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[13] 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
[12] 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
[11] 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
[10] 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
[9] 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
[8] 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
[7] 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
[6] 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
[5] 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
[4] 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
[3] 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
[2] 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
[1] 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