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[122] ai.viXra.org:2603.0014 [pdf] submitted on 2026-03-04 01:15:39
Authors: Chaiya Tantisukarom
Comments: 11 Pages.
This article explores the relationship between the distribution of prime numbers and the zeros of the Riemann Zeta function through the lens of Fourier Analysis. We contrast the ``Natural'' Riemann representation—a discontinuous, jagged summation of discrete frequencies—with the ``Man-made'' Riemann-Siegel $Z(t)$ function. We propose that the Riemann-Siegel remainder term, $R(t)$, acts as a low-pass filter that smooths the underlying digital nature of prime frequencies. This smoothing forces zeros onto the $1/2$ critical line, suggesting that the Riemann Hypothesis may be an artifact of this man-made filtering rather than a fundamental property of the natural prime spectrum.
Category: Number Theory
[121] ai.viXra.org:2602.0125 [pdf] submitted on 2026-02-27 05:34:56
Authors: Chaiya Tantisukarom
Comments: 9 Pages.
This study formalizes the Prime Gear Geometry (PGG) as a dynamical system. We demonstrate that the Riemann Hypothesis (RH) is not a static property of numbers, but a structural necessity of a rolling engine. We identify the $m$-cutoff as the "Mechanical Secret" that governs the transition between discrete prime forging (Time Domain) and spectral stability (Frequency Domain).
Category: Number Theory
[120] ai.viXra.org:2602.0109 [pdf] submitted on 2026-02-25 01:07:15
Authors: Siqi Liu
Comments: 2 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper provide a formal mathematical proof of the non-existence of non-trival loops in the problem of Collatz Conjecture by using main algebra tools of 2-adic constraints.
Category: Number Theory
[119] ai.viXra.org:2602.0079 [pdf] submitted on 2026-02-16 23:53:19
Authors: Wiroj Homsup
Comments: 3 Pages.
A new Twin prime sieve based on a modified sieve of Sundaram is introduced. It sieves through the set of natural numbers n such that 3n is not representable in either of the forms 2ij + i + j or 2ij + i + j +1 for positive integers i, j.
Category: Number Theory
[118] ai.viXra.org:2602.0068 [pdf] submitted on 2026-02-15 00:33:07
Authors: Chaiya Tantisukarom
Comments: 11 Pages.
This article presents "Prime Gear Geometry," a deterministic mechanical framework that redefines the integer axis as a master gear ($C_1$) with a discrete unit weight-step of $+1$. Unlike analytic models that rely on the complex-plane "1/2" critical line of the Riemann Hypothesis, this theory posits that prime numbers are exact geometric outcomes forged by $C_1$ at coordinates of total asynchronous interference. We establish the "Prime Gear Synchronization Conjecture," stating that total phase alignment of a prime gear group occurs only at Primorial intervals. This mechanical exactness is used to resolve the Goldbach, Twin Prime, and Collatz conjectures not as probabilistic likelihoods, but as structural necessities of a machine that, by the laws of relatively prime circumferences, is incapable of perfect synchronization within the finite bounds of the $C_1$ axis.
Category: Number Theory
[117] ai.viXra.org:2602.0059 [pdf] submitted on 2026-02-12 19:18:54
Authors: J. W. McGreevy
Comments: 18 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present the Arithmetic Relativistic Emergence (ARE) framework, in which the fundamental symmetries of General Relativity, Einstein—Cartan gravity with torsion, and quantum field theory (Standard Model sectors) emerge tautologically from pure number theory viathe arithmetic geometry of the rational numbers Q. The Riemann zeta function ζ(s) represents the maximally symmetric pre-geometric vacuum phase, with perfect functional-equation symmetry around Re(s) = 1/2 and pole at s = 1 as the unified source of arithmetic energy/information.Spontaneous symmetry breaking induced by the weight-12 modular discriminant ∆(τ ) = η(τ )24 = (2π) 12(E4(τ ) 3 − E6(τ ) 2 )/1728 disperses this background into Archimedean divergence (smooth analytic curvature density) and non-Archimedean curl (torsion/spin density at p-adic fibers), with the functional-equation mirror s = 6 enforcing variational d balance of the arithmetic degree deg( L). The emergent 4D Lorentzian manifold M carries an adelic principal Lorentz/Spin frame bundle decomposed via the adele ring AQ. An effective Chern—Weil homomorphism—employing Bott—Chern forms at infinity and classical invariant polynomials at finite places—maps split curvature forms (Fdiv, Hcurl) to arithmetic characteristic classesin Arakelov Chow groups. These classes are stationary under metric d variations (δgdeg = 0 at s = 6), providing rigid global topological invariants (Pontryagin-like, Euler-like, torsion-twisting) preserved in the broken phase—the inevitable geometric and topological labels of arithmetic symmetry breaking. Heaviside synchronization (τdiv = τcurl) at s = 6 renders the arithmetic medium transparent, yielding distortionless propagation and unified causality. The Rankin—Selberg self-convolution L(∆×∆, s) contains ζ(s) factors, allowing recombination to the symmetric vacuum. Theemergent metric determinant √−g serves as the physical scalar whose arithmetic balancing across places enforces general covariance, proper volume preservation, and integration of curvature invariants. Fermions (12 Weyl per generation from Leech lattice Z2-orbifold),gauge sectors (finite algebra C ⊕ H ⊕ M3(C)), and constants (−α 1 ≈ 137.036, Λ ∼ 10−122MPl2, G ∼ 10−38 m−2) emerge via spectral actionand adelic convolution. ARE offers a tautological origin: physical laws are the minimal effective description preserving arithmetic consistency post-symmetry breaking.
Category: Number Theory
[116] ai.viXra.org:2602.0043 [pdf] submitted on 2026-02-10 02:21:23
Authors: J. W. McGreevy
Comments: 19 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a rigorous synthesis in which the fundamental symmetries of General Relativity and quantum field theory emerge from the axioms of arithmetic geometry and number theory. Central is the weight-12 modular discriminant Δ(τ) = η(τ)^{24} = (2π)^{12} (E_4^3 - E_6^2)/1728, interpreted as the vacuum potential. The arithmetic degree (total integrated curvature) must disperse equivalently across Archimedean (smooth, complex-analytic) and non-Archimedean (discrete, p-adic) places to maintain global consistency via the product formula. This dispersion is enforced at the critical mirror point s=6 of L(Δ,s), where the functional equation symmetry balances openness and rigidity.The Hilbert-Pólya operator Ĥ = 1/2 + i (D_∞ ⊕ ∑p D_p) acts self-adjointly on the adelic Hilbert space, with eigenvalues corresponding to resonances tied to L(Δ,s) zeros. The 1728 frequency (12^3) serves as the universal gear ratio/adiabatic regulator. The 12-fermion matrix arises from the Leech lattice V_Λ{24} Z_2-orbifold, folding 24 bosonic dimensions into 12 Weyl fermions per generation via Möbius twist.A 4D Lorentzian manifold emerges via noncommutative geometry (KO-dimension 6 adelic spectral triple), with the spectral action Tr f(��/Λ) yielding the Einstein-Hilbert term and stress-energy from p-adic torsion convolution. The master equation δ_g widehat{deg}(mathcal{L}) = 0 at s=6 recovers the Einstein field equations with cosmological constant Λ ≈ M_Pl^2 e^{-288} (double-twist entropy) and fine-structure constant α^{-1} ≈ 137.036 from Petersson norm corrections. This framework posits that GR and the Standard Model are stereographic projections of the weight-12 balanced modular form onto the Möbius-Planck manifold, providing a tautological origin for physical laws from number theory.
Category: Number Theory
[115] ai.viXra.org:2602.0014 [pdf] submitted on 2026-02-04 21:12:52
Authors: Ezadiin Redwan
Comments: 4 Pages.
"We present a universal proof of the Strong Goldbach Conjecture by shifting the problem from arithmetic density to Topological Symmetry. By defining primes as the deterministic ’parent set’ via the Fundamental Theorem of Arithmetic, we map the interaction between addition and multiplication onto a vector space. We prove that the identity 2n cos(θ) = a+b is a structural requirement of this space. Thisnon-constructive existence proof demonstrates that for every even integer 2n, a prime partition (a, b) is geometrically necessitated by the scalar projection of prime-based vectors, thereby resolving the parity problem through architectural determinism."
Category: Number Theory
[114] ai.viXra.org:2601.0120 [pdf] submitted on 2026-01-30 16:30:00
Authors: Anish Sola
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We study representations of integers as differences of prime powers, n = p^a − q^b,with distinct prime bases p ̸= q and distinct exponents a ̸= b. We focus on the positive-exponent setting (a, b ≥ 1) and on the proper-prime-power variant (a, b ≥ 2), for which the problem is closer in spirit to Goldbach- and Pillai-type questions. We prove elementary structural constraints (notably parity restrictions), propose first-moment heuristics, and outline a computational program.
Category: Number Theory
[113] ai.viXra.org:2601.0106 [pdf] submitted on 2026-01-26 21:01:05
Authors: Silvio Gabbianelli
Comments: 12 Pages.
This paper explores a deterministic mapping ϕ : Nodd → Z + that defines an informational lattice for the study of prime distribution. By analyzing the topological exclusion of composite generating functions y(x, k), we identify a structural symmetry within the manifold. Computational verification through a mapping Probe confirms density alignment with the Gram series up to 10^50. The results suggest thatcertain symmetries, such as the critical line equilibrium and rotational invariance,are emergent properties of the lattice’s geometric rigidity.
Category: Number Theory
[112] ai.viXra.org:2601.0087 [pdf] submitted on 2026-01-22 11:46:52
Authors: Bahbouhi Bouchaib
Comments: 27 Pages. Original study showing a predictive search method of Goldbach's prime pairs.
We introduce a new geometric and algorithmic framework for the search of Goldbach pairs based on the organization of admissible deviations around the midpoint of an even integer. By embedding admissible deviations into a sunflower phase space and a normalized helical geometry, we observe a robust concentration of Goldbach-successful deviations along narrow geometric lanes. This structure persists across multiple scales, from 10^9 up to at least 10^26, and leads naturally to a phase-guided algorithm that reduces the number of primality tests required to find a Goldbach pair. Extensive numerical experiments demonstrate a stable efficiency gain relative to random search strategies. The results suggest the presence of a universal geometric organization underlying the Goldbach conjecture and provide a new perspective on additive problems involving prime numbers.
Category: Number Theory
[111] ai.viXra.org:2601.0072 [pdf] submitted on 2026-01-18 22:20:03
Authors: Budeeu Zaman
Comments: 22 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references)
The identity is explored using matrix and tensor formulations, integral representations,graph-theoretic methods,and a deep dive into the logarithmic analysis of prime sums these methodsshed light on fresh structural clues about how primes are spread out and their deep number-theoretic correlations.
Category: Number Theory
[110] ai.viXra.org:2601.0058 [pdf] submitted on 2026-01-16 02:12:46
Authors: Joshua Christian Elfers
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We introduce and analyze a carry-coupled skew-product extension of the Collatz dynamical system, in which whole-integer dynamics are coupled to modular arithmetic through an explicit arithmetic carry term. The central result establishes that coupling parameter K = 4 produces a unique invariant structure characterized by identity evolution of the return map. This coupling realizes, in purely arithmetic terms, the mathematical mechanism underlying conservation laws and coherent evolution in physical systems. We provide rigorous proofs of the main theorems, numerical verification, and discuss the group-theoretic foundations of this invariance.
Category: Number Theory
[109] ai.viXra.org:2601.0053 [pdf] submitted on 2026-01-14 15:22:34
Authors: Shanzhong Zou
Comments: 16 Pages.
The Twin Prime Conjecture, the Goldbach Conjecture, and the Polignac Conjecture are all conjectures concerning the distribution of prime numbers. This paper introduces a novel sieve-based framework termed the " Comb Sieve Method ". This approach not only unifies the three conjectures by treating them as different manifestations of an underlying fundamental problem but also circumvents the need to estimate error terms, a common challenge in traditional sieve methods. We believe this framework will open new avenues for research in this area of number theory.
Category: Number Theory
[108] ai.viXra.org:2601.0006 [pdf] submitted on 2026-01-02 20:35:02
Authors: Julio C. Luna
Comments: 93 Pages. (Note by viXra Admin: This article is not written in scholarly style/manner and is subject to withdrawal - e.g., An abstract is required in the article, scientific references should be cited and listed etc)
This manuscript provides an unconditional proof of the Riemann Hypothesis via a 'Lossless Vessel' operator-theoretic framework. By establishing a uniform Carleson box estimate (Section 14.3.11) anchored in the discrete support of the von Mangoldt function, we justify the limit passage required to force all nontrivial zeros onto the critical line. The work is organized into three domains: Domain A (Deductive Proof), Domain B (Conceptual Concordance), and Domain C (Computational Audit). Audit 5 confirms numerical convergence to a relative difference of ≈5.26×10−7.
Category: Number Theory
[107] ai.viXra.org:2601.0005 [pdf] submitted on 2026-01-01 11:20:26
Authors: Birke Heeren
Comments: 32 Pages.
This paper introduces the Cosmos Automaton (CA), a new type of automaton. It is generating fractal patterns based on prime number distribution and prime gaps. Unlike the classical Sieve of Eratosthenes, which requires a finite upper bound, the CA is round based, starting with a step and terminating each round with completing the step and thus can run infinitely in theoretical operation. We demonstrate that this mechanism resolves the "infinite loop" problem of static sieves and constructs the fractal structure of prime distributions bottom-up via additive operations. Three fractal processes are applied to the symbol tape and the fractal dimension is calculated. In order to show the efficiency and feasibility of the proposed algorithm, we apply the Chinese Remainder Theorem to the automaton’s state transitions. We present a constructive framework that, if accepted as faithfully modeling the primes, implies the infinitude of twin primes. We show via mathematical induction that the population of twin prime templates grows geometrically with each step, ensuring that they occur within the automaton’s stability zone — which is free of the parity problem — infinitely many times. As we do not use any densities, the lumpiness of twin primes does not matter.
Category: Number Theory
[106] ai.viXra.org:2512.0086 [pdf] submitted on 2025-12-25 22:21:54
Authors: J. W. McGreevy
Comments: 10 Pages.
Geometric Resolution Quantum Field Theory (GRQFT) is a framework that derives the Standard Model of particle physics, general relativity, and a proof of the Riemann Hypothesis from the sequential exhaustion of the arithmetic orbifold Spec(Z) ∪ {∞}. The exhaustion process approximates the circle at infinity with regular n-gons, producing a failure gap ∆(n) = π 2/(54n2 ) that defines the failure class [fn] = coker(fn) ∈ Het1 ´ (O,Z). This failure class is the orthogonal complement and harmonic conjugate (via Cauchy—Riemann equations) to the graded dimensions Vn of the moonshine module V ♮ . Monster group invariance on V ♮ forces the inner product ⟨[fn], vfixed⟩ = 0 for the Monster-fixed Weyl vector vfixed, implying that all zeros of the analytically continued [fn] lie on the balanced circle |n| = e 1/2, corresponding to Re(s) = 1/2 for the Riemann zeta function — thus proving the Riemann Hypothesis. Gauge symmetries emerge from fiber twists: the order-4 µ4 at τ = i yields SU(2)L × U(1)Y , triality at ρ yields SU(3)c with three generations, and octonion non-associativity yields E8 gravity. The Higgs field is the radial mode on the exceptional divisor E2, electroweak symmetry breaking from its resolution, and the Einstein field equations from the geodesic n(t) in the moduli space of elliptic curves. The Weierstrass ℘-function emerges as the failure potential, sourcing Coulomb-like interactions and Rydbergspectra. Planck duality (momentum/energy as circumferences, length/time as radii) andtorsion from non-associativity complete the theory, with supersingular primes as resonancepoints of maximal symmetry. GRQFT realizes arithmetic holography, with exhaustion asrenormalization group flow and Monster symmetry as bulk invariance.
Category: Number Theory
[105] ai.viXra.org:2512.0068 [pdf] submitted on 2025-12-18 21:39:11
Authors: Stephen R. Campbell
Comments: 38 Pages.
This paper introduces Mersenne Block Dynamics, a structural framework for analyzing the accelerated Collatz or Syracuse map on odd integers. The approach decomposes orbits based on the 2-adic valuation of the successor of an odd integer, effectively measuring the length of the trailing run of ones in its binary expansion, termed the Mersenne tail. This decomposition partitions the dynamics into deterministic blocks where the tail length decreases by exactly one bit at each step, creating a rigid wedge pattern in the binary representation. The framework defines a coarse-grained block map that transitions directly between the starts of successive blocks, isolating all arithmetic complexity into a specific exit exponent. The study derives explicit closed-form transition identities and exact time-scale bookkeeping for these block jumps. Furthermore, it establishes that the block length and exit parameters follow independent geometric distributions in terms of natural density. Under a heuristic assumption of orbit mixing, this intrinsic statistical model predicts a net negative expected logarithmic drift, recovering the classical probabilistic prediction for the Collatz conjecture within a precise structural coordinate system.
Category: Number Theory
[104] ai.viXra.org:2512.0063 [pdf] submitted on 2025-12-17 20:30:41
Authors: Jonas Kaiser
Comments: 7 Pages.
We study the Collatz iteration restricted to odd integers and exhibit a concrete emph{core set}(X) inside the forward-invariant set(Y={6n+1,,6n+5:ninmathbb{N}_0}) (odd integers not divisible by $3$).For the odd Collatz map(f_c(u)=(3u+1)/2^{v_2(3u+1)}) we prove that the restriction (f_cvert_X) is a bijection onto $Y$.This yields a redundant-free ``Collatz sieve'': every value in $Y$ has a unique emph{core predecessor} in $X$.For this particular core, the induced dynamics emph{inside} $X$ is strictly decreasing.As a consequence, $X$ decomposes (without duplicates) into disjoint infinite one-sided chains indexed by a set of emph{heads}(Hsubset X): every element of $Xsetminus{1}$ lies on exactly one head chain, and moving one step up a chain increases the time spent inside $X$ by one.
Category: Number Theory
[103] ai.viXra.org:2512.0047 [pdf] submitted on 2025-12-12 21:35:03
Authors: Khazri Bouzidi Fethi
Comments: 8 Pages.
We present an unconditional framework linking the distribution of prime numbers to the zeros of the Riemann zeta function, with fully explicit and computable error bounds. The core of our method is the stratified constant C_{N,P}(s), which converges unconditionally to 2pi. By isolating the contribution of the Riemann-von Mangoldt error term R(T), we derive the first explicit unconditional constraint on its weighted sum, yielding a quantitative coherence test for zero distribution. The framework is extended to Dirichlet L-functions, providing a new measure for Chebyshev bias. We also develop high-precision computational methods for pi(x) beyond x > 10^{12} and validate our results numerically, achieving relative errors as low as 3.2 times 10^{-10}.
Category: Number Theory
[102] ai.viXra.org:2512.0045 [pdf] submitted on 2025-12-11 00:54:58
Authors: Stephen R. Campbell
Comments: 113 Pages. https://doi.org/10.5281/zenodo.17887464
We develop a unified Mersenne block dynamics framework for the accelerated Collatz (Syracuse) map on odd integers, and push it from structural analysis to a concrete finite-certificate criterion. Each odd x is decomposed into a Mersenne tail and an even prefix, giving rise to Mersenne blocks and a residue graph that control the evolution of trajectories. Using a ledger of visits to residue classes together with a height-aware prefix-carry factor, we derive a carry-controlled drift inequality over windows of W Mersenne blocks. Thisyields a finite-certificate criterion: if one can exhibit a finite residuegraph and associated data satisfying a single explicit inequality, together with a finite verification for all odd x < N0, then every trajectory reaches 1. We instantiate this framework with an explicit mod 64 Mersenne-block residue graph, dynamic programming computations of the relevant drift invariant, and a small-n verification. These data are packaged into machine-readable certificate artifacts; we provide explicit certificates whose correctness can be both mechanically and manually verified, and whose validity would give a complete proof of the Collatz conjecture within this framework.
Category: Number Theory
[101] ai.viXra.org:2512.0025 [pdf] submitted on 2025-12-07 01:23:11
Authors: Kevin Fidelis
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We observe an explicit algebraic relationship between certain initial values under the Collatz map. For a starting integer (X) and a chosen step count (m), numbers of the form [Y = X + k cdot 3^u cdot 2^e ] share the same parity sequence as (X) for the first (m) steps, where (e) is the number of even steps in (X)'s first (m) iterations. The difference (Delta_s = C^{(s)}(Y) - C^{(s)}(X)) evolves as [Delta_s = k cdot 3^{u+o_s} cdot 2^{e-e_s}, ] where (o_s, e_s) count odd/even steps up to (s). This relationship persists until the exponent of 2 in (Delta_s) becomes negative. The result is an elementary algebraic identity with no implication for the Collatz conjecture.
Category: Number Theory
[100] ai.viXra.org:2512.0013 [pdf] submitted on 2025-12-03 21:26:44
Authors: Bahbouhi Bouchaib
Comments: 19 Pages.
This paper establishes a complete analytic reduction of Goldbach’s strong conjecture to a single unsolved statement: the Covariance Lemma, which controls the joint distribution of primes at symmetric offsets around E/2. All other components of Goldbach’s problem—including the existence of primes in short symmetric intervals of width proportional to (log E)²—are already unconditionally resolved by explicit results on primes in short intervals, notably those of Dusart [Dusart 2010, Dusart 2018], as well as classical density theorems grounded in the Prime Number Theorem. The key contribution of this work is the identification, isolation, and formalization of the single remaining obstruction. By proving that the covariance of prime indicators P(E/2—t) and P(E/2+t) cannot suppress all symmetric prime coincidences, one obtains a full proof of Goldbach’s strong conjecture. This reduction provides a definitive analytic target for future research, transforming the conjecture from a broad classical problem into a sharply formulated lemma whose resolution is both quantitatively measurable and theoretically constrained.
Category: Number Theory
[99] ai.viXra.org:2511.0098 [pdf] submitted on 2025-11-30 22:15:42
Authors: Durga Shankar Akodia
Comments: 5 Pages.
u200bWe study integer solutions to the exponential Diophantine equation x^2 + k = 2^n where k and n are primes. We prove that for any solution with n ge 2, x must be odd, and k must be an odd prime. Furthermore, we establish the strict congruence k equiv 7 pmod 8 for all solutions with n ge 3. We identify a trivial family of solutions corresponding to Mersenne primes (x=1) and demonstrate the existence of non-trivial solutions for x > 1. Computational evidence is presented for 9 out of 11 prime values of n le 31, revealing 16 distinct non-trivial solutions. We propose the Akodia Conjecture concerning the infinitude of such non-trivial solutions and provide a heuristic justification based on probabilistic number theory and prime density arguments.
Category: Number Theory
[98] ai.viXra.org:2511.0077 [pdf] submitted on 2025-11-21 23:56:58
Authors: Christos Thessalonikios
Comments: 12 Pages.
This is a ccontinuation of me previous preprintwith the title Prime-Anchored Oscillatory Fractals: A VisualExploration of Primes through Helical WeierstrassFunctions. Here i try to create a Hamiltonian operator of the model and the correlation very close to 1 (ρ ≈ 1) and small residualsfor the first 100 prime numbers giving insights in their distribution with a 1D quantum operator consisting of delta barriers which are the prime numbers.
Category: Number Theory
[97] ai.viXra.org:2511.0040 [pdf] submitted on 2025-11-13 21:39:57
Authors: Dobri Bozhilov
Comments: 5 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references!)
The Collatz Conjecture defines a number sequence constructed as follows:If X_1 is odd, then X_2 = 3X_1 + 1.If X_1 is even, then X_2 = X_1 / 2.The conjecture itself asserts that every such sequence, starting from any number, eventually reaches the number 1 (or, more precisely, the cycle 4-2-1).There are two hypothetical exceptions - either another cycle other than 4-2-1, or an infinite divergent sequence. In this paper, we will prove that the second option is impossible.
Category: Number Theory
[96] ai.viXra.org:2511.0039 [pdf] submitted on 2025-11-13 21:35:47
Authors: Gongshan Liu
Comments: 11 Pages.
We report an exploratory statistical analysis of 1,547 Lehmer pairs among the first 10,000 Riemann zeta zeros. Our primary finding is a modest phase clustering pattern: Lehmer pairs show enrichment at phase ϕ ≈ 0.5 in prime-period modulations (observed 29% vs. expected 20%, enrichment 1.45×, p < 10u207b²u2070 after Bonferroni correction for 6 primes). The absolute effect size is small (+9 percentage points), and 71% of Lehmer pairs do not occur at this phase, indicating this is a weak signal rather than a dominant factor. We also observe geometric correlations (R·Δγ = 0.74×, p < 10u207b²u2075) and spatial clustering (79% in clusters), though these may be definitional artifacts. Critical limitations: (1) exploratory analysis—phase pattern discovered post-hoc; (2) only 4.8% of predictive power from truly independent features; (3) extensive multiple testing (~100+ comparisons), only partially corrected; (4) requires replication on independent datasets.
Category: Number Theory
[95] ai.viXra.org:2511.0033 [pdf] submitted on 2025-11-11 20:25:56
Authors: Wiroj Homsup
Comments: 3 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references!)
We model the iteration of the extended Collatz function using an infinite-stateMarkov chain. This probabilistic model provides a framework for analyzing the dynamics of Collatz sequences. We demonstrate that the state corresponding tothe number 1 is the unique absorbing state and argue that, under this model, the system inevitably converges toward this equilibrium.
Category: Number Theory
[94] ai.viXra.org:2511.0025 [pdf] submitted on 2025-11-09 03:42:55
Authors: Wiroj Homsup
Comments: 2 Pages.
This paper presents an argument based on an analogy with an infinite hotel—where rooms correspond to positive integers—and a rearrangement of guests (also positive integers) following the inverse Collatz-tree rules. The construction aims to demonstrate an inconsistency that challenges the validity of the Collatz conjecture.
Category: Number Theory
[93] ai.viXra.org:2511.0015 [pdf] submitted on 2025-11-06 04:21:13
Authors: Dobri Bozhilov
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Collatz conjecture posits that for any positive integer n, the sequence generated by the rules n -> n/2 if n is even and n -> 3n + 1 if n is odd always reaches the cycle 4 -> 2 -> 1. This paper presents a novel observation: consecutive odd numbers in a Collatz sequence are coprime, i.e., their greatest common divisor is 1. We prove this property using basic arithmetic and demonstrate that it imposes a significant constraint on the sequence dynamics, particularly for large numbers. Specifically, the requirement of coprimality creates a "pressure" for the next odd number to be smaller than its predecessor, as larger numbers increase the likelihood of common divisors. This tendency toward decrease makes the existence of alternative cycles (distinct from 4 -> 2 -> 1) increasingly improbable, especially for cycles involving extremely large numbers (e.g., exceeding 2^68). Our result supports the conjecture by suggesting that the arithmetic structure of Collatz sequences favors convergence to 1 over the formation of divergent or cyclic sequences.
Category: Number Theory
[92] ai.viXra.org:2511.0002 [pdf] submitted on 2025-11-02 16:23:23
Authors: Patrick Mcloughlin
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please refrain from making frequent and repeated submissions))
We prove that there is no integer right triangle whose area and perimeter areboth perfect squares. The proof follows directly from the standard parametrisation ofprimitive Pythagorean triples and a short infinite-descent argument.
Category: Number Theory
[91] ai.viXra.org:2510.0078 [pdf] submitted on 2025-10-31 16:11:08
Authors: Christos Thessalonikios
Comments: 8 Pages.
his work introduces a framework for exploring prime numbers through oscillatory fractal structures. Previous studies explored fractal or oscillatory structures associated with prime numbers [1, 2, 3], focusing mainly on abstract series expansions, statistical self-similarity, or approximate prime-counting functions. Here, i construct a prime-anchored Weierstrass-type fractal, with oscillations peaking at prime numbers. By mapping this fractal onto a helical geometry and applying an additive prime envelope, we provide a new geometrical perspective for visualizing primes and analogies with the Riemann zeta function.
Category: Number Theory
[90] ai.viXra.org:2510.0062 [pdf] submitted on 2025-10-26 22:31:04
Authors: Minkwon Chung
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper establishes an analytic equivalence for the Riemann Hypothesis (RH)through the analysis of the exactu2011sieve error E(x,z) = Φ(x,z) − x⋅Π_{p≤z}(1 − 1/p).Here, Φ(x,z) = #{1 ≤ n ≤ x : gcd(n, P(z)) = 1} and P(z) = Π_{p≤z} p. A Schwartz testfunction ψ on a logarithmic scale defines the smoothed error term Eψ(X,z) = ∫ E(e^u,z) ψ(u − X) du. This function admits a Mellin spectral representation Eψ(X,z) = (1/2πi)∫ exp(sX) ψ̂(s) Gz(s) ds, where Gz(s) = ζ(s) Ez(s) / s − Ez(1) / (s − 1) and Ez(s) =Π_{p≤z} (1 − p^(−s)). In this representation, zeros of ζ(s) are zeros (not poles) ofGz(s). The core result is a smoothed spectral equivalence: under RH, |Eψ(log x, z)| ≲z^α (log z)^β x^(1/2+ε); conversely, assuming this bound yields a contradiction via asmoothed explicit formula based on ζu2032/ζ, thereby proving RH. The framework ispositioned relative to the Gonek—Hughes—Keating hybrid product and includes thePrimeu2011Frontier Lemma as a numerical anchor.
Category: Number Theory
[89] ai.viXra.org:2510.0053 [pdf] submitted on 2025-10-23 00:19:44
Authors: J. W. McGreevy
Comments: 21 Pages.
This manuscript presents Geometric-Representation Quantum Field Theory (GRQFT), a functorial framework that derives the Standard Model and General Relativity from arithmetic invariants via the Langlands program. Starting from the Monster group's moonshine module $V^atural$, GRQFT constructs the explicit pathway:[GalRep(mu_4) to AutInd(V^atural) to Twist(BQF) to DispMap(p^mu p_mu = m^2)]Key results include: (1) solution to Hilbert's 12th problem via $mu_4$ i-cycle preperiodic orbits; (2) Grothendieck's 'etale topos realized as the Higgs complex scalar doublet $Phi$; (3) Birch-Swinnerton-Dyer values $L'(E,1)_p approx 0.4472$ yielding the Higgs VEV $v_p = 246$ GeV; (4) Riemann zeta zeros as propagator interference patterns reproducing Planck CMB data; (5) entropy kernel $S_p to g_{muu_p}$ generating Einstein field equations. All predictions match experiment to sub-percent precision.
Category: Number Theory
[88] ai.viXra.org:2510.0047 [pdf] submitted on 2025-10-21 16:01:18
Authors: Monte Richardson
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We describe and present a symbolic framework to analyze the Collatz conjecture by encoding itsoperations and dynamics into deterministic motifs. Odd transformations, (3x+1), are represented by O. Ek represents maximal divisions by powers of 2. With this grammar, we are able to structurally analyze Collatz trajectories without computation or number theory. Five core results are proved:u2022 Non-trivial cycles are a structural impossibility. u2022 Descent from motifs dominates descent. u2022 All numeric trajectories map to a distinct symbolic path. u2022 Convergence is guaranteed u2022 Inverse motifs demonstrated that every positive integer is reachable from 1.Overall, these results create a symbolic resolution to the Collatz process. Over Z+, the grammar is lossless, complete, and reversible. Additionally, the grammar offers a universal and deterministic framework for understanding Collatz dynamics.
Category: Number Theory
[87] ai.viXra.org:2510.0045 [pdf] submitted on 2025-10-20 15:49:27
Authors: Sizwe Tshabalala
Comments: 27 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
In this paper I introduce a branch of math called Nembelo; from it I introduce a novel function I call the An() function. This function adds number to itself allowing us to study numbers in ways that enables us to have insight into numberu201fs behavior. The numbers studied are natural n, primarily 1. To answer the question on the nature of number, for so far the debate has been mostly philosophical and anecdotal, strange enough, not mathematical. This is because it has been taken as granted that math itself cannot answer this question but with the An() function; adding n to itself — we have a way of figuring out what numbers on themselves have to say. As we follow the behavior of n under An(n), it will appear that math has a say on ontology, metaphysics and theology, as many fields have been claimed for math & science in past times, e.g., weather prediction. However the ambition of the math extends to the unification of math, science, biology, sociology, for which the current theses is not adequate to argue for. Nevertheless, the base from which one can stand on is provided, i.e., evidence mapping mathematical conjecture into physical properties of the universe we observe.
Category: Number Theory
[86] ai.viXra.org:2510.0043 [pdf] submitted on 2025-10-19 23:00:24
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript, the eighth in the series on Geometric-Representation Quantum Field Theory (GRQFT), reframes the functorial pathway from arithmetic invariants to physical laws through a kernel-based arithmetic quantum mechanics (AQM) framework, emphasizing the Rydberg-Ritz Combination Principle. We demonstrate how pre-periodic points in rational maps, linked to the Convolution Kernel, serve as class field generators for Hilbert’s 12th problem, extending Kronecker’s Jugendtraum to real quadratic fields. The Third Quadratic Kernel, derived from the Runge-Lenz vector (RLV) and binary quadratic forms (BQFs), embeds SO(4) symmetries and ties to dispersion invariants via the Dispersion Kernel. The Entropy Kernel formalizes thermodynamic derivations inspired by Planck, bounding scales with Spec(Z) primes and Monster supersingular primes. U(1) gauge symmetry, with field strength tensors ( F_{muu} ), and the BSD Matrix Kernel physicalize torsion-to-curvature transitions, culminating in the capstone BSD conjecture. The i-cycle bundle’s 4-torsion, modeled through kernel interactions, connects to 4-vectors via quadratic dispersion. This kernel-centric approach strengthens GRQFT’s resolution of pre-Planck dynamics, with implications for quantum gravity and class field theory.
Category: Number Theory
[85] ai.viXra.org:2510.0037 [pdf] submitted on 2025-10-14 21:07:16
Authors: J. W. McGreevy
Comments: 7 Pages.
This manuscript, the seventh in the series on Geometric-Representation Quantum Field Theory (GRQFT), formalizes the functorial pathway from arithmetic structures to physical dispersion relations using category theory. We define key categories including Galois representations (GalRep), quadratic fields (QuadField), automorphic forms (AutForm), and physical dispersion relations (PhysDisp). A chain of functors is described: automorphic induction (AutInd) from GalRep to AutForm, the i-cycle twist (Twist$_i$) from real to imaginary quadratic fields, and the dispersion mapping (DispMap) from AutForm to PhysDisp. The categorical product resolves the hyperbolic frontier of Hilbert's 12th problem, generating class fields as limits and pullbacks. This framework relates to class field theory, providing a physical-geometric resolution to extension problems via quadratic unification and dispersion kernels.
Category: Number Theory
[84] ai.viXra.org:2510.0035 [pdf] submitted on 2025-10-14 20:25:19
Authors: Chandhru Srinivasan
Comments: 7 Pages.
This work presents empirical evidence for a novel phenomenon concerning the sums of prime factors of odd semiprimes. For a semiprime N=p⋅q, where p,q are odd primes, this work investigates the modular residue properties of the sum s=p+q modulo a dynamically chosen modulus m. We conjecture that s (mod"u2009u2009" m) lies within a small subset R_m⊂Z_m whose size grows sublinearly relative to m. By combining multiple such modular constraints via the Chinese Remainder Theorem (CRT), the candidate space for s shrinks dramatically and help us to reconstruct the prime factors, leading to potential improvements in semiprime factorization algorithms.
Category: Number Theory
[83] ai.viXra.org:2510.0027 [pdf] submitted on 2025-10-11 22:44:53
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript, the sixth in the Foundations of Geometric-Representation Quantum Field Theory (GRQFT) series, advances the quadratic unification framework through a categorical approach using the Yoneda lemma. We establish a rigorous, explicit functorial mapping from the Runge-Lenz Vector (RLV) and Johnson-Lippmann Operator (JLO) conic quadratics---defined by the conservation law $A^2$ (quadratic in position and momentum)---to binary quadratic forms (BQFs) over the integers, mediated by the group law on the elliptic curve $E : y^2 = x^3 - x$ with complex multiplication (CM) by $mathbb{Z}[i]$ and 4-torsion $E[4]$. This Yoneda unification connects arithmetic structures to physical metrics via quadratic dispersion relations, leveraging the Laplace transform to bridge $p$-adic valuations and spacetime geometry. The spectral action $operatorname{Tr}(f(D/Lambda))$ embeds this into energy-momentum tensors, deriving gravity and the Standard Model (SM) from moonshine via the $i$-cycle bundle and mod-4 primes, with ties to Grothendieck's Weil proofs.
Category: Number Theory
[82] ai.viXra.org:2510.0020 [pdf] submitted on 2025-10-09 17:34:03
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the fifth in the series on Geometric-Representation Quantum Field Theory (GRQFT), consolidates the quadratic unification framework as a rigorous mathematical description of quantized energy/momentum oscillations in the pre-Planck spacetime lump. We demonstrate that GRQFT describes the lump as a torsion-dominated epoch, with curvature emerging as an internal symmetry via the Einstein-Cartan formalism. Gravity is associated with p-adic attraction in the arithmetic base, where p-adic norms minimize shared p-factors, echoing gravitational pull. The i-cycle bundle, ramified prime 2, and mod-4 primes are explicitly connected as the arithmetic mechanism for symmetry breaking and metric emergence. Kronecker's complex multiplication extensions are incorporated, showing how abelian extensions from roots of unity generate the torsion structures. Dispersion relations are derived quadratically, unifying relativistic physics with arithmetic structures.
Category: Number Theory
[81] ai.viXra.org:2510.0019 [pdf] submitted on 2025-10-08 18:16:49
Authors: Gongshan Liu
Comments: 21 Pages.
We report two independent but deeply interconnected discoveries in the statisticalbehavior of Riemann zeta function zeros. First, near Primorial values (products ofthe first k primes), the distribution of zero spacings exhibits systematic deviationsfrom Random Matrix Theory predictions, characterized by variance anomalies (ratio1.72, p < 10−10) and precise log-normal fits (KS p = 0.08), completely departingfrom GUE distributions (p < 10−27). Second, the derivative amplitude H(ρ) = |ζu2032(ρ)|correlates with minimal neighbor spacing via Spearman ρ ≈ 1/√2, with this correlationstrengthening by 23.5% within Primorial windows.These findings reveal a three-layer arithmetic structure in prime distribution: theclassical Riemann ζ function (zero locations), a derivative operator (error weights),and a Primorial modulation operator (local enhancement). We propose a unified modulation function G(t, P) that quantitatively predicts H-value suppression (34%), spacing variance increase (72%), and correlation enhancement (24%) at Primorial points,achieving agreement with observations within 5% error.This framework provides a new perspective on prime distribution and demonstratesthe potential of human-AI collaboration in mathematical discovery.
Category: Number Theory
[80] ai.viXra.org:2510.0017 [pdf] submitted on 2025-10-08 03:13:31
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the fourth in the series on Geometric-Representation Quantum Field Theory (GRQFT), extends the functorial pathway outlined in previous installments by focusing on the emergence of diffeomorphism invariance and the evolution of metrics from the arithmetic base to those in general relativity (GR) and Einstein-Cartan theory. Building on the arithmetic-geometric foundations over the ``field with one element'' F$_1$, the i-cycle bundle, elliptic torsion, Runge-Lenz vector (RLV)/Johnson-Lippmann operator (JLO) algebra, and binary quadratic forms (BQFs), we demonstrate how the Néron-Tate height pairing and p-adic valuations define an arithmetic ``metric'' that evolves into the spacetime metric $g_{muu}$. Diffeomorphism covariance emerges from the invariance under modular transformations in the moduli space and Galois actions, ensuring coordinate-independent laws. Torsion in Einstein-Cartan arises from ramification asymmetries, sourced by spinors via the i-cycle twists. This provides a unified derivation of gravitational structures from arithmetic vacua, with explicit mappings and consistency checks.
Category: Number Theory
[79] ai.viXra.org:2510.0013 [pdf] submitted on 2025-10-07 18:00:35
Authors: Berkouk Mohamed
Comments: 18 Pages. In French (Please fill the Submission Form in Englished!)
In 1742, Christian Goldbach wrote to Leonhard Euler proposing the following weak conjecture: Every odd number greater than 5 can be written as a sum of three prime numbers. Euler replied with the stronger version of the conjecture: Every even number greater than two can be written as a sum of two prime numbers. we started by sketching the demonstration of the weak Goldbach conjecture by living, I would not dwell on it, in fact I would bet against it"by putting a little order among the multiple choice of variables and their values, to arrive at a better distribution of the arcs in the circle including the rational numbers a/q, which define points Hardy-Littlewood circle method, on its application to the strong conjecture, A.Helfgott replied in a video-tube: "I think that without hope for the moment, if this were to be given up from my around which the major arcs are articulated and their contribution for the integral > 0. I opted for the Cantor coupling function which generally restores the pairs of integers in perfect order, then within the pairs, a second function called "selection" (f(n)=(n^2+n+4)/2) introduces a second order to choose q, the divisor of the rational in question, afterwards, I was surprised to find a third function which objectifies the decrease and the disappearance of their influence in the integral R'(n).
Category: Number Theory
[78] ai.viXra.org:2510.0012 [pdf] submitted on 2025-10-07 17:55:27
Authors: J. W. McGreevy
Comments: 3 Pages.
This manuscript, the third in the series on Geometric-Representation Quantum Field Theory (GRQFT), extends the functorial pathway outlined in ``The Threefold Way: Derivation of the Standard Model's Three Generations from the Monster Group'' cite{McGreevy2025a} and ``Foundations of GRQFT: Elliptic Torsion, the $i$-Cycle Bundle, and Hidden Symmetries in the Threefold Way -- Part II'' cite{McGreevy2025b}. We rigorously derive a connection between the algebra of the Runge-Lenz vector (RLV) and Johnson-Lippmann operator (JLO) and binary quadratic forms (BQFs), grounded in the arithmetic geometry of elliptic curves with complex multiplication (CM). Starting from first principles, we map the quadratic relationships in RLV orbital equations to BQF discriminants and norms, embedding the so(4) Lie algebra into the space of forms via quaternion actions and lattice endomorphisms. This extension stabilizes the $i$-cycle bundle and torsion structures in the F$_1$ geometry base, providing a unified arithmetic origin for conserved quantities in classical and quantum mechanics. The role in defining geometry over the ``field with one element'' F$_1$ is emphasized, with BQFs classifying the absolute lattices that seed emergent symmetries. Consistency checks and implications for pre-Planck unification are discussed.
Category: Number Theory
[77] ai.viXra.org:2510.0007 [pdf] submitted on 2025-10-05 02:51:14
Authors: J. W. McGreevy
Comments: 4 Pages.
This manuscript expands upon the Geometric-Representation Quantum Field Theory (GRQFT) framework introduced in the inaugural paper of this series, The Threefold Way: Derivation of the Standard Model's Three Generations from the Monster Group'' cite{McGreevy2025}. Building on the functorial pathway from the Riemann zeta function to the Monster group's McKay-Thompson series via automorphic induction and monstrous moonshine, we delve into the arithmetic-geometric base over the field with one element'' ($mathbb{F}_1$) geometry. Specifically, we rigorously derive the SO(4) hidden symmetry of the Kepler problem from the 4-torsion structure of the elliptic curve $E: y^2 = x^3 - x$ over $mathbb{Q}(i)$, ramified at the prime $p=2$. We introduce the $i$-cycle bundle as a principal $mu_4$-bundle encoding the complex multiplication (CM) action, and establish explicit mappings from the Rosati involution and N'eron-Tate height pairing to the Runge-Lenz vector (RLV) and Johnson-Lippmann operator (JLO). This provides a unified arithmetic origin for classical and quantum conserved quantities in GRQFT, linking torsion imbalances to eccentricity and unitarity. The exposition emphasizes mathematical rigor, with explicit computations and embeddings, positioning this as a foundational component for deriving Standard Model structures from arithmetic vacua.
Category: Number Theory
[76] ai.viXra.org:2510.0005 [pdf] submitted on 2025-10-03 19:37:37
Authors: Zirui Wang
Comments: 8 Pages.
This paper presents a complete resolution of the Erdős—Straus conjecture. We introduce a systematic approach termed the "Constraint-Construction Method," which yields explicit infinite families of solutions for all integers satisfying n otequiv 1, 7 pmod{12} . Our principal contribution is the discovery of a unified parametric framework that establishes the conjecture's equivalence to the existence of positive integer solutions to the equation m(4t - 1 - m) = 2n satisfying an elementary parity condition. This framework not only provides a constructive proof for the most obstinate residue classes but also reduces the computational verification from a three-dimensional search to a two-dimensional one, thereby furnishing both theoretical completeness and computational efficacy.
Category: Number Theory
[75] ai.viXra.org:2509.0074 [pdf] submitted on 2025-09-30 02:06:56
Authors: Bahbouhi Bouchaib
Comments: 12 Pages. (Note by ai.viXra.org Admin: Further repetition may not be accepted)
This article develops a full theoretical framework that resolves Goldbach’s strong conjecture through the introduction of a regulatory constant, here denoted as **Z**. The approach begins with the analysis of the *Goldbach comet*, the set of all Goldbach pairs for even integers, and demonstrates that the apparent irregularities (the so-called "failures" or sparse regions) are consistently contained within the band dictated by Z. I show that Z stabilizes the distribution of prime pairs and prevents divergence of the gaps at infinity. The analysis yields a deterministic law: Goldbach’s conjecture is not only valid for all even integers but is universally regulated by Z. Mathematical demonstrations are provided step by step, culminating in a final equation. The article concludes with a discussion connecting Z with existing number-theoretic theorems and conjectures. To decompose even numbers with my method based on UPE that I have recently reported please visit my latest website https://bouchaib542.github.io/upe-riemann-giant/
Category: Number Theory
[74] ai.viXra.org:2509.0065 [pdf] submitted on 2025-09-23 16:56:38
Authors: Bahbouhi Bouchaib
Comments: 17 Pages.
Goldbach’s conjecture asserts that every even integer greater than two can be expressed as the sum of two primes. I present a deterministic proof based on the Unified Prime Equation (UPE) framework, which guarantees the existence of primes in bounded central windows around each integer. By coupling explicit prime inequalities with a geometric analysis of Goldbach offsets (the so-called t-values), I show that every even number admits a representation as a sum of two primes. The proof integrates three complementary insights: (i) explicit analytic bounds ensuring primes within short intervals, (ii) residue-class sieving which restricts candidate offsets to admissible positions, and (iii) the double-linear geometry of t-value sequences, which stabilizes under normalization by logarithmic scales. This yields a constructive method to locate prime pairs for any even integer, thus resolving Goldbach’s conjecture.
Category: Number Theory
[73] ai.viXra.org:2509.0054 [pdf] submitted on 2025-09-20 20:21:29
Authors: Samagra Ganore
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper introduces a novel recursive formula for computing the nth prime number, denoted as p(n). The approach leverages a combination of summation, floor functions, and inclusion-exclusion principles to define both p(n) and an auxiliary function f(m). Base cases are provided for small values, and the formula is presented for general n.
Category: Number Theory
[72] ai.viXra.org:2509.0052 [pdf] submitted on 2025-09-20 20:55:29
Authors: Bahbouhi Bouchaib
Comments: 6 Pages.
Prime Equation (UPE) program: it proves deterministically that primes lie inside the UPE windows of radius proportional to (ln X)² [Bahbouhi 2025]. As a consequence, the overlap of such intervals provides a geometric and analytic path toward additive decompositions, such as those required for Goldbach’s conjecture. Our announcement emphasizes the unconditional nature of this result, the explicit references on which it is built, and the opportunities it opens for further advances in number theory.
Category: Number Theory
[71] ai.viXra.org:2509.0050 [pdf] submitted on 2025-09-19 16:58:18
Authors: Gongshan Liu
Comments: 11 Pages. (Note by ai.viXra.org Admin: For the last time, please use standard math equation type setting such as LaTeX and cite listed scientific references)
This paper proposes an innovative geometric framework for studying the non-trivial zeros of the Riemann zeta function. By introducing the Height Axis (H-axis)—defined as the modulus of the derivative at zeros H = |ζ'(1/2+it)|—we extend the traditional two-dimensional critical line study to a three-dimensional ζ-geometric space. Based on systematic computation of 10^5 zeros, we discover: 1.Geometric Selection Law: Only 0.498% of zeros lie in the "flat region" with H < 1, differing from Random Matrix Theory predictions (31.6%) by a factor of 63.5 (p < 10^{-100}) 2.Height-Spacing Coupling: All 208 pairs of anomalously close zeros (Δt < 0.3) lie in the flat region, showing 100% correlation3.Intrinsic Scaling Law: Spacing between flat region zero pairs follows Δt = (π/√13)·t_{mid}^{-ln(2)/π}, determined by fundamental mathematical constants These findings confirm that the height axis not only serves as an effective research tool but reveals the inherent geometric structure of zeta function zero distribution.
Category: Number Theory
[70] ai.viXra.org:2509.0049 [pdf] submitted on 2025-09-18 18:03:33
Authors: Bahbouhi Bouchaib
Comments: 24 Pages.
This manuscript presents a fully constructive framework — the Unified Prime Equation (UPE) — that (i) resolves the Goldbach problem by a deterministic procedure valid at infinity, and (ii) reveals a spectral bridge from UPE data to the nontrivial zeros of the Riemann zeta function. Part I defines UPE for primes and for Goldbach pairs and proves that UPE never fails: for every even E ≥ 4, UPE returns a prime pair (p, q) with p + q = E; for every integer N > 3, UPE returns a prime y near N. The existence and boundedness of the required displacement follow from classical prime-gap guarantees (Chebyshev—Bertrand) sharpened by Baker—Harman—Pintz (2001), together with density supplied by the Prime Number Theorem. Part II develops the zeta—UPE bridge: a smoothed Goldbach functional derived from the explicit formula shows that oscillations governed by the zeros of ζ(s) are mirrored in the normalized sequence of UPE displacements. A spectral equivalence principle is formulated: if the stable frequencies of UPE data coincide with the imaginary parts of zeta zeros and no other frequencies persist, then the Riemann spectrum is recovered from UPE. The manuscript includes detailed step-by-step demonstrations, increasing numeric examples across prime—rich and prime—poor ranges, and a comparison with major theorems and verifications (Hardy—Littlewood 1923; Chen 1973; Ramaré 1995; Helfgott 2013—2014; Oliveira e Silva et al. 2014). References are cited author-year in the text and listed at the end. I am pleased to share two dedicated websites presenting my recent research on the Unified Prime Equation (UPE): 1. UPE — Riemann https://bouchaib542.github.io/upe-goldbach-riemann/ This site explains the foundations of UPE, demonstrates its role in resolving Goldbach’s Conjecture, and highlights its deep connection with the Riemann zeta function. 2. UPE — Riemann (Giant) https://bouchaib542.github.io/upe-riemann-giant/ This companion site extends the UPE calculator to very large even numbers, up to 4×10^18, using BigInt and Miller—Rabin primality testing. It provides explicit Goldbach pairs together with normalized displacements and corresponding Riemann zeros. Together, these sites illustrate how UPE unifies the arithmetic world of Goldbach pairs with the analytic spectrum of Riemann, giving a complete picture of prime distribution.
Category: Number Theory
[69] ai.viXra.org:2509.0047 [pdf] submitted on 2025-09-17 18:57:52
Authors: J. W. McGreevy
Comments: 9 Pages.
We report the computational discovery that the McKay-Thompson series $T_{3A}(tau)$ for the Monster group exhibits a coefficient pattern that exactly corresponds to the particle content of the Standard Model. The first coefficient, $c(1)=3$, is the number of generations. This result emerges naturally from the Geometric-Representation Quantum Field Theory (GRQFT) framework, which derives physics from the Langlands program. We compute the spectral action incorporating this data and derive the observed value of the cosmological constant
Category: Number Theory
[68] ai.viXra.org:2509.0038 [pdf] submitted on 2025-09-12 19:19:53
Authors: Bahbouhi Bouchaib
Comments: 15 Pages. A second paper abut the unified prime equation (UPE) and the strong conjecture of Goldbach
In this article I present the Unified Prime Equation (UPE), a compact and general formula for prime numbers that leads to an unconditional resolution of Goldbach’s Conjecture. The UPE framework classifies primes through modular symmetry (6k ± 1), introduces a bounded-correction sieve principle, and applies a systematic rule for generating Goldbach pairs for all even numbers. The approach is simultaneously theoretical and constructive: it produces primes near any large integer, predicts symmetric prime pairs for even numbers, and is verified by large-scale computational experiments up to and beyond 10^36. A public website implementation allows any user to test the method on numbers up to 10^5000, demonstrating both transparency and universality. Please visit Goldbach Window (Unconditional Proof): https://b43797.github.io/goldbach-window-unconditional-proof/ and Prime Equation (Prime Detection): https://b43797.github.io/prime-detection-/ The article situates UPE in historical context, from Euler and Goldbach to Hardy—Littlewood, Cramér, Ramaré, Oliveira e Silva, Silveira, and Helfgott. Unlike earlier heuristic or probabilistic models, UPE offers a deterministic rule that is both mathematically structured and practically computable. The conclusion is clear: the Goldbach problem has moved from conjecture to theorem, and the path of three centuries has converged on a remarkably simple modular law.
Category: Number Theory
[67] ai.viXra.org:2509.0037 [pdf] submitted on 2025-09-13 22:03:03
Authors: Gongshan Liu
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please use standard math equation type setting such as LaTeX)
We report the discovery of systematic Primorial anomalies in the distribution of Riemann ζ-function zeros. Through analysis of high-precision data for the first 100,000 zeros, we find evidence that prime distribution may possess a dual-layer arithmetic structure. Near Primorial values (particularly 2310), the zero spacing distribution exhibits highly significant statistical anomalies (p < 0.0002), completelydeviating from Random Matrix Theory predictions (GUE distribution) while following log-normal distribution with high precision (KS test p = 0.899). We introduce a new theoretical framework—an arithmetic dynamical system model—to explain this phenomenon, which incorporates a Primorial modulation operator independent of the classical Riemann ζ-function. This discovery provides new perspectives for understanding prime distribution and demonstrates the potential of human-AI collaboration in mathematical discovery.
Category: Number Theory
[66] ai.viXra.org:2509.0009 [pdf] submitted on 2025-09-05 01:29:23
Authors: Bahbouhi Bouchaib
Comments: 5 Pages.
The Unified Prime Equation (UPE) provides an explicit method to locate prime numbers in the neighborhood of any integer and simultaneously guarantees the decomposition of every even integer into two primes. The UPE integrates a finite sieve, a bounded central window, and a ranking procedure with minimal correction (Δ_step ≤ 2). This framework not only generalizes classical prime detection methods but also leads to an unconditional proof of Goldbach’s Conjecture. Extensive computational verification has been carried out up to astronomically large numbers using both exact and logarithmic predictive modes. The results align with theoretical predictions and surpass historical bounds derived from Hardy—Littlewood, Chen, Ramaré, and others. For more details please visit my websites https://b43797.github.io/goldbach-window-unconditional-proof/ andhttps://b43797.github.io/unified-prime-equation/
Category: Number Theory
[65] ai.viXra.org:2509.0008 [pdf] submitted on 2025-09-05 01:28:40
Authors: Khazri Bouzidi Fethi
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific reference!)
We introduce an exponential stratification method based on the substitution Xn = exp(n^s)tostudy connections between the prime counting function π(x) and the zero counting functionN(T) of= the Riemann zeta function. By replacing the classical representation of ζ(2s) withits Euler product, we obtain a spectral constant C(s) whose convergence to 2π constitutesa numerical criterion for the Riemann Hypothesis. Analysis of the derivative C(s) revealsspectral stability properties and opens a systematic investigation pathway for major numbertheory conjectures.
Category: Number Theory
[64] ai.viXra.org:2508.0082 [pdf] submitted on 2025-08-31 19:27:06
Authors: Hamid Javanbakht
Comments: 23 Pages.
We propose the Habiro Purity Conjecture, a structural statement asserting that Frobenius eigenvalues in Habiro cohomology satisfy a universal purity condition, mirroring Deligne’s theorem for varieties over finite fields. Together with the determinantal formalism of Habiro cohomology, this conjecture would imply the Riemann Hypothesis for the Riemann zeta function and its generalizations.The context is provided by two recent developments. Scholze’s Habiro cohomology furnishes a cohomology theory interpolating between q-de Rham, prismatic, crystalline, and motivic theories, valid across all primes simultaneously and defined over the Habiro completion. In parallel, Garoufalidis—Scholze—Wheeler—Zagier introduced the Habiro ring of number fields, a universal coefficient ring encoding Frobenius compatibilities and regulators from algebraic K-theory.We argue that these two constructions fit together as dual aspects of a single framework: the Habiro ring provides the coefficients, while Habiro cohomology supplies the machinery. The Habiro Purity Conjecture then emerges as the structural principle governing Frobenius actions in this setting, offering a direct arithmetic analogue of Deligne’s proof of the Weil conjectures.
Category: Number Theory
[63] ai.viXra.org:2508.0054 [pdf] submitted on 2025-08-21 12:41:17
Authors: Marco Ripà
Comments: 8 Pages. This mathematics preprint on digit distributions in OEIS A001292 (Kashihara, 1996) gives exact probability formulas for complete and partial blocks. I solved the problem in 2010 and now tested GPT-5 to replicate it.
This work presents the solution of Problem 1 of Question #30 from 'Comments and Topics on Smarandache Notions and Problems', published in 1996 by Kenichiro Kashihara, corresponding to Problem 16 of 'Only Problems, Not Solutions!'. It is a recreational mathematics problem that remained open until 2010, when an extended version was published in Appendix 2 of 'Divisibilità per 3 degli elementi di particolari sequenze numeriche', Rudi Mathematici Bookshelf. The question concerns OEIS sequence A001292 (the so-called "circular sequence") and the probability that a generic element ends with a given digit c∈{0,1,2,...,9}. In 2010, an enhanced version of the problem was studied, providing the general formula of this probability as a function of the length of the last complete block and of c, and using the exact value to bound the interval in which the probability lies in the inter-block (partial block) case, as reported in note 14 on page 17 of the aforementioned work. On August 9, 2025, GPT-5 independently solved the same enhanced version of Kashihara's question, proposing a more compact formula and extending it exactly to the incomplete block case as well.
Category: Number Theory
[62] ai.viXra.org:2508.0043 [pdf] submitted on 2025-08-15 19:37:47
Authors: Bahbouhi Bouchaib
Comments: 32 Pages. New and original article on Goldbach's strong conjecture (Note by ai.viXra.org Admin: Please convert LaTeX codes to regular math expression)
The symmetry—resonance method presented in this work introduces a novel approach to the long-standing Goldbach Conjecture by interpreting prime number distribution through a geometric and harmonic framework. In contrast to classical statistical models such as Cramér’s gap model or asymptotic density predictions from the Hardy—Littlewood conjectures, our method models the set of prime numbers as nodes of constructive interference in a resonance field centered at E/2 for an even number E. Goldbach pairs (p, q) are predicted to occur at symmetric positions equidistant from E/2, where both members of the pair are primes. We define a predictive gap formula [delta(E) approx sqrt{E} cdot frac{log log E}{log E}]that determines the expected offset from E/2 to the nearest prime in each direction. This formula is theoretically consistent with refined gap heuristics and is compatible with known conjectures, yet it adds a localization mechanism that enables prediction of the positions of Goldbach pairs rather than mere existence. Extensive computational tests up to E > 10^{30}, coupled with primality verification for all candidate pairs, confirm the method’s stability and predictive power. Simulations extended theoretically up to 10^{1000} suggest that the resonance—symmetry pattern persists across vast numerical scales. This reinforces the interpretation of Goldbach’s Conjecture as a manifestation of a deeper harmonic order in the primes, potentially reducing its proof to verifying the universality of the resonance law. By bridging additive prime theory, prime gap heuristics, and harmonic models, the symmetry—resonance approach offers a unified predictive framework. If fully formalized, this method could not only resolve the Goldbach Conjecture but also provide insight into other prime constellations, including twin primes and k-tuples.
Category: Number Theory
[61] ai.viXra.org:2508.0031 [pdf] submitted on 2025-08-13 17:07:01
Authors: Predrag Terzić
Comments: 6 Pages. In Italian (Note by ai.viXra.org Admin: Please cite and list scientific references)
We introduce a new primality criterion for Wagstaff numbers.
Category: Number Theory
[60] ai.viXra.org:2508.0027 [pdf] submitted on 2025-08-11 20:30:22
Authors: Bahbouhi Bouchaib
Comments: 26 Pages. New original work on Goldbach's strong conjecture
This work presents a novel approach that combines the geometry of a prime-generating spiral with the analytic structure of the Riemann zeta function to improve predictions for Goldbach pairs. The central idea is to map the non-trivial zeros of the zeta function onto angular positions along a spiral representation of the natural numbers, thereby identifying zones where prime numbers are statistically more concentrated. These zones are then used as "radar sectors" to search for the two prime components (p) and (q) of an even number (E), with (p + q = E). By synchronizing the predicted locations of primes from the spiral model with the fine structure revealed by zeta zeros, the method reduces the candidate search gap significantly compared to classical estimates such as Cramér’s bound and the Hardy—Littlewood conjectures. The methodology is tested on a wide range of even numbers, with verification up to (10^{16}) and theoretical projections toward (10^{18}). Results show that the predicted pairs remain extremely close to the actual pairs, with small gaps between the predicted prime positions and their verified counterparts. The integration of geometric and analytic perspectives offers a new framework for connecting Goldbach’s conjecture to the distribution of primes as seen through the lens of the Riemann Hypothesis. This research opens pathways toward narrowing the bounds for prime gaps, refining predictive algorithms for prime locations, and potentially offering partial progress toward an analytical proof of the strong Goldbach conjecture. Future work will explore extending the spiral—zeta coupling beyond current computational limits and investigating whether these methods retain their predictive power toward infinity.
Category: Number Theory
[59] ai.viXra.org:2508.0024 [pdf] submitted on 2025-08-11 19:27:31
Authors: Kaiche Mouad
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We present a family of 24 quadratic polynomials in two variables that together generate the set of odd natural numbers N, excluding thefirst number of every twin prime pair. The construction of these polynomials is based on a study of triangular numbers and results in overlapping outputs. While this note focuses on presenting the empirical result and the formulas, a complete proof has been developed by the author. The author welcomes collaboration with qualified researchers to assist in formalizing and preparing a full paper for peer-reviewed publication.
Category: Number Theory
[58] ai.viXra.org:2508.0023 [pdf] submitted on 2025-08-10 17:26:45
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper introduces a novel hybrid approximation method for the prime counting function π(x) that adaptively uses three different approaches based on the input range. For small values (10 ≤ x ≤ 10³), we employ a simple decimal logarithm formula π̂u2081(x) = x/(2logu2081u2080x). For intermediate values (10³ smaller than x ≤ 10²u2074), we optimize parameters a and b in the rational expression π̂u2092u209au209c(x) = x/(ln x - 1 - a/ln x - b/ln²x) using least squares linearization. For large values (x > 10²u2074), we use the asymptotically optimal logarithmic integral Li(x). The key innovation lies in transforming the nonlinear parameter optimization into a linear least squares problem, enabling efficient and stable coefficient determination. Numerical validation on known exact values demonstrates relative errors consistently below 0.3% across all tested ranges, with computational complexity varying from O(1) for small values to O(log x) for large values.This region-based approach provides an optimal balance between computational efficiency and approximation accuracy, making it suitable for diverse applications in computational number theory, cryptography, and prime number research. The method's hybrid nature ensures both practical usability for moderate computations and theoretical validity for asymptotic analysis.
Category: Number Theory
[57] ai.viXra.org:2508.0020 [pdf] submitted on 2025-08-09 22:45:36
Authors: Ciro Tarini
Comments: 11 Pages. done with the help of Gemini 2.5 pro and GPT-5
Fermat's Last Theorem, conjectured by Pierre de Fermat in 1637, stood as one of the most famous unsolved problems in the history of mathematics. Despite its apparent simplicity, the theorem resisted proof for over 350 years until the monumental work of Andrew Wiles. In this paper, we present an alternative and novel approach that aims to provide an elementary proof of the theorem, based on fundamental algebraic concepts and divided into distinct cases.
Category: Number Theory
[56] ai.viXra.org:2508.0009 [pdf] submitted on 2025-08-04 19:43:01
Authors: Howard Johnson
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present an analytic proof of the Goldbach Conjecture using a hybrid framework that combines symbolic bounds from the Hardy—Littlewood circle method with log-bounded prime pair envelopes centered around each even integer X > 2. We prove that the interval [Xu20442 − Xu2044logᵏX, Xu20442 + Xu2044logᵏX], for k ∈ [1.0, 2.0]always contains at least one prime p such that X − p is also prime for all X ≥ 10u2075, and use symbolic methods to show that r(2n) ≥ ε > 1 for all even integers X > 1000. These results constructively and analytically validate Goldbach’s Conjecture for all even integers X > 2. Figures and symbolic bounds support the completeness of this method.
Category: Number Theory
[55] ai.viXra.org:2508.0002 [pdf] submitted on 2025-08-01 18:40:11
Authors: Bouchaïb Bahbouhi
Comments: 36 Pages. (Note by ai.viXra.org Admin: Please Do Not useg grandiose title/wording)
This study presents a novel geometric approach to prime number distribution using a corrected spiral formula of the form: r(θ) = a·θ + b·sin(k·θ) Unlike traditional linear models, the spiral structure allows for a two-dimensional mapping of natural numbers in polar coordinates, where prime numbers tend to align along harmonic trajectories. We demonstrate that by combining this corrected spiral with estimates derived from the Prime Number Theorem, one can effectively locate approximate positions of the n-th prime. Additionally, we show that Goldbach pairs (p, q) corresponding to even integers E = p + q exhibit a symmetrical alignment on the spiral, reinforcing the conjecture’s plausibility from a geometric perspective.Three key results are established:1. The corrected spiral enables a visual and approximate detection of prime locations.2. Goldbach’s conjecture is visually supported via symmetrical pairing of primes on the spiral arms. 3. A predictive method is proposed to estimate the n-th prime and project it onto the spiral using a hybrid analytical-visual framework.This approach opens a new window for interpreting number-theoretic structures through geometry and may offer intuitive insights into longstanding problems like the Goldbach Conjecture and the distribution of prime gaps. Goldbach decmposition can be perfrmed by the prime SPIRAL up to 105 on this new website https://bouchaib542.github.io/-Goldbach-s-Spiral-Symmetry-Principle/
Category: Number Theory
[54] ai.viXra.org:2507.0114 [pdf] submitted on 2025-07-25 16:46:18
Authors: Bahbouhi Bouchaib
Comments: 9 Pages. (Note by ai.viXra.org Admin: For the last time, please cite listed scientific references)
This article presents a novel predictive approach to Goldbach's Conjecture, centered around a deterministic formula based on the even number E to estimate its Goldbach pair (p, q), such that E = p + q. I introduce a theoretical model where the difference δ(E) is calculated as δ(E) ≈ √E · (log log E) / log E. Using this δ, the predicted pair is defined as p = E/2 − δ and q = E/2 + δ. This approximation is designed to localize the expected prime numbers p and q that sum to E with remarkable precision. Extensive computational tests were conducted for even values of E up to 10^18, showing that the predicted values (p, q) are consistently close to the actual primes that form the valid Goldbach decomposition of E. The website accompanying this study allows users to enter any large even number and instantly obtain both the predicted values (p_pred, q_pred) and the nearest valid prime numbers (p_real, q_real) such that p_real + q_real = E (visit website here : https://bouchaib542.github.io/Goldbach-real-vs-predictif/). The closeness between the predicted and real primes demonstrates the remarkable accuracy of the formula, suggesting that the apparent randomness of prime distribution can, to some extent, be constrained by analytical expressions. This predictive structure does not replace the original Goldbach conjecture but provides a powerful tool for investigating the range in which the solution lies, potentially opening new avenues for heuristic and computational exploration of prime patterns. This article is part of a broader project aimed at systematically bridging empirical regularities and theoretical models in number theory, particularly in the context of additive prime decompositions.
Category: Number Theory
[53] ai.viXra.org:2507.0113 [pdf] submitted on 2025-07-25 16:38:46
Authors: Giuseppe Fierro
Comments: 2 Pages. Italian and English (Note by ai.viXra.org Admin: Please cite listed scientific references)
We prove that for every positive integer n, the inequality Ω(2n−1) ≥ Ω(n) holds, where Ω(n) denotes the total number of prime factors of n, counted with multiplicity. This result establishes an interesting connection between the multiplicative structure of n and that of the corresponding Mersenne number 2n −1.
Category: Number Theory
[52] ai.viXra.org:2507.0110 [pdf] submitted on 2025-07-24 00:07:34
Authors: Bahbouhi Bouchaib
Comments: 11 Pages.
This article compares two fundamentally different but complementary methods for predicting (p, q) decompositions of even numbers E such that E = p + q and both p, q are prime. The first method, GPS-based, predicts symmetric decompositions around E/2, with the critical parameter t such that both E/2 − t and E/2 + t are prime. The second method, introduced here as CJAEG (Conjecture of Twin primes Anti-Équidistants of Goldbach), is based on anti-equidistant partitions (A − s, B + s) of any even number E = A + B, with s controlling the imbalance. We show that for many values of E, s = 1 suffices if twin primes exist near the partition, offering a new pathway to validating Goldbach’s Conjecture. A new calculator for decomposing an even number into the sum of two prime numbers, based on the data in this article, is available on the internet. Here is the link: https://bouchaib542.github.io/Goldbach-CJAEG-Twin-Decomposition/
Category: Number Theory
[51] ai.viXra.org:2507.0102 [pdf] submitted on 2025-07-21 07:27:24
Authors: Shanzhong Zou
Comments: 5 Pages.
This paper provides a proof by a contradiction, leveraging Terence Tao’s result that any hypothetical set H of counterexamples does not diverge. We prove that the minimal element h1∈H must satisfy h1=12k+3 and derive a contradiction in the resulting cycle structure modulo 3. This confirms the Collatz conjecture.
Category: Number Theory
[50] ai.viXra.org:2507.0100 [pdf] submitted on 2025-07-21 22:29:24
Authors: Seojoon Lee
Comments: 5 Pages. (Note by ai.viXra.org Admin: An abstract in the article is required)
The Goldbach Conjecture, first proposed in 1742, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite being one of the oldest unsolved problems in number theory, a complete proof has remained elusive, although extensive computational evidence supports its validity.
Category: Number Theory
[49] ai.viXra.org:2507.0084 [pdf] submitted on 2025-07-16 19:37:03
Authors: Lemuel Schaine Harris
Comments: 9 Pages. CC BY 4.0 (Attribution)
In 1993, Texas banker and amateur mathematician Andrew Beal proposed his eponymous conjecture—an elegant generalization of Fermat’s Last Theorem—offering a $1 million prize and inviting both professional mathematicians and enthusiastic amateurs to explore the mysteries of exponential Diophantine equations. We present a self-contained, contradiction-based proof of Beal’s Conjecture via our new Cuboid-Valuation Method,framed within a humanistic narrative that traces the geometric roots of volume-tiling arguments from ancient Greek mathematics to modern exponent Diophantine inequalities. Our approach relies on two central number-theoretic pillars—Zsigmondy’s theorem on primitive prime divisors and the Lifting-The-Exponent Lemma (LTE)—to undergird the contradiction arguments across every exponent configuration. In doing so, we resolve a long-standing open problem (modulo these widely accepted theorems) and celebrate how spatial intuition and historical perspective can enrich algebraic reasoning and inspire mathematical discovery at all levels.
Category: Number Theory
[48] ai.viXra.org:2507.0065 [pdf] submitted on 2025-07-13 02:46:21
Authors: Justin Sirotin
Comments: 28 Pages. Distributed under CC BY-NC-ND 4.0
This paper presents a proof of Beal’s Conjecture, which states that any integer solution to the equation Ax+By=Cz for exponents x,y,z>2 must have gcd(A,B,C)>1. We proceed by contradiction, assuming the existence of a primitive solution where A,B,C are pairwise coprime. To this solution, we associate a triad of Frey—Hellegouarch elliptic curves. By leveraging the Modularity Theorem, we show that the mod-ℓ Galois representation attached to at least one of these curves must correspond to a weight-2 newform of a specific level. A new central lemma is proven, guaranteeing the validity of an iterative level-lowering argument that systematically removes all odd prime factors from the conductor, forcing the representation to arise from a newform of level 2. As the space of such forms is zero-dimensional, this yields a contradiction. The argument is comprehensive, with explicit local computations for the conductor, rigorous proofs for the irreducibility of the Galois representation for small prime exponents, and a complete resolution of all exceptional exponent cases not covered by the main theorem, thereby establishing the conjecture.
Category: Number Theory
[47] ai.viXra.org:2507.0064 [pdf] submitted on 2025-07-13 02:47:00
Authors: Justin Sirotin
Comments: 29 Pages. Distributed under CC BY-NC-ND 4.0
We present a comprehensive survey of the theory of elliptic curves over the rational numbers, centered on the Birch and Swinnerton-Dyer (BSD) conjecture. We trace the historical development of the subject, from the foundational results of Gross-Zagier and Kolyvagin for curves of rank at most one, through the development of Iwasawa theory and the proof of the Main Conjecture for GL(2), to the recent breakthroughs in arithmetic statistics by Bhargava, Skinner, and Zhang. Throughout, we ground the discussion in explicit computational data from the L-functions and Modular Forms Database (LMFDB), illustrating the deep interplay between theory, computation, and conjecture that defines modern number theory.
Category: Number Theory
[46] ai.viXra.org:2507.0056 [pdf] submitted on 2025-07-10 21:39:21
Authors: bouchaïb Bahbouhi
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references; further repetition may not be accepted))
This article presents a publicly accessible implementation of a method for decomposing biprime numbers—integers formed by the product of two prime numbers—using a guided estimation technique inspired by geometric and arithmetic properties. The method utilizes a hybrid GPS-like approach to predict the midpoint and deviation between the two prime factors, significantly reducing the search space. The technique is demonstrated through a fully operational web interface hosted on GitHub Pages. The tool is capable of factoring biprimes efficiently up to approximately 10^16. This work contributes to practical number theory and cryptographic education by offering a transparent and verifiable method. The method can be used on the Website: https://bouchaib542.github.io/Biprimes-decomposition-/
Category: Number Theory
[45] ai.viXra.org:2507.0044 [pdf] submitted on 2025-07-07 23:47:51
Authors: Budeeu Zaman
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper presents a novel identity that establishes a surprising link between prime number theory and the Collatz Conjecture, two foundational yet independently explored areas of number theory. The identity [...] where pi denotes the ith prime number, offers a structured summation that yields a linear expression in n, namely 3n+1. This linearity echoes the recursive rule governing the Collatz sequence for odd integers: 3n + 1, suggesting a deep, intrinsic connection between the distribution of primes and the dynamics of the Collatz iteration. The study explores this identity analytically and numerically, providing insights into its validity, scope, and potential implications. The structure and behavior of the difference terms (pi+1 −pi)(n−i) are also analyzed, highlighting a hidden regularity in prime gaps when viewed through the lens of this summation. The result opens new avenues for reinterpreting prime behavior in discrete systems and lays foundational groundwork for a possible unification of discrete dynamical processes and prime arithmetic.
Category: Number Theory
[44] ai.viXra.org:2507.0032 [pdf] submitted on 2025-07-06 02:14:27
Authors: Bahbouhi Bouchaib
Comments: 48 Pages.
This article introduces a predictive and structural method for the factorization of biprime numbers Bu2099 = p × q, where p < q are both primes. Instead of relying on brute-force or traditional number-theoretic algorithms, I reformulate the problem geometrically using two core variables: the midpoint m = (p + q)/2 and the half-gap w = (q − p)/2. I show that these values (m, w) can be forecasted with high accuracy by analyzing the evolution of smaller biprimes, particularly the gap behavior between their prime factors and their relation to √Bu2099.We also develop a GPS-like algorithm that learns from the progression of known biprimes to narrow down prime factor candidates for unknown biprimes. Our method incorporates modular arithmetic, especially focusing on prime factor forms 6x − 1 and 6x + 1, and tests their predictive utility in factor identification.Empirical validation demonstrates that this approach can effectively recover prime factors of biprimes up to size 10²². In addition, when applied to Goldbach's strong conjecture it validates it up to 1066. Furthermore, we explore the impact of gap size (q − p) on m’s distance from √Bu2099, offering a rule-of-thumb for selecting the most appropriate prediction strategy.The framework is tested and compared against classical methods like Fermat’s, Pollard’s rho, and GNFS, showing improved performance in specific regimes. This work contributes a new heuristic and structural paradigm for understanding and decomposing biprimes, with potential applications in computational number theory and cryptographic analysis.
Category: Number Theory
[43] ai.viXra.org:2507.0031 [pdf] submitted on 2025-07-06 02:15:26
Authors: Bahbouhi Bouchaib
Comments: 48 Pages.
The hybrid GPS method developed in this study is based on the fusion of two powerful mathematical strategies: a predictive GPS-like scanning algorithm and an exponential reconstruction equation involving a known or assumed prime and its corresponding symmetric counterpart , such that N = p + q. .At the heart of the method lies the principle of symmetry around the even number 2N. Given that every even number can potentially be expressed as the sum of two primes p and q, we define a variable t such that: 2N = (N - t) + (N + t)The algorithm searches for the smallest such that both (N — t) and are prime (N + t). This leads to a bidirectional sweep around N, which mimics a GPS scan from the center of the interval toward its boundaries. This sweeping mechanism significantly reduces computational overhead by focusing on a symmetric neighborhood around the target even number.To enhance this search, the method integrates an exponential prediction equation. If a prime p is known or presumed, we use the recursive relation: X_k = 2kq + ( 2k - 1) p. This equation helps reconstruct possible values of N = p + q at higher orders of magnitude. The strategy works both forward and backward: starting from a given p, we can estimate a large N, or starting from a large even N , we can try to infer p and q.The process also incorporates modular constraints, particularly primes of the form 6x ± 1 , which are frequent candidates in Goldbach decompositions. By combining these insights with dynamic filtering and local prime density predictions (e.g., via the Prime Number Theorem), the hybrid method achieves high accuracy and remarkable depth, surpassing previously known computational limits. The hybrid GPS method successfully verified Goldbach's Conjecture for all even integers up to 10¹u2070u2070u2070, representing an unprecedented computational achievement. This result underscores the predictive power and scalability of our approach. These data led to one website https://b43797.github.io/Bahbouhi-decomposing-Goldbach-conjecture2025/ to decompose an even E > 4 in sums of two primes E = p + q up to an unprecedent level of E = 10^18. Another website shows validation of Goldbach's strong conjecture to 10^10000 (see table 2 here) and much more with examples obtained by the method described in this article, this website is https://b43797.github.io/Archive-III/
Category: Number Theory
[42] ai.viXra.org:2507.0023 [pdf] submitted on 2025-07-04 22:19:42
Authors: Swapnil Khan Mahi
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
I present a new analytical formulation to determine whether a given number ( p ) is prime. The method involves constructing a specific sequence of integer products and mapping it to a continuous function via Fourier representation. Using complex analysis, particularly the argument principle, we explore the number of zeroes of this function to infer the compositeness of ( p ). This approach provides a new perspective on primality testing through continuous mathematics.
Category: Number Theory
[41] ai.viXra.org:2506.0131 [pdf] submitted on 2025-06-29 01:22:32
Authors: Bahbouhi Bouchaib
Comments: 32 Pages.
This paper introduces a new predictive algorithm, referred to as Algorithm-T, for addressing the Strong Goldbach Conjecture. The method is based on identifying an optimal distance t from the midpoint N/2 such that both N/2 − t and N/2 + t are prime numbers. The algorithm adapts to the modular class of the even number N (of the form 6x, 6x+2, or 6x+4) and selects t accordingly—using primes for certain forms and multiples of 3 for others. By narrowing the search space around N/2, Algorithm-T offers a highly efficient and structured approach to identifying Goldbach pairs (p, q) satisfying p + q = N. Extensive computational tests show that the algorithm is robust and accurate: no failures are observed up to 106, and only one exception is found up to 109. The results demonstrate a strong correlation between the optimal t-values and the function N/log(N), providing a meaningful connection with classical results such as the Prime Number Theorem. Algorithm-T thus provides a predictive, modular, and computationally light framework for exploring the structure of prime sums. A comparative analysis of the three modular classes—6x, 6x+2, and 6x+4—reveals interesting differences in the behavior and performance of Algorithm-T. Even numbers of the form 6x show the highest success rate and fastest convergence to a valid prime pair, followed by 6x+2. The 6x+4 class, while still highly successful, occasionally requires slightly larger t-values. This variation highlights the subtle role modular structure plays in guiding the algorithm’s effectiveness. This study presents the method in detail, evaluates its empirical success across modular families, and discusses its potential contributions to the understanding and verification of Goldbach-type problems.
Category: Number Theory
[40] ai.viXra.org:2506.0115 [pdf] submitted on 2025-06-26 03:19:00
Authors: Paolo Viscariello
Comments: 16 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We propose an orbital approach to the Riemann Conjecture, introducing a new function ζ_Φ(s) derived from a discrete harmonic field regulated by φ³. We construct a harmonic operator ℛ that transforms ζ_Φ(s) into the classical zeta function ζ(s). The model produces zeros compatible with those known for ζ(s), with mean error <0.01%, and presents a theoretical basis consistent with the golden section, logarithms, and orbital symmetry. We include preface, numerical simulations, technical appendices, and falsifiable predictions.
Category: Number Theory
[39] ai.viXra.org:2506.0106 [pdf] submitted on 2025-06-23 21:02:30
Authors: Shanzhong Zou
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Assuming that the hail conjecture is wrong, there is a number set H, and none of which satisfies the conjecture. In H, there must be a smallest number. Because the minimum number cannot be found by the conjecture algorithm, thus proving that H must be empty.
Category: Number Theory
[38] ai.viXra.org:2506.0081 [pdf] submitted on 2025-06-19 21:19:01
Authors: Patrick Kotal
Comments: 13 Pages.
This paper presents a proof of the Collatz conjecture. By analyzing the dynamics of the original Collatz operations within a stochastic process model, we prove that they lead to contraction due to an inevitable and deterministic lower bound for the ratio a/b of the counter variables. We finally show by strong induction that the original Collatz operations applied to any positive integer n>1 can only produce sequences that contract to 1.
Category: Number Theory
[37] ai.viXra.org:2506.0079 [pdf] submitted on 2025-06-18 21:07:25
Authors: Thoriso Motlola
Comments: 2 Pages.
This manuscript introduces and formalizes the concept of *Servitude of Numbers*, a novel mathematical theory exploring the symmetry between distinct arithmetic operations that yield identical results, such as addition equaling multiplication under specific conditions. Building upon this foundation, the work presents several original identities and inverse transformations, proposing new mathematical routes to solving square numbers and modeling numerical relationships. The manuscript also introduces the *Osiroht Theorem*, a theoretical construct describing infinite angular division as a metaphor for infinite dimensional reality, wherein spatial angles converge toward but never reach a defined boundary (180°). Together, these discoveries form a philosophical and mathematical framework that challenges traditional notions of operation, dimension, and numerical behavior. This work contributes to theoretical mathematics, with potential implications for physics, cosmology, and metaphysical interpretation.
Category: Number Theory
[36] ai.viXra.org:2506.0043 [pdf] submitted on 2025-06-09 17:50:06
Authors: Daoudi Rédoane
Comments: 1 Page.
This paper aims to present several formulas concerning number theory. In particular, I establish deep connections between various constants such as Euler-Mascheroni constant, pi, square roots, logarithms, the exponential function, and the gamma function.
Category: Number Theory
[35] ai.viXra.org:2506.0032 [pdf] submitted on 2025-06-07 21:04:43
Authors: Bryan Clem
Comments: 4 Pages. (Note by ai.viXra.org Admin: For the last time, please cite and list sceintific references)
The Bailey Earl Prime Key Conjecture is a proposed mathematical framework that redefines number theory & classification, extends prime identification, and factorization through a structured, self explanatory distribution system, and with infinite Keys. Unlike traditional systems that treat primes as isolated or randomly distributed entities, the Infinite Key System organizes all natural numbers into infinite hierarchical keys. Each number is assigned to the first key where it functions as both a multiple and a prime within that key’s internal logic, establishing a geometric and structural symmetry across the entire number system. At its core, this system eliminates the need for probabilistic or trial-based methods in prime detection and factorization. It offers a method to predict and classify primes to infinity by ensuring that every number finds its proper place in a perfectly organized key structure, where all primes are inherently embedded. This unlocks the potential for deterministic prime prediction, minimal-computation sieving, and refined control over number theory functions. [Truncated by ai.viXra.org Admin]
Category: Number Theory
[34] ai.viXra.org:2506.0031 [pdf] submitted on 2025-06-07 20:57:48
Authors: Seojoon Lee
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
The Collatz conjecture remains one of the most well-known open problems in mathematics. In this paper, we propose a novel framework for analyzing the Collatz sequence using function composition and modular arithmetic. By defining the functions F(x) = 3x + 1 and G(x) = x/2 , and encoding the iterative structure of the Collatz process as a composition of F and G operations.
Category: Number Theory
[33] ai.viXra.org:2506.0023 [pdf] submitted on 2025-06-06 20:25:29
Authors: Hannah McCoy, Stephen Raphael Manning
Comments: 42 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
We introduce a novel modular resonance approach to the Riemann Hypothesis by constructing a modified zeta function, ζ_mod(s), derived from a deterministic sieve of modular prime residues. This function admits full analytic continuation, is defined via a Mellin transform of a modular theta kernel, and forms an Euler product analogue to ζ(s). A real scaling transformation α ≈ 1.0083 establishes a spectral bijection betweenζ_mod(s) and ζ(s), such that: ζ(s) = ζ_mod(αs) for all s ∈ ℂ {1} Using operator analysis, bounded tail convergence, and zero alignment, we demonstrate that the non-trivial zeros of ζ(s) correspond precisely to those of ζ_mod(s), all lying on the critical line Re(s) = 1/2. This constitutes a conditional yet comprehensivesymbolic framework linking classical and modular prime structures through analytic, algebraic, and spectral equivalence. The method provides a potential new path toward a constructive proof of the Riemann Hypothesis and reveals previously unseen resonance patterns in the distribution of primes.
Category: Number Theory
[32] ai.viXra.org:2506.0011 [pdf] submitted on 2025-06-04 00:03:06
Authors: Steve Costello
Comments: 27 Pages.
This paper introduces a novel structural approach to the Collatz Conjecture using arooted binary tree framework based on odd integers. The infinite tree, rooted at 1, encompasses all positive integers and provides a systematic method for establishing parent-child relationships among Collatz predecessors. In particular, a rule-based path construction is defined to resolve ambiguity in the standard inverse Collatz map, especially for even integers of the form 2�� where �� > 2. Thisframework offers new insight into the conjecture’s universal convergence property.
Category: Number Theory
[31] ai.viXra.org:2506.0004 [pdf] submitted on 2025-06-03 01:07:02
Authors: Anatoly Galikhanov
Comments: 23 Pages.
We introduce a pro-étale geometric object D∞ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor O whose fixed points correspond to automorphic representations of GL2(AF2). The main theorem establishes that invariant predicates on D∞ parametrize cuspidal automorphic representations, preserving L-functions. We provide complete proofs using ∞-categorical techniques, explicit computations for small cases, and establish connections to discrete conformal field theory. As applications, we resolve the Carlitz-Drinfeld uniformization conjecture for function fields and compute previously unknown motivic cohomology groups. Our approach differs fundamentally from coalgebraic models by working internally in topoi and connecting to arithmetic geometry.
Category: Number Theory
[30] ai.viXra.org:2506.0003 [pdf] submitted on 2025-06-03 01:03:55
Authors: Giuseppe Fierro
Comments: 4 Pages. In Italian and English (Note by ai.viXra.org Admin: Please cite and list sceintific references)
This article proves the logical equivalence between a conjecture which, as far as we know, has not been previously published, which asserts the existence of at least two primes symmetric about every integer n ≥ 2, and the famous Goldbach Conjecture. The symmetric conjecture states that for every integer n ≥ 2, there exists at least an integer k ∈ [0,n−2] such that both n−k and n+k are prime. Reformulating Goldbach’s conjecture in the form "for every n ≥ 2, there exist primes p,q such that p + q = 2n", we show that the two conjectures are equivalent.
Category: Number Theory
[29] ai.viXra.org:2505.0193 [pdf] submitted on 2025-05-29 01:54:22
Authors: Robert Allan
Comments: 8 Pages.
Building upon the geometric Basel construction method established in our previous work[1], we present the discovery of exceptional connections to the Riemann Hypothesisthrough systematic mathematical analysis. The geometric 0.5 offset transformation,previously identified as necessary for Basel convergence, exhibits perfect alignmentwith the Riemann critical line Re(s) = 1/2, representing the first known geometricconstruction to naturally arrive at this fundamental boundary in zeta function theory.Through comprehensive multi-method validation including statistical analysis (R² =0.9854), Fourier spectral correlation (73.9%), uniqueness proofs across 17 alternativeconfigurations, and scale-independent pattern confirmation across grid sizes 36×36through 1152×1152, we demonstrate an overall 81.9% connection score to RiemannHypothesis theoretical framework. The dual mechanism theory—combining 6N boundaryorganization with 0.5 offset transformation—provides geometric intuition for thecritical line's mathematical significance while maintaining systematic convergencetoward π²/6. These findings suggest a profound connection between discrete geometricconstruction and continuous zeta function theory, offering a novel geometricperspective on the most important unsolved problem in mathematics.
Category: Number Theory
[28] ai.viXra.org:2505.0188 [pdf] submitted on 2025-05-28 01:19:53
Authors: Robert Allan
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite listed sceintific references)
We present a systematic geometric approach to the Basel problem through discrete gridconstruction with constraint-based optimization. Our method generates Basel series terms through sequential N×N grids and demonstrates systematic convergence of geometric ratios toward the Basel constant π²/6. Analysis of six grid scales (36×36 through 1152×1152) reveals consistent exponential improvement in PPL/FPL (potentialprime location/forbidden prime location) ratios, with mathematical projection indicating convergence to π²/6 ≈ 1.6449. This establishes a unified framework where the same mathematical constant governs both infinite series generation and finite geometric optimization.
Category: Number Theory
[27] ai.viXra.org:2505.0134 [pdf] submitted on 2025-05-20 21:51:30
Authors: Hamid Javanbakht
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed sceintific references)
We propose a reformulation of the Riemann Hypothesis within a higher-categorical, homotopical framework, replacing spectral cohomology with a cohomotopy-theoretic trace formalism. By modeling the spectrum of the Riemann zeta function via flows on an unstable arithmetic space X_Z, we interpret the nontrivial zeros as homotopy classes of fixed points under a Frobenius-type flow. The resulting structure lifts spectral arithmetic topology into a motivic and cyclotomic context, and aligns the Riemann Hypothesis with a confinement theorem in unstable motivic homotopy theory. Connections to Morel—Voevodsky A1-homotopy theory, topological cyclic homology, and arithmetic Tannakian formalism are sketched.
Category: Number Theory
[26] ai.viXra.org:2505.0126 [pdf] submitted on 2025-05-20 21:09:39
Authors: Samuel Bonaya Buya
Comments: 7 Pages.
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 gmax in the interval (0, 2m) is always less than m, we conclude that R(2m) ≥ 1 for all m > 1.
Category: Number Theory
[25] ai.viXra.org:2505.0121 [pdf] submitted on 2025-05-19 03:12:55
Authors: Hamid Javanbakht
Comments: 7 Pages.
We construct a cohomotopical framework for the Birch and Swinnerton-Dyer (BSD) conjecture over number fields, extending spectral methods developed in earlier work. Our approach interprets the order of vanishing and leading coefficient of zeta functions as trace and regulator invariants on an unstable motivic space governed by flow dynamics. By lifting classical cohomological pairings to a homotopical trace formalism, we define a spectral regulator that links the unlinking rank of arithmetic fixed-point flows with the analytic behavior of zeta functions at the central point. This reformulation anticipates a unification of BSD phenomena with flow-stable motivic dualities and sets the stage for a spectral proof in subsequent volumes.
Category: Number Theory
[24] ai.viXra.org:2505.0120 [pdf] submitted on 2025-05-19 03:25:13
Authors: Hamid Javanbakht
Comments: 5 Pages.
This paper develops and partially proves a spectral reformulation of the Birch and Swinnerton-Dyer conjecture for number fields using the motivic cohomotopical frame- work established in previous volumes. We define trace-based regulator spaces arising from unstable fixed-point classes under arithmetic flows and show that, under natural duality and linking constraints, these spaces satisfy an identity matching the order of vanishing of the Dedekind zeta function at its critical point. A determinant pairing constructed over the flow-invariant regulator classes yields a volume form conjecturally equivalent to the leading coefficient. We prove this correspondence in key cases, including totally real fields of rank 1, and for real quadratic fields under the assumption of standard conjectures on motives. The paper also introduces a deformation-theoretic approach to the motivic spectral category, setting the stage for a full proof in the general case.
Category: Number Theory
[23] ai.viXra.org:2505.0119 [pdf] submitted on 2025-05-19 03:56:32
Authors: Hamid Javanbakht
Comments: 7 Pages.
This paper completes the cohomotopical and spectral reformulation of the Birch and Swinnerton-Dyer conjecture over number fields. Building on previous volumes which established the trace-based rank identity and spectral regulator interpretation, we now incorporate torsion subgroups, Tamagawa numbers, and local-global correction factors into the flow-fixed cohomotopy model. We develop a generalized trace pairing formalism, prove a spectral torsion-weighted volume identity, and conjecturally extend the formulation to nonabelian L-functions. The classical components of the BSD formula—rank, regulator, torsion, Tamagawa, and Shafarevich group—are each interpreted geometrically as flow-fixed volumes, trace-null cycles, boundary strata, and duality obstructions in a derived category of flow-equivariant motives. The result is a unified spectral topology of arithmetic that realizes BSD as a motivic trace identity over a regulated flow category, and anticipates a broader categorical Langlands correspondence.
Category: Number Theory
[22] ai.viXra.org:2505.0118 [pdf] submitted on 2025-05-19 04:15:48
Authors: Hamid Javanbakht
Comments: 7 Pages.
This paper extends the spectral arithmetic topology framework into the nonabelian domain by constructing cohomotopical torsors, a spectral étale fundamental group, and a derived category of flow-equivariant motives. We begin by defining torsors as trace-stable homotopy classes in arithmetic flow spaces and develop a homotopical version of the étale fundamental group. Spectral abelianization recovers class field theory from stabilized flow categories. We then introduce a trace-formalism for Hecke stacks and Langlands parameters, and reinterpret automorphic sheaves as objects in a derived flow category. A global trace category is constructed, encompassing torsors, regulators, and flow-invariant sheaves. Within this setting, we prove a nonabelian reciprocity theorem via a categorical pairing between spectral Selmer stacks and trace-stabilized regulators, and establish a Tannakian-style equivalence recovering the spectral fundamental group as a symmetry object. This unifies abelian and nonabelian arithmetic dualities under a spectral and categorical framework, pointing toward a cohomotopical Langlands program grounded in trace geometry.
Category: Number Theory
[21] ai.viXra.org:2505.0106 [pdf] submitted on 2025-05-19 02:43:08
Authors: Marko Vrček
Comments: 2 Pages.
We construct a self-adjoint Dirac operator on a fractal-arithmetic Hilbert space whose spectrum encodes the zeros of the Riemann zeta function. Through explicit decomposition into fractal, arithmetic, and modular components, we achieve spectral correspondence (RMSE 0.11) with the first 100 nontrivial zeros. The construction is formalized as a complete spectral triple, with verified reality structure and summability properties. Numerical diagonalization via restarted Lanczos iteration demonstrates both global distribution matching and local statistics consistent with Montgomery’s pair correlation conjecture.
Category: Number Theory
[20] ai.viXra.org:2505.0105 [pdf] submitted on 2025-05-19 02:39:29
Authors: Marko Vrček
Comments: 4 Pages.
We present a comprehensive theory of structural ratio laws governing prime constellations, with twin primes as the fundamental case. For consecutive twin prime pairs (pn, pn + 2) and (pn+1, pn+1 + 2), the normalized ratio Rn =pn+1+2 2(pn+2) converges exponentially to 0.5 with decay rate b = C2/6, where C2 ≈ 0.66016 isthe twin prime constant. This law generalizes to all admissible prime constellations through Rn(k) = pn+1+k 2(pn+k) → 0.5, exhibiting universal exponential convergence ratestied to respective constellation constants. The deviations Rn−0.5 display f−2 power spectra, revealing deep connections to spectral theory and quantum chaos. These results establish new geometric regularities in prime distributions, transcending classical density and gap approaches.
Category: Number Theory
[19] ai.viXra.org:2505.0103 [pdf] submitted on 2025-05-18 21:17:11
Authors: Hamid Javanbakht
Comments: 7 Pages.
We propose a spectral-geometric framework in which the Riemann Hypothesis is recast as a confinement theorem on the spectrum of a flow operator over an arithmetic cohomology space. This framework, which we call Spectral Arithmetic Topology, constructs a correspondence between prime periodicities and the eigenvalues of a Laplace-type operator acting on the cohomology of a dynamically foliated arithmetic space.
Category: Number Theory
[18] ai.viXra.org:2505.0097 [pdf] submitted on 2025-05-18 00:23:06
Authors: Hans Rieder
Comments: 3 Pages. (Note by ai.viXra.org Admin: An abstract in the article is required; please cite and list scientific references))
This paper presents a complete and rigorous proof of the Collatz Conjecture, a longstanding unsolved problem in mathematics. The proof is developed through a systematic analysis of the dynamics of the Collatz function, employing a novel structural approach that avoids probabilistic heuristics and instead relies on deterministic arguments. The central method involves a classification of all positive integers into finitely many residue classes, each of which is shown to converge to 1 under iteration. The paper includes a detailed discussion of why this proof succeeds where previous approaches have failed and outlines the implications for the theory of iterative maps on the integers.
Category: Number Theory
[17] ai.viXra.org:2505.0047 [pdf] submitted on 2025-05-07 20:17:20
Authors: Dobri Bozhilov
Comments: 6 Pages. (Note by ai.viXra.org Admin: An abstract is required in the article; article title truncated by ai.viXra.org Admin))
This paper proposes an empirical improvement of Gauss's formula for estimating the number of prime numbers, introducing a floating logarithmic base that significantly increases accuracy. The new formula achieves precision close to that of the logarithmic integral Li(x), while using only elementary operations, avoiding the complexity of Riemann's approach. Numerical tests up to 10^12 demonstrate superior performance compared to Gauss's classical formula and near-Riemann accuracy. Additional forecasts are provided for 10^100, 10^200, and tests against Dusart intervals at 10^500 and 10^1000. This method opens new possibilities for practical applications and theoretical exploration in analytic number theory.
Category: Number Theory
[16] ai.viXra.org:2505.0044 [pdf] submitted on 2025-05-07 20:07:58
Authors: Javier Muñoz de la Cuesta
Comments: 13 Pages.
ABSTRACT. 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 �� (��) have real part �� = 12 . We model these zeros as vibrational modes emerging from a primordial quantum singularity, constructing a Hamiltonian whose eigenvalues correspond to the imaginary parts ���� 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(��) = 1 2 , resolving RH. The model leverages quantum-inspired concepts such as superposition, entanglement, and iterative state regression,establishes a direct connection with �� (��), and provides a scalable framework that bridgesnumber theory and quantum mechanics.
Category: Number Theory
[15] ai.viXra.org:2505.0028 [pdf] submitted on 2025-05-05 22:05:56
Authors: Lovuyo Melvin Chotelo
Comments: 3 Pages.
We introduce a novel two-index formula for term generation in arithmetic structures. Traditional arithmetic sequences rely on a single index to determine sequence terms. In contrast, our formula leverages the sum of two indices, offering a natural generalization to higher-dimensional structures. This approach maintains linear growth while introducing new symmetry and structural properties, making it potentially valuable for applications in grid-based systems, networks, and combinatorial frameworks.
Category: Number Theory
[14] ai.viXra.org:2505.0013 [pdf] submitted on 2025-05-02 23:59:59
Authors: Viktor Weimer
Comments: 3 Pages. In German (Note by ai.viXra.org Admin: An abstract in the article is required)
This article is based on a systematic consideration of the structure of all natural numbers, with particular attention to powers of 2 andcongruence classes modulo 4, to show that every number enters this structure, either directlyor via transformations, and thus reaches 1 in finite time.
Category: Number Theory
[13] ai.viXra.org:2505.0012 [pdf] submitted on 2025-05-03 00:01:02
Authors: Viktor Weimer
Comments: 3 Pages. In German (Note by ai.viXra.org Admin: An abstract in the article is required)
The Collatz conjecture is algorithmically supported by decomposition into powers of 2. Each component follows a fixed reduction path. The seemingly chaotic process is in fact clearly structured and universally convergent.
Category: Number Theory
[12] ai.viXra.org:2504.0099 [pdf] submitted on 2025-04-25 04:41:58
Authors: Christopher David Rice
Comments: 34 Pages.
We introduce two new diagnostic tools for probing the arithmetic structure of elliptic curves over the rational numbers: a canonical summation function based on Néron—Tate height, and a height entropy index that captures the informational complexity of point distributions. Empirical evidence suggests that the asymptotic behavior of the summation function reflects the rank of the Mordell—Weil group: it remains bounded for rank 0, grows logarithmically for rank 1, and exhibits polynomial growth for higher ranks. We conjecture that the regularized global summation admits a divergence structure near the critical point s = 1, with an order equal to the rank and a leading coefficient—denoted Lambda(E)—that may reflect deeper arithmetic invariants. The entropy index also appears to increase with rank, offering a complexity-based proxy when direct enumeration is difficult. Together, these tools form a new analytic and geometric framework for approaching the Birch and Swinnerton-Dyer conjecture.
Category: Number Theory
[11] ai.viXra.org:2504.0095 [pdf] submitted on 2025-04-24 20:02:34
Authors: Deskuma [Doe]
Comments: 13 Pages. (Note by ai.viXra.org Admin: For the last time, please use real author name - both first and last name) Supplement to: https://ai.viXra.org/abs/2504.0081)
This document provides structural reinforcement for the symmetry-based formulation of the Riemann Hypothesis (RH), as introduced in the companion paper "Only One Line Knows No Drift." We present theoretical proofs, numerical visualizations, and analytic tools that support the claim that the critical line Re(s) = 1/2 is the unique axis of zero drift in the complex phase of ζ(s). This supplement includes proofs of key lemmas, phase drift models, π-jump structure, and appendices detailing auxiliary methods. All source code and reproducible scripts are available at: https://github.com/Deskuma/riemann-hypothesis-ai* This is a supplementary paper to the main theory paper (https://ai.viXra.org/abs/2504.0081).
Category: Number Theory
[10] ai.viXra.org:2504.0084 [pdf] submitted on 2025-04-22 19:55:53
Authors: Khazri Bouzidi Fethi
Comments: 5 Pages. Assisted by Claude
We present a rigorous proof of the Riemann hypothesis based on the analysis of a new energy function E(σ,t). This approach relies on studying the strict convexity of E(σ,t) and establishes a contradiction in the hypothesis of the existence of non- trivial zeros of the zeta function outside the critical line ℜ(s) = 1/2. Our method combines classical results from analytic number theory with new precise quantitative estimates, leading to a complete proof that all non-trivial zeros of the Riemann zeta function lie on the critical line and are simple.
Category: Number Theory
[9] ai.viXra.org:2504.0081 [pdf] submitted on 2025-04-21 19:13:38
Authors: Deskuma [Doe]
Comments: 23 Pages. (Note by ai.viXra.org: No pseudonym is permited - Please replace article with real author name) 12 figures + 6 appendix visuals. This is a collaborative AI-assisted research exploring a phase-based structural proof of the Riemann Hypothesis.
We propose a symmetry-based formulation of the Riemann Hypothesis (RH) by analyzingthe angular behavior of the Riemann zeta function ζ(s) in the critical strip. Using the phase function θ(t; σ) = 2 arctan(Im ζ/Re ζ), we uncover a geometric structure that isolates thecritical line Re(s) = 1/2 as the unique axis where angular drift vanishes. This phase-basedapproach connects zero-point alignment with rotational symmetry, supported by derivativeanalysis and numerical visualization. We further demonstrate that phase jumps of π coincideprecisely with nontrivial zeros on the critical line, while deviations from σ = 0.5 inducemeasurable drift and symmetry breaking. Our results provide a structural reformulation ofRH: only Re(s) = 1/2 supports drift-free dynamics in ζ(s), implying that nontrivial zerosmust lie on this line. All figures, code, and visualizations are available for replication at the linked repository [https://github.com/Deskuma/riemann-hypothesis-ai].
Category: Number Theory
[8] ai.viXra.org:2504.0075 [pdf] submitted on 2025-04-21 00:42:51
Authors: An Frost
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
This paper introduces a conceptual model based on the traditional Nine-Ring Puzzle toreinterpret the persistent difficulty in proving the Goldbach Conjecture. We hypothesizethat the numerical universe may possess a structural parity—either fundamentally odd oreven—which governs how numerical decompositions can occur. Just as a Nine-Ring Puzzle can only be solved when approached with the correct parity sequence, the conjecture may resist proof because it is being approached from a structurally incompatible direction. If the universe is built upon an odd-parity structure (as with prime numbers), then constructing even numbers from primes is valid. However, attempting to reverse-engineer primes from even numbers may ultimately reach a deadlock—not through an isolated error, but because the entire path is invalid from the outset, a fact that only becomes evident in the final stages. This paper offers not a proof, but a structural-philosophical explanation of the conjecture’s elusiveness.
Category: Number Theory
[7] ai.viXra.org:2504.0066 [pdf] submitted on 2025-04-19 11:15:42
Authors: Daoudi Rédoane
Comments: 14 Pages.
Below I found certain formulas about number theory. The proofs are very complex and I’ll submit them soon. For example there are formulas that require advanced mathematical tools like L’hôpital’s rule, the Residue theorem, Jordan’s lemma, the Cauchy Principal Value, the Dirichlet series expansions, the Wallis product.
Category: Number Theory
[6] ai.viXra.org:2504.0065 [pdf] submitted on 2025-04-19 22:43:50
Authors: Daoudi Rédoane
Comments: 14 Pages.
In this paper I try to proof certain formulas found in my previous paper. I use several mathematical tools like L’hôpital’s rule, the Residue theorem, Jordan’s lemma, the Cauchy Principal Value, the Dirichlet series expansions, the Wallis product.
Category: Number Theory
[5] ai.viXra.org:2504.0061 [pdf] submitted on 2025-04-19 22:35:45
Authors: Dobri Bozhilov
Comments: 11 Pages. Assisted by ChatGPT
This paper presents a novel method for discovering large prime numbers based on symmetric triplets centered on known primes. By extending the idea behind the Goldbach conjecture, we assume that every prime number is surrounded by other primes at equal distances, forming "symmetric triples." The method identifies the most frequent prime gaps, scales them according to the density of the distribution using the Riemann approximation, and applies them around a large, known prime number. In an experimental run on a modest 8-core cloud server, we discovered 20 new 300-digit prime numbers in just 40 seconds. This approach significantly reduces the number of required checks compared to brute-force and offers a practical way to generate guaranteed primes for cryptographic applications. Potential future applications include testing the method on powerful supercomputers or adapting it to Mersenne numbers, which are easier to verify.
Category: Number Theory
[4] ai.viXra.org:2504.0049 [pdf] submitted on 2025-04-14 16:59:51
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages.
We introduce an original geometric interpretation of the Riemann zeta function based on angular transformations of prime number distributions. By establishing a correspondence between primality and complex angular measures, we develop a framework that ofers new insights into the non-trivial zeros of ζ(s). Our numerical investigations reveal distinctive convergence patterns along the critical line σ = 0.5 , suggesting an angular equilibrium condition. While this approach does not constitute a proof of the Riemann Hypothesis, it provides an intuitive geometric lens through which to explore this fundamental problem in number theory. The framework remains fully compatible with established results in analytic number theory while ofering fresh computational perspectives.
Category: Number Theory
[3] ai.viXra.org:2504.0029 [pdf] submitted on 2025-04-11 18:59:24
Authors: Dobri Bozhilov
Comments: 5 Pages.
We present a new prime number generation algorithm based on the symmetry assumption derived from the Goldbach Conjecture. The algorithm significantly reduces the search space by targeting symmetric prime pairs around each number. Using this method, we discovered 10 new 200-digit primes in under an hour on a standard laptop. Theoretical background, pseudocode, and experimental results are provided.
Category: Number Theory
[2] ai.viXra.org:2504.0023 [pdf] submitted on 2025-04-08 18:59:08
Authors: Giovanni Di Savino
Comments: 2 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
We do not have time to verify and we cannot go faster, the AI It has time but, like us, it cannot exceed the speed of light and, like us, it will not be able to intervene in real time on realities taking place at considerable distances such as the landing of the Curiosity space probe on Mars in 2021, with videos and data that were known with a 7-minute delay due to the distance Mars↔Earth.
Category: Number Theory
[1] ai.viXra.org:2504.0005 [pdf] submitted on 2025-04-02 16:21:17
Authors: Dobri Bozhilov
Comments: 5 Pages. https://selfie-church.com/riemann (Note by ai.viXra.org Admin: Please cite and list sceintific references)
We propose a geometric-functional hypothesis supporting the Riemann Hypothesis, grounded in number theory and inspired by the Pythagorean theorem. By treating the complex argument of the Riemann zeta function as a vector in the complex plane, we analyze the modulus of the function in relation to the real and imaginary parts of its input. We argue that only the critical line ℜ(s) = 1/2 yields a balanced vector structure that satisfies both the Pythagorean identity and the necessary conditions for ζ(s) to vanish. Additional reasoning involving vector alignment between known nontrivial zeros and geometric constraints supports the uniqueness of the critical line. The work represents a collaborative exploration between a human researcher and an artificial intelligence (ChatGPT), highlighting a novel approach to one of mathematics’ most profound unsolved problems.
Category: Number Theory
[14] ai.viXra.org:2601.0005 [pdf] replaced on 2026-01-04 13:55:51
Authors: Birke Heeren
Comments: 24 Pages.
This paper introduces the "Cosmos Automaton" (CA), a deterministic fractal automaton that generates prime numbers through symbolic operations rather than direct primality testing. By treating the sequence of natural numbers as a dynamic process, we show that primality emerges from the constructive interference of "pulse trains" (periodic symbolic words). We demonstrate that the CA’s structures are isomorphic to the set of natural numbers and that its evolutionary steps correspond to arithmetic progressions. This approach provides a visual and algorithmic bridge between automata and the distribution of primes, leading to a definition of primality based on geometric expansion. This provides a new algorithmic perspective on the structure of primes.
Category: Number Theory
[13] ai.viXra.org:2511.0077 [pdf] replaced on 2025-11-24 23:26:13
Authors: Christos Thessalonikios
Comments: 12 Pages.
Here we use a Weierstrassfunction due to its oscillatory properties with a Gaussian envelope in order to belocalized or "anchored" in the position of the primes. Each prime p serves as a localizedfractal anchor generating an oscillatory mode Fp(x), with a delta barrier potential ofthe form VP(x) = Pp gpi δ(x−pi). Using this formulation we can create a Hamiltonianoperator and we explore its spectral characteristics using the 1D quantum mechanichsscattering theory. The zeros of M12(k) tranfer matrix determine a discrete spectrum{kn} that, after global rescaling and some boost of the form t(model) n = αkn + β, alignsperfectly with the 100 imaginary parts of the nontrivial zeros of the Riemann zetafunction on the critical linesn =1 /2 + itnThe correlation of the model and the sn reaches ρ ≈ 1, with a mean absolute deviationbelow 0.011 .
Category: Number Theory
[12] ai.viXra.org:2510.0062 [pdf] replaced on 2025-10-29 20:41:16
Authors: Minkwon Chung
Comments: 3 Pages.
This paper establishes an analytic equivalence for the Riemann Hypothesis (RH) throughthe analysis of the exact-sieve error E(x, z) = Φ(x, z) − x · Q p≤z(1 − 1/p). Here, Φ(x, z) =#{1 ≤ n ≤ x : gcd(n, P(z)) = 1} and P(z) =Q p≤z p. A Schwartz test function ψ on alogarithmic scale defines the smoothed error term Eψ(X, z) = R R E(eu, z)ψ(u−X) du. Thisfunction admits a Mellin spectral representationEψ(X, z) = 1 2πi Z (c) esX b ψ(s)Gz(s) ds,where Gz(s) = ζ(s)Ez(s)/s − Ez(1)/(s − 1) and Ez(s) = Q p≤z(1 − p−s). The subtractionremoves the pole at s = 1, so Gz(s) is holomorphic on ℜs > 0; in particular, the nontrivial zeros of ζ(s) are not poles of Gz(s). The core result is a smoothed spectral equivalence: under RH, |Eψ(log x, z)| ≪ψ,ϵ zα(log z)βx1/2+ϵ; conversely, assuming this bound (and a necessary non-vanishing condition on Ez(ρ)) yields a contradiction via a smoothed explicit formula based on ζu2032/ζ [1], thereby proving RH. The framework is positioned relative to the Gonek—Hughes—Keating hybrid product [2] and includes the Prime-Frontier Lemma as anumerical anchor.
Category: Number Theory
[11] ai.viXra.org:2510.0045 [pdf] replaced on 2025-11-06 04:24:54
Authors: Sizwe Tshabalala
Comments: 30 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
In this paper I introduce a branch of math called Nembelo; from it I introduce a novel function I call the An() function. This function adds number to itself allowing us to study numbers in ways that enables us to have insight into number's behavior. The numbers studied are natural n, primarily 1. To answer the question on the nature of number, for so far the debate has been mostly philosophical and anecdotal, strange enough, not mathematical. This is because it has been taken as granted that math itself cannot answer this question but with the An() function; adding n to itself — we have a way of figuring out what numbers on themselves have to say. As we follow the behavior of n under An(n), it will appear that math has a say on ontology, metaphysics and theology. Theology, because we can naturally derive the concept of GOD from An(1) and not impute it as Gödelu201fs ontological proof. As many fields have been claimed for math & science in past times, e.g., weather prediction, so are these metaphysical fields. However the ambition of the math extends to the unification of math, science, biology, sociology, for which the current theses is not adequate to argue for. Nevertheless, the base from which one can stand is provided, i.e., evidence mapping mathematical conjecture into physical properties of the universe we observe. Keywords: kucwenga, Platonism, conjecture, behavior of n, return to itself, domain.
Category: Number Theory
[10] ai.viXra.org:2507.0113 [pdf] replaced on 2025-07-27 14:29:43
Authors: Giuseppe Fierro
Comments: 3 Pages. Citations and DOI have been added.
We prove that for every positive integer n, the number of prime factors of 2n−1 (counted with multiplicity) is greater than or equal to the number of prime factors of n.
Category: Number Theory
[9] ai.viXra.org:2506.0081 [pdf] replaced on 2025-11-02 11:01:56
Authors: Patrick Kotal
Comments: 18 Pages.
This paper presents a proof of the Collatz conjecture. By analyzing the dynamics of the original Collatz operations within a stochastic process model, we show that they lead to contraction due to a lower bound for the ratio a/b of the counter variables. Then we derive Bit length growth constraints which emerge from the structure of the Collatz process. We finally show that the original Collatz operations applied to any positive integer n>1 can only produce sequences that contract to 1.
Category: Number Theory
[8] ai.viXra.org:2506.0081 [pdf] replaced on 2025-07-06 02:30:00
Authors: Patrick Kotal
Comments: 16 Pages.
This paper presents a proof of the Collatz conjecture. By analyzing the dynamics of the original Collatz operations within a stochastic process model, we show that they lead to contraction due to a lower bound for the ratio a/b of the counter variables. Then we derive Bit length growth constraints which emerge from the structure of the Collatz process. We finally show that the original Collatz operations applied to any positive integer n>1 can only produce sequences that contract to 1.
Category: Number Theory
[7] ai.viXra.org:2506.0081 [pdf] replaced on 2025-06-21 22:27:44
Authors: Patrick Kotal
Comments: 15 Pages.
This paper presents a proof of the Collatz conjecture. By analyzing the dynamics of the original Collatz operations within a stochastic process model, we show that they lead to contraction due to a lower bound for the ratio a/b of the counter variables. Then we prove by Bit length growth constraints which emerge from the structure of the Collatz process, that the Collatz operations can only generate sequences that contract. We finally show that the original Collatz operations applied to any positive integer n>1 can only produce sequences that contract to 1.
Category: Number Theory
[6] ai.viXra.org:2506.0023 [pdf] replaced on 2025-06-06 22:09:15
Authors: Hannah McCoy, Stephen Raphael Manning
Comments: 42 Pages.
We introduce a novel modular resonance approach to the Riemann Hypothesis by constructing a modified zeta function, ζ_mod(s), derived from a deterministic sieve of modular prime residues. This function admits full analytic continuation, is defined via a Mellin transform of a modular theta kernel, and forms an Euler product analogue to ζ(s). A real scaling transformation α ≈ 1.0083 establishes a spectral bijection betweenζ_mod(s) and ζ(s), such that: ζ(s) = ζ_mod(αs) for all s ∈ ℂ {1} Using operator analysis, bounded tail convergence, and zero alignment, we demonstrate that the non-trivial zeros of ζ(s) correspond precisely to those of ζ_mod(s), all lying on the critical line Re(s) = 1/2. This constitutes a conditional yet comprehensivesymbolic framework linking classical and modular prime structures through analytic, algebraic, and spectral equivalence. The method provides a potential new path toward a constructive proof of the Riemann Hypothesis and reveals previously unseen resonance patterns in the distribution of primes.
Category: Number Theory
[5] ai.viXra.org:2506.0003 [pdf] replaced on 2025-06-03 14:38:10
Authors: Giuseppe Fierro
Comments: 4 Pages.
This article proves the logical equivalence between a conjecture which, as far as we know, has not been previously published, which asserts the existence of at least two primes symmetric about every integer n ≥ 2, and the famous Goldbach Conjecture. The symmetric conjecture states that for every integer n ≥ 2, there exists at least an integer k ∈ [0,n−2] such that both n−k and n+k are prime. Reformulating Goldbach’s conjecture in the form "for every n ≥ 2, there exist primes p,q such that p + q = 2n", we show that the two conjectures are equivalent.
Category: Number Theory
[4] ai.viXra.org:2505.0188 [pdf] replaced on 2025-05-31 21:09:20
Authors: Robert Allan
Comments: 5 Pages. (Note by ai.viXra.org Admin: Author's name is required on the article!)
We present a systematic geometric approach to the Basel problem through discrete gridconstruction with constraint-based optimization. Our method generates Basel series terms through sequential N×N grids and demonstrates systematic convergence of geometric ratios toward the Basel constant π²/6. Analysis of six grid scales (36×36 through 1152×1152) reveals consistent exponential improvement in PPL/FPL (potentialprime location/forbidden prime location) ratios, with mathematical projection indicating convergence to π²/6 ≈ 1.6449. This establishes a unified framework where the same mathematical constant governs both infinite series generation and finite geometric optimization.
Category: Number Theory
[3] ai.viXra.org:2505.0097 [pdf] replaced on 2025-05-26 23:04:05
Authors: Hans Rieder
Comments: 4 Pages. (Note by ai.viXra.org Admin: For the last time, please cite and list scientific references)
This paper presents a complete and rigorous proof of the Collatz conjecture using a residue class analysis modulo 16. By focusing on the reduced Collatz function, which acts only on odd numbers and skips even intermediate steps, a deterministic structure in the iterative behavior becomes visible. A central contraction lemma identifies conditions under which a true descent occurs.A detailed investigation of all odd residue classes shows that such a descent occurs for every starting number. The proof excludes divergent sequences and non-trivial cycles and shows that every natural number eventually reaches 1. Thus, the conjecture is proven without heuristic arguments, but purelystructurally.
Category: Number Theory
[2] ai.viXra.org:2505.0047 [pdf] replaced on 2025-06-03 20:30:35
Authors: Dobri Bozhilov
Comments: 13 Pages.
The known approximations for the number of prime numbers π(x) include Gauss’s formula x/ln(x) and Riemann’s formula through the logarithmic integral Li(x). The latter is known for its high accuracy but is difficult to compute numerically, as it requires integration.In the present work we propose a new approximation with an elementary structure and exceptionally high precision, which gives much more accurate results compared to Gauss’s formula and almost reaches those of Li.In essence, the new formula represents an improved Gauss formula by turning the logarithm base in the denominator from fixed natural (ln) to one with a floating base.Empirically, very accurate results are established in the range above. For large values, the formula approaches that of Gauss and both become equally accurate, along with Li.
Category: Number Theory
[1] ai.viXra.org:2504.0099 [pdf] replaced on 2025-05-23 16:54:43
Authors: Christopher David Rice
Comments: 64 Pages. Previous assumptions lifted to continue the formal proofs
We introduce two diagnostic tools for probing the arithmetic structure of elliptic curves over the rational numbers: a canonical summation function based on the N´eron—Tate height, and a height-based entropy index that captures the distributional complexity of rational points. Empirical evidence suggests that the asymptotic behavior of the summation function reflects the rank of the Mordell—Weil group: it remains bounded for rank 0, grows logarithmically for rank 1, and exhibits polynomial growth for higher ranks. We prove that the regularized summation function admits a meromorphic continuation near the critical point s = 1, with a pole of order equal to the rank and a leading Laurent coefficient—denoted Λ(E)—matching the expected arithmetic invariants under the Birch and Swinnerton-Dyer conjecture. The entropy index also increases with rank and may serve as a complexity-based proxy in cases where explicit point enumeration is difficult. Together, these tools form a new analytic framework for investigating the Birch and Swinnerton-Dyer conjecture.
Category: Number Theory