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[11] 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
[10] 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
[9] 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
[8] 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
[7] 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
[6] 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
[5] 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
[4] 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
[3] 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
[2] 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
[1] 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