonometric functions. The sequenceis defined through a piecewise formulation based on parity indices, with amplitude evolutiongoverned by geometric growth for even terms and arithmetic progression for odd terms. Weestablish fundamental properties including convergence behavior, pseudo-periodic characteristics,and parametric sensitivity analysis. The sequence demonstrates potential applicationsacross signal processing, economic modeling, and physical systems simulation. Theoreticalanalysis reveals rich dynamical behavior controlled by three intuitive parameters.
Category: General Mathematics
[9] ai.viXra.org:2510.0050 [pdf] submitted on 2025-10-22 23:08:22
Authors: Joseph Dalsasso
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Great Pyramid encodes 25 mathematical/geodetic constants (π, ϕ, e, Earth’s polar circumference)with <0.3% error. Novel Dalsasso Height Divisions (146.608 m) yield 21/25 perfect chamber/void matches (2023 muon tomography). 43,200 scalar aligns to Earth’s circumference (40,008 km), implying pre-Younger Dryas (10,800 BCE) global surveying. Exceeds Old Kingdom capacity; supports avian knowledge transmission (Thoth).
Category: General Mathematics
[8] ai.viXra.org:2507.0137 [pdf] submitted on 2025-07-31 19:34:52
Authors: Lukáš Vik
Comments: 8 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present symbolic and p-adic reformulations of the Lonely Runner Conjecture (LRC),introducing auxiliary constructions to analyze visibility dynamics. By encoding runners assymbolic sequences and shifted p-adic expansions, we define new criteria for loneliness andexplore methods to suppress or structurally control its duration. This formulation allows forentropy-like analysis, symbolic compression, and potential finite verification heuristics
Category: General Mathematics
[7] ai.viXra.org:2507.0111 [pdf] submitted on 2025-07-25 16:54:54
Authors: Abdullah Bin Usman
Comments: 21 Pages. (Note by viXra.org Admin: Please cite and list scientific references)
Dual State Analysis (DSA) is presented as a novel mathematical framework aimed at addressing the interpretability and symbolic limitations of standard quantum computing formalisms. In contrast to the Hilbert-space approach, which relies on complex amplitudes and probabilistic measurement, DSA represents logical and computational systems through the simultaneous quantification of presence (P) and absence (A), extended by a symbolic phase parameter (Θ). We rigorously establish the axiomatic foundations of DSA and prove that structurally distinct expressions (e.g., x − x and y−y for x̸ = y) remain inequivalent, thus resolving a key cognitive and algebraic paradox associated with contextual zero. The Dual State Number (DSN) formalism is fully developed, including definitions for addition, subtraction, scalar multiplication, and novel dual-sum operations. Using this framework, we symbolically emulate quantum-like phenomena such as superposition, entanglement analogues, and phase-based interference entirely through deterministic symbolic arithmetic. We also propose a conceptual photonic implementation, mapping DSA operations to optical components. Comparative analysis highlights DSA’s strengths in interpretability, deterministic evolution, and symbolic scalability, with potential applications in algorithm design, finance, education, and symbolic AI. This work lays a formal foundation for a new class of deterministic, phase-aware, and interpretable symbolic computation inspired by quantum behavior.
Category: General Mathematics
[6] ai.viXra.org:2507.0079 [pdf] submitted on 2025-07-14 11:41:04
Authors: Maxim Govorushkin
Comments: 89 Pages. 10.5281/zenodo.15837292
We present a complete, self—contained demonstration of the Riemann Hypothesis built around a new "local operatoru2009+u2009OS—analyticity" framework. Starting on L²(0,∞) we define a family of compact integral operators Ku209b whose Fredholm determinant exactly equals the ratio Ξ(s)/Ξ(1—s). We then derive an absolutely convergent cluster expansion for lnu2009det(I—Ku209b) in the continuous polymer gas, extend it to the critical strip ℜu2009su2009≥u2009½, and establish rigorous Borel summability via sharp Carleman estimates and contour—shift bounds that exclude renormalon—type singularities.Next, we verify all five Osterwalder—Schrader axioms for the Euclidean correlators generated by lnu2009det(I—Ku209b), and apply GNS reconstruction to obtain a contracting, strongly continuous semigroup whose self—adjoint generator D serves as the Hilbert—Pólya operator. Spectral analysis of D—compact resolvent, absence of continuous spectrum, monotonicity of eigenvalues under s—derivatives and Kreĭn—Rutman positivity—yields a one—to—one correspondence between its discrete spectrum and the nontrivial zeros of ζ(s), proving they lie on ℜu2009s=½ and are all simple.A key technical innovation is the "Nomadic Method" (Homeless), which uses overlapping local charts and transition maps to localize cluster bounds, Borel expansions and OS positivity into a single uniform proof. All analytic constants, operator—norm estimates and s—derivative controls are made explicit, closing gaps that prevented previous global approaches from reaching ℜu2009s=½.deposit includes:— the full LaTeX source with appendix J (lemmas J.1—J.14, J.3u2032, J.9u2032), — Python scripts for discretizing Ku209b and computing first eigenvalues, — Jupyter notebooks reproducing the first twenty nontrivial zeros, — a pytest—based CI workflow ensuring everything compiles and tests pass out of the box. Linking these elements provides both the mathematical rigor and computational reproducibility needed to validate our Hilbert—Polya construction and settle the Riemann Hypothesis.
Category: General Mathematics
[5] ai.viXra.org:2506.0059 [pdf] submitted on 2025-06-16 01:49:18
Authors: Edrianne Paul Casinillo
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
The Reserve Arithmetic System (RAS) presents a novel framework for addressing division by zeroby encapsulating the numerator within a symbolic zero, termed a reserve. This proof-of-concept paper outlines the foundational mechanics of RAS, including its arithmetic operations and key algebraic properties. The notation 0⟨x⟩ denotes a "zero with reserve x," signifying that while the numerical result is zero, the original numerator is retained symbolically. This approach extends classical arithmetic to include well-defined division by zero, preserving informational content and enabling new algebraic structures
Category: General Mathematics
[4] ai.viXra.org:2506.0014 [pdf] submitted on 2025-06-04 00:12:26
Authors: Edrianne Paul B. Casinillo
Comments: 8 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Division by zero is traditionally seen as undefined, causing discontinuities in boththeoretical mathematics and practical computation. The Reserve Arithmetic System(RAS) offers an alternative framework: instead of rendering division by zero undefined,it represents such cases with symbolic values that encapsulate the numerator in aconceptual structure called a reserve. This paper presents a formal framework for RAS,showing that it constitutes a unital commutative semiring-like structure, preserving algebraic properties such as closure, associativity, commutativity, distributivity, and identity. The reserve preserves symbolic information about operations involving zero denominators, enabling traceable, consistent reasoning in symbolic computation and related domains.
Category: General Mathematics
[3] ai.viXra.org:2506.0010 [pdf] submitted on 2025-06-04 00:01:52
Authors: Charles R. Tibedo
Comments: 42 Pages. Solvers Included in PDF for public verification
The Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Riemann zeta function ��(��) lie on the critical line Re(��) = 1/2, stands as the most consequential unsolved problem in pure mathematics. Its resolution would not only deepen our understanding of prime number distribution but also unify disparate domains of mathematics and physics. This work resolves RH through a novel synthesis of septimal-adelic spectral synthesis, hypotrochoidic geometry, and modular stress conservation, anchored by the normalization of the Riemann Siegel framework to cyclic boundary conditions (modulo 1) and validated computationally (��(10−80)). This work provides a proposed formal resolution of the Riemann Hypothesis, validated through axiomatic proofs and computational syntheses. The synthesis of hypotrochoidic geometry, septimal cohomology, and adelic spectral theory establishes a new potential field for exploration in analytic number theory.
Category: General Mathematics
[2] ai.viXra.org:2505.0123 [pdf] submitted on 2025-05-20 17:33:24
Authors: Hamid Javanbakht
Comments: 92 Pages.
This volume introduces temporal cohomology, a new framework in algebraic topology that integrates cohomological structures with internalized temporal dynamics. Rather than treating time as an external parameter, we define a category of sheaves indexed by trace-evolving sites, where homological invariants stabilize under Frobenius-like flows. The core construction involves a tower of arithmetic sites equipped with transition functors encoding temporal descent. Within this enriched setting, we develop fixed-point theories, modal regulators, and trace pairings that generalize classical cohomology and open pathways toward a spectral reformulation of zeta invariants. The formalism unifies temporal logic, topos theory, and motivic descent into a cohesive cohomotopical topology. This volume lays the categorical and spectral foundation for later applications to L-functions, field theories, and the Millennium problems.
Category: General Mathematics
[1] ai.viXra.org:2505.0050 [pdf] submitted on 2025-05-09 21:42:35
Authors: Bryan Clem
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references)
I’m very happy to present a new prime number solution that I have discovered with you all. I’ve discovered that our entire number system is completely reactionary and prime numbers are not random. And the proof was lying in how prime number 2 eliminates all even integers. This causes a reactionary effect where 3n removes all odd composites of 3 to infinity. Where N is 3,5,7,9, & 11. And you increase these 5 multiples of prime number 3 by 10 to infinity. This in turn causes prime number 5 to have a reactionary effect that forces the equation 5n. Where n equals 5 & 7. And you increase these 2 multiples by 6 to infinity simultaneously for prime number 5’s exact composites. This causes a reactionary effect that forces all prime numbers past 5 to end in a last digit of 1,3,7, or 9. Next to completely remove all of prime numbers 2,3, & 5’s composites from the number line you lay out the next 8 prime numbers of 7,11,13,17,19,23,29, & 31 out on paper and increase these 8 primes by the amount of 30 apiece to infinity. This naturally filters out all composites of 2,3, & 5 while catching all primes past 5. I have found that each prime number past 5 has 8 recursive prime multiples. One for each row and they are last digit locked to infinity. And each of these 8 prime multiples for all primes are increased by 30 to infinity with no calculating. The 1st of their 8 recursive multiples are always its square. So you multiply 7u20227 =49. Which is the second number down in row 5. (Truncated by ai.viXra,org Admin)
Category: General Mathematics
[1] ai.viXra.org:2506.0059 [pdf] replaced on 2025-06-21 02:47:13
Authors: Edrianne Paul B. Casinillo
Comments: 6 Pages.
The Reserve Arithmetic System (RAS) presents a novel framework for addressing division by zeroby encapsulating the numerator within a symbolic zero, termed a reserve. This proof-of-concept paper outlines the foundational mechanics of RAS, including its arithmetic operations and key algebraic properties. The notation 0⟨x⟩ denotes a "zero with reserve x," signifying that while the numerical result is zero, the original numerator is retained symbolically. This approach extends classical arithmetic to include well-defined division by zero, preserving informational content and enabling new algebraic structures.
Category: General Mathematics