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[9] ai.viXra.org:2604.0097 [pdf] submitted on 2026-04-27 23:44:21
Authors: Felipe A. Wescoup
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper presents a proof of the Beal Conjecture by means of a geometric and set-theoretic argument. Each term in the conjecture — A^x, B^y, and C^z — is interpreted as a volume composed of three-dimensional seed cubes whose side lengths are determined by the prime factors of the base. From this foundation, the prime factor sets of A and B are shown to be either disjoint or overlapping. The disjoint case is eliminated by contradiction: if A and B share no common prime, the equation A^x + B^y = C^z cannot be satisfied for any integer C and exponent z ≥ 3. The overlapping case directly implies that A, B, and C share a common prime factor, which is precisely what the conjecture asserts. Because these two cases are exhaustive, the conjecture is proven.
Category: Number Theory
[8] ai.viXra.org:2604.0087 [pdf] submitted on 2026-04-26 18:09:16
Authors: Liam Isaac
Comments: 8 Pages. Creative Commons Attribution 4.0 International (CC-BY 4.0) (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Collatz function is one of the simplest difficult problems in modern mathematics. For any positive integer, multiply any odd integer by 3 and add 1, while any even integer is divided by 2. Take the result and re-insert it into the function. Every integer will eventually fall to 1, and begin a loop of the sequence 1-4-2-1. Will this function produce another loop at some point? Will the jumps (3n+1) overtake the drops (/2) and climb to infinity? Through a nested fractal implementation as well as the reduction principle set in Catalan's Conjecture, it is shown that both of these questions are topologically impossible within the system.
Category: Number Theory
[7] ai.viXra.org:2604.0085 [pdf] submitted on 2026-04-26 17:56:29
Authors: Sonny Thorgren
Comments: 2 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper presents a novel and closed-form formula for the calculation of the mathematical constant Pi. By examining an infinite series of a specific rational function, it is shown that the sum converges directly to Pi. I provide a complete step-by-step mathematical proof utilizing partial fraction decomposition to break down the formula and connect it to well-known series. The resulting expression offers an elegant and educational bridge between rational functions and number theory.
Category: Number Theory
[6] ai.viXra.org:2604.0083 [pdf] submitted on 2026-04-25 13:38:30
Authors: Yufei Liu
Comments: 3 Pages.
We derive an explicit arithmetic formula for L'(x)=∑_ρ (1/ρ) e^{-x/ρ}, where ρ runs over non-trivial zeros of the Riemann zeta function ζ(s) in the symmetric pairing. Using the Mellin form of the Guinand--Weil explicit formula, we prove L'(x) = e^{-x} - ∑_{n=2}^∞ (Λ(n)/n) J_0(2√(x log n)) + (1/(2π)) ∫_{-∞}^∞ (e^{-x/(1/2+it)}/(1/2+it)) (Γ'/Γ)(1/4+it/2) dt. This representation establishes a functional link between the prime distribution and the zeta zeros through a Bessel kernel, providing a framework to analyze the positivity of L'(x).
Category: Number Theory
[5] ai.viXra.org:2604.0068 [pdf] submitted on 2026-04-18 01:00:30
Authors: J. W. McGreevy
Comments: 18 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a unified arithmetic field theory in which a single topological defect at prime691 on the modular curve X(1) forces a singular Legendre point — the arithmetic breakdown of the classical map from canonical velocity to canonical momentum. Eta refraction disperses analytic volume from the flat Eisenstein background, which concentrates at the high-symmetry elliptic points of X(1). Conductor-9 triality clutching performs principalization, restoring an effective invertible Legendre map and allowing the standard variationalprinciples of Lagrangian and Hamiltonian mechanics to hold on a stable low-energy soliton.= The Ramanujan τ (n) coefficients supply the matrix elements of the quantized curvaturetorsion potential on the clutched bundle. The Leech lattice provides the global analytic volume capacity, while temperature scaling via the Boltzmann constant turns the arithmetic engine thermodynamic. Stationary action (Hamilton’s principle) selects the physical trajectories, and acceleration (time derivative of constraints) enforces global smoothness via the product formula. Explicit 1-to-1 mappings are given between RFTP arithmetic objects, classical mechanics, and quantum structures. Gravity and electromagnetism emerge as different projections of the same clutched density. The Higgs field is the radial principalization process, and the hydrogen soliton is a concrete low-temperature realization whose Balmer spectrum is quantized by the clutched modes with τ (n) transitions. The zeta zeros appear as the self-adjoint spectrum of the hamiltonized soliton, giving variational realizations of the Riemann Hypothesis and the Birch—Swinnerton-Dyer conjecture. Historical context forLagrangian/Hamiltonian mechanics, Jacobi, Liouville, and Dirac is provided throughout.
Category: Number Theory
[4] ai.viXra.org:2604.0054 [pdf] replaced on 2026-04-16 12:49:23
Authors: Yufei Liu
Comments: 4 Pages.
We define a function L(x)=∑_ρ(1-e^{-x/ρ}) for x>0, where the sum runs over all non-trivial zeros ρ of the Riemann zeta function ζ(s), taken in the symmetric pairing ρ and 1-ρ to ensure absolute convergence. We prove that L(x) converges absolutely for every x>0, that it is differentiable, and that its derivative is given by L'(x)=∑_ρ (1/ρ) e^{-x/ρ} (with the same pairing). We also show that L(x) is real-valued. This functional serves as a continuous analogue of the Li coefficients. We state the conjecture that the Riemann Hypothesis is equivalent to the strict positivity L'(x)>0 for all x>0. All results are unconditional and rely only on standard zero-density estimates.
Category: Number Theory
[3] ai.viXra.org:2604.0022 [pdf] submitted on 2026-04-06 11:14:31
Authors: Birke Heeren
Comments: 44 Pages.
We present a deterministic, endogenous, non-stationary S-adic automaton thatmodels the Sieve of Eratosthenes as a dynamical system over a finite symbolic alphabet. Theautomaton operates through three operators — shift, expansion, and filtering — applied sequentiallyto a growing symbolic tape, and provably reproduces the classical prime-compositeclassification for every integer n ≥ 2. Unlike algorithmic sieves, the automaton generatesan internal symbolic representation of the number line whose structure can be analyzed atevery step.Our first focus is: Can this new framework reproduce known mathematical knowledge?We demonstrate that this representation is not arbitrary: the tape exhibits a four-lettersubstructure {a, b, c, d} governed by an explicit substitution morphism and an upper triangulartransition matrix Mp. The dominant eigenvalue p−2 controls the population dynamicsof twin prime templates, yielding a recursive growth formula consistent with OEIS sequenceA059861 and with the combinatorial factors underlying the Hardy—Littlewood k-tuple conjecture.A central structural result is the Stability Zone SZn = [n+1, 2n−1], a provably immutableinterval in which prime candidates survive all prior filtering steps. Using a Frozen Windowtechnique, we verify the persistence of the symbolic structure experimentally up to n =250,000.Our second focus is: Can this new framework lead to new mathematical knowledge?Finally, we discuss the local Hausdorff—Besicovitch dimension of the prime candidateset within the Stability Zone. It begins near 0.92 and increases monotonically toward 1as n → ∞, following D(p) = ln(p − 1)/ ln(p). This process — vanishing fractality —unfolds dynamically inside the growing, advancing Stability Zone as it travels through thenumber line, and provides a deterministic, structural perspective on the transition from theordered regime of small primes to the apparent randomness observed in large-scale primedistributions.The automaton is offered not as a computational tool for generating primes, but as aresearch instrument: a symbolic framework in which arithmetic properties of the naturalnumbers emerge from the internal dynamics of the system.
Category: Number Theory
[2] ai.viXra.org:2604.0005 [pdf] submitted on 2026-04-02 20:51:07
Authors: Dhayaa Hussein Razzaq
Comments: 9 Pages.
This research establishes a rigorous algebraic and spectral framework for studying prime distribution. The paper proves that the primality criterion c_n = -mu(text{rad}(n))varphi(text{rad}(n))—which is the additive inverse of OEIS A063659—generates the meromorphic ratio F(s) = -zeta(s)/zeta(s-1) via a Dirichlet series. We construct a self-adjoint Hankel operator M and derive an exact trace identity Tr(M) = L(2i_0 - 1) linking it to the logarithmic derivative of the Riemann zeta function. Furthermore, the "Pentagonal Balance" is presented as a structural equilibrium of five exact algebraic identities for logarithmic sums over primes. The entire mathematical architecture is verified numerically with 50-digit precision and generalized to Dirichlet characters. This work was assisted by Google Gemini (LLM) for LaTeX formatting and structural suggestions
Category: Number Theory
[1] ai.viXra.org:2604.0004 [pdf] submitted on 2026-04-01 20:29:40
Authors: Birke Heeren
Comments: 41 Pages.
We present a deterministic, endogenous, non-stationary S-adic automaton that models the Sieve of Eratosthenes as a dynamical system over a finite symbolic alphabet. The automaton operates through three operators — shift, expansion, and filtering — applied sequentiallyto a growing symbolic tape, and provably reproduces the classical prime-compositeclassification for every integer n ≥ 2. Unlike algorithmic sieves, the automaton generatesan internal symbolic representation of the number line whose structure can be analyzed atevery step. Our first focus is: Can this new framework reproduce known mathematical knowledge? We demonstrate that this representation is not arbitrary: the tape exhibits a four-letter substructure {a, b, c, d} governed by an explicit substitution morphism and upper triangular transition matrix Mp. The dominant eigenvalue p − 2 controls the population dynamics oftwin prime templates, yielding a recursive growth formula consistent with OEIS sequenceA059861 and consistent with the combinatorial factors underlying the Hardy-Littlewoodk-tuple conjecture. A central structural result is the Stability Zone [n + 1, 2n − 1], a provably immutable interval in which prime candidates survive all prior filtering steps. Using a Frozen Window technique, we verify the persistence of symbolic structure experimentally up to n = 250,000. Our second focus is: Can this new framework lead to new mathematical knowledge? Finally, we discuss a new way of fractal dimension (self similarity), fitting for the prime candidate set within the Stability Zone. It begins near 0.92 and increases toward 1 as n → ∞, following D = ln(p − 1)/ ln(p). This process — vanishing fractality — unfoldsdynamically inside the growing, advancing Stability Zone as it travels through the numberline, and provides a structural perspective on the transition from the ordered structure ofsmall primes to the apparent randomness observed in large-scale prime distributions. The automaton is offered not as a computational tool for generating primes, but as a research instrument: a symbolic framework in which arithmetic properties of the natural numbers emerge from the internal dynamics of the system.
Category: Number Theory