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[31] viXra:2503.0201 [pdf] submitted on 2025-03-31 07:43:56
Authors: Kohji Suzuki
Comments: 7 Pages.
We try interpreting the Riemann hypothesis as something three-dimensional.
Category: Number Theory
[30] viXra:2503.0197 [pdf] submitted on 2025-03-31 20:26:38
Authors: Yibing Xiong
Comments: 13 Pages.
Three variables X,Y,Z are expressed in terms of undetermined coefficients —— variables, binary, and then put into Fermat's last theorem equations. In the equations composed of undetermined coefficients, the number of independent equations is no less than the number of variables, and the solution is no less than zero, so Fermat's last theorem is established.
Category: Number Theory
[29] viXra:2503.0176 [pdf] submitted on 2025-03-28 04:04:55
Authors: Vasant Jayasankar
Comments: 58 Pages.
This paper introduces the RTA Framework for Mathematics, a dimensional projection model that proposes to redefine mathematics as the structured emergence of symbolic, geometric, and harmonic patterns constrained by information-theoretic principles. Rather than treating mathematics as a purely axiomatic or abstract system, RTA posits that all mathematical structures arise from projection constraints on higher-dimensional information spaces, governed by principles of entropy minimization, harmonic balance, and recursive self-similarity. This paper begins with the simplest symbolic expressions in one dimension, showing how numerical and algebraic structures emerge from fundamental constraints. These expressions are then projected into higher-dimensional geometric spaces—revealing the role of oscillations, harmonics, and resonance in shaping more complex mathematical behavior. From this foundation, I examine three historically significant problems—Riemann, Collatz, and Kayeka—and propose that each is a manifestation of a distinct dimensional geometry: harmonic resonance, entropic spiral collapse, and recursive symbolic structuring, respectively. Together, these results suggest that mathematics itself is not a human invention, but the natural consequence of a structured universe operating under a universal projection law. This framework reinterprets mathematical complexity as the layered expression of dimensional geometry, bounded by constraints imposed by entropy and symmetry. The RTA framework potentially offers not only solutions to long-standing mathematical puzzles but a new foundational theory for understanding what mathematics is, where it comes from, and how it governs all emergent structure across domains.
Category: Number Theory
[28] viXra:2503.0175 [pdf] submitted on 2025-03-27 02:47:25
Authors: Song Fei
Comments: 42 Pages.
This paper introduces the Langlands Watch (LW) framework, a novel approach that maps automorphisms phiintext{Aut}(X) of a variety X/mathbb{Q} to a dynamic time representation—comprising a second hand, minute hand, and hour hand—to unify arithmetic and geometric insights across number theory. Initially designed for elliptic curves E/mathbb{Q} , LW enhances the predictive power of the Birch-Swinnerton-Dyer (BSD) conjecture by precisely determining the order of vanishing text{ord}_{s=1}L(E,s)=r and bounding the Tate-Shafarevich group text{III}(E/mathbb{Q}) , validated across low-rank (ensuremath{r=0,1}) , high-rank (ensuremath{r=2}) , and non-trivial text{III} scenarios. Extending beyond elliptic curves, LW adapts to higher-dimensional Abelian varieties, demonstrating its versatility in predicting ranks and L-function behavior for complex structures. By integrating local traces, analytic forms, and global cohomology, LW refines BSD’s arithmetic predictions while forging a robust bridge to the Geometric Langlands Program (GLP) via moduli stacks like ensuremath{text{Bun}_{text{GL}_{2}}}. Theoretical advancements include symmetry-driven constraints on L-function singularities, offering a fresh perspective on Langlands Program challenges. Concrete examples—ranging from a rank 2 elliptic curve to a CM curve with non-trivial ensuremath{text{III}}, and a rank 2 Abelian surface—underscore LW’s practical efficacy. We conclude by affirming LW’s independence as a tool, its necessity within the Langlands Program, and its potential to generalize across varieties, paving the way for future explorations into Iwasawa theory, Shimura varieties, and beyond.
Category: Number Theory
[27] viXra:2503.0167 [pdf] submitted on 2025-03-27 02:19:03
Authors: Bahbouhi Bouchaib
Comments: 24 Pages. Original article
This paper presents detailed analyses of congruences modulo in the case of even sum S = A + B. These analyses were performed in order to design a way to demonstrate GSC in an elementary logical way. Even if we succeed with the rules of congruence in putting an even number in the sum of two prime numbers, this does not constitute a definitive mathematical proof, which is why the GSC remains unprovable. This is why we must resort to a logical reasoning which consists of eliminating false propositions and keeping only one which is true. The one which is true must lead us to the truth of the GSC and thus we succeed in demonstrating it mathematically. This paper provides an elementary mathematical proof by deciding between four propositions such that the GSC is the only true one (logical reasoning by an indirect proof). This conclusion is reached by taking into account established facts in mathematics about prime numbers in [0 - n] and [n - 2n] intervals.
Category: Number Theory
[26] viXra:2503.0165 [pdf] submitted on 2025-03-26 03:30:32
Authors: Joseph L. Pe
Comments: 7 Pages. This paper appeared in the (now defunct) Journal of Recreational Mathematics 31(3), 2002
The definition of perfect numbers is generalized relative to arbitrary arithmetical functions. Some functions are exhibited that generate odd perfect numbers at large scales.
Category: Number Theory
[25] viXra:2503.0164 [pdf] submitted on 2025-03-25 07:42:55
Authors: Joseph L. Pe
Comments: 14 Pages. This paper was published in the (now defunct) journal Mathematical Specturm, 40(1), 2008, 40(1), 2008.
This paper reports on the research efforts of a group formed in 2002 to work on a problem introduced by the author. The group studied picture-perfect numbers, which are numbers defined using reversal and the usual definition of perfect numbers, and were initially thought to be quite rare. The paper announces the discovery of a remarkable theorem by one of the group's members, Andersen's theorem, that generates a (probable) infinity of PP numbers at large scales.
Category: Number Theory
[24] viXra:2503.0161 [pdf] submitted on 2025-03-25 13:02:32
Authors: Jochen Kiemes
Comments: 17 Pages.
This paper presents an approach that reinterprets the Collatz sequence by transforming it into a new sequence, highlighting the dynamic relationship between the trailing and leading bits of its elements. This mapping enables the study of a "bit-race," whose well-defined statistical properties rigorously guarantee the convergence of all Collatz sequences to 1.
Category: Number Theory
[23] viXra:2503.0153 [pdf] submitted on 2025-03-26 02:47:51
Authors: Joseph Mbelawadzai, Mohamadou Ousmanou Abdou, Yannick Kouakep Tchaptchie
Comments: 25 Pages. In French
This article aims to present a new mathematical theory around the concept we call "recursive sum." The notion of recursive sum, in our view, is defined as the process of reducing an integer to a single digit between 0 and 9 by repeatedly adding its digits. Once defined, we explore the fundamental arithmetic properties of this operation, including its periodicity, its relationship to congruence modulo 9 and congruence modulo 3, and the determination of a prime number. In addition, we define, for an integer a between 0 and 9, another concept related to recursive sum called "inverse recursion," defined as the set of all integers whose recursive sum is equal to a. We discuss the theoretical implications of these concepts and their relationship to numerical arithmetic. Finally, we present practical examples illustrating the application of this theory in solving mathematical problems.
Category: Number Theory
[22] viXra:2503.0152 [pdf] submitted on 2025-03-26 04:01:29
Authors: Danilka H. Urena
Comments: 3 Pages.
In number theory, one of the fundamental concepts is the prime number, a natural number greater than 1 that has no positive divisors other than 1 and itself. However, in certain mathematical contexts, this classical notion of primality can be extended or adapted to study more complex numerical structures. A particularly interesting and novel concept is that of artificial prime numbers, a variation that allows us to examine primality within specific numerical sets, beyond the conventional set of integers.
Category: Number Theory
[21] viXra:2503.0151 [pdf] submitted on 2025-03-25 02:27:24
Authors: Dinh Van Tien
Comments: 54 Pages.
This paper presents a comprehensive overview of extended Diophantine equations—high-degree equations that admit infinitely many solution sets, with each set comprising infinitely many elements. We focus on the effective application of the formula: ab = k(a+b) + c, where a, b, k, and c are integers, to transform complex equations into symmetric forms. This transformation facilitates the derivation of polynomial expansions and enables the systematic construction of solution sets with distinct and primitive elements.
Category: Number Theory
[20] viXra:2503.0135 [pdf] submitted on 2025-03-22 20:36:27
Authors: Jose Acevedo Jimenez
Comments: 5 Pages. In Spanish (Note by viXra Admin: Listed scientific references should be cited in the article)
This article explores the concept of artificial prime numbers, defined within specific sets of integers. A number is considered an artificial prime if it has no divisors other than itself within the given set. Through this definition, we seek to extend the idea of primality beyond traditional prime numbers, offering new ways to analyze numbers in number theory. Additionally, some important properties of these numbers are discussed, as well as their relationship with classical primes.
Category: Number Theory
[19] viXra:2503.0131 [pdf] replaced on 2025-04-04 13:13:48
Authors: Lincoln French
Comments: 5 Pages.
We attempt to prove that no order-3 magic square can be constructed using 7, 8, or9 distinct nonzero perfect square numbers. We then extend this result to eliminate allstructurally valid configurations, regardless of the number of repeated values. The ar-gument combines parity constraints, algebraic identities, convexity properties, infinitedescent, and complete structural enumeration. This aims to resolve a long-standingopen question in recreational number theory.
Category: Number Theory
[18] viXra:2503.0122 [pdf] submitted on 2025-03-20 21:42:37
Authors: Oliver Couto
Comments: 5 Pages.
There are numerical solutions available on Wolfram world of mathematics website (ref. # 1) for the equation, (a^6+b^6+c^6+d^6)=(e^6+f^6+g^6+h^6). In this paper the author has arrived at numerical solution by algebra instead of elliptical theory. There are methods for the (6-4-4) equation in (ref #6) providing numerical solutions but parametric solutions are not shown. Also on the internet the author has not come across a similar method (given in this paper) for the above mentioned equation.
Category: Number Theory
[17] viXra:2503.0110 [pdf] submitted on 2025-03-18 20:57:39
Authors: Khazri Bouzidi Fethi
Comments: 4 Pages. (Note by viXra Admin: Listed scientific references should be cited in the article; AI assisted/generated content is in general not acceptable)
This paper proposes a more accurate approximation to the prime counting function, especially for small numbers, using a base-10 logarithm. It relates this approximation to the Riemann zeta function, providing geometric interpretations of its terms.
Category: Number Theory
[16] viXra:2503.0105 [pdf] replaced on 2025-06-04 15:24:15
Authors: Wolfgang Mückenheim
Comments: 2 Pages.
It is shown by the intersection of the complements of all Kempner series belonging to definable natural numbers that not all natural numbers can be defined. We call them dark.
Category: Number Theory
[15] viXra:2503.0094 [pdf] submitted on 2025-03-16 01:22:43
Authors: Jinhua Fei
Comments: 4 Pages.
In this paper, the abc conjecture is negated under certain conditions.
Category: Number Theory
[14] viXra:2503.0092 [pdf] submitted on 2025-03-15 11:43:09
Authors: Andrey V. Voron
Comments: 5 Pages.
The article shows the possibility of a formula calculation of patterns of ideal magic squares of the 4th and 8th order, magic cubes of the 2nd and 4th order, as well as substantiates the hypothesis that the laws of dialectics apply to numbers as ideal objects. In particular, the presence of mathematical patterns can characterize the dialectical law of "unity and struggle of opposites", when the sums of the numbers of the pattern act as opposites. The presence of the golden division constant or its derivatives between the opposite patterns is probably due to the property of the unity of the elements that make up such a whole. This unity of the elements that make up the whole is probably due to the greatest number of structural mathematical connections obtained in connection with the unique properties of the golden proportional relations.
Category: Number Theory
[13] viXra:2503.0088 [pdf] submitted on 2025-03-14 12:58:15
Authors: Jasmine Burns
Comments: 15 Pages.
The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zeta function lie on the critical line ℜ(s) = 1/2. In this paper, we prove that RH is false by demonstrating that the evolution of zeta zeros under the de Bruijn-Newman heat equation is fundamentally unstable. We establish that irregularities in prime gaps introduce an unbounded forcing term in the heat equation, leading to a necessary shift in the location of zeta zeros and forcing Λ > 0, contradicting RH. Furthermore, we resolve the Pair Correlation Conjecture independently of RH, showing that the statistical structure of zeta zeros remains unchanged under the heat evolution. This result confirms that the known statistical properties of the zeta function are not contingent on RH but instead arise from a deeper structural phenomenon tied to prime number modularity and diffusion dynamics. Our findings necessitate a fundamental reevaluation of the role of RH in analytic number theory, shifting focus toward a more geometrically and dynamically informed understanding of the zeta function's zeros.
Category: Number Theory
[12] viXra:2503.0083 [pdf] submitted on 2025-03-13 02:38:18
Authors: Joseph Pe
Comments: 7 Pages.
We call an integer sequence thick if the quotients formed from its terms are dense in the set of real numbers. To find thick sequences, we consider the geometric, Fibonacci, power, and prime sequences. We show that the sequence of primes is thick provided that a conjecture DC-2 holds. DC-2 says that certain pairs of linear Dirichlet conditions have infinitely many solutions. It is a weak form of Dickson’s conjecture, which states that a finite system of linear Dirichlet conditions has infinitely many solutions and generalizes Dirichlet’s well-known result on primes in arithmetic progressions. Also, we obtain partial results for the general thickness problem for an arbitrary sequence and look at heuristic evidence for the validity of DC-2. We conclude with a short list of problems for further research.
Category: Number Theory
[11] viXra:2503.0070 [pdf] submitted on 2025-03-11 21:10:41
Authors: Song Li
Comments: 4 Pages.
This paper investigates the convergence and divergence of the $qx+r$ problem (Crandall conjecture), which is a generalization of the $3x+1$ problem (Collatz conjecture). Through probabilistic analysis, we establish that the convergence condition for the $qx+r$ problem is $q<4$,and predict that it converges to the precise value $frac{r}{4 - q}$ ; when $q>4$, the transformation sequence diverges to infinity. The results of this study provide new insights into understanding the dynamic behavior of the $qx+r$ problem.
Category: Number Theory
[10] viXra:2503.0069 [pdf] submitted on 2025-03-11 21:09:29
Authors: Hyeon Jun Ahn
Comments: 10 Pages. (Note by viXra Admin: AI assisted article is in general not acceptable)
Based on heuristics related to Cramér's conjecture, this paper proposes a suitable hypothesis and investigates its implications. The study encompasses prime gaps, Andrica's conjecture, the mean of consecutive prime numbers, and a detailed analysis of Oppermann's conjecture.
Category: Number Theory
[9] viXra:2503.0064 [pdf] submitted on 2025-03-10 21:11:01
Authors: Daniil Beliavkyi
Comments: 4 Pages. (Note by viXra Admin: AI assisted article is in general not acceptable)
We analyze eigenvalue spacings from quantum simulations and refined untwisted zeta zeros—Riemann zeta zero approximations adjusted with prime-based oscillations—achieving near-convergence to the Gaussian Unitary Ensemble (GUE). Quantum spacings yield a Kolmogorov-Smirnov (KS) statistic D = 0.1159 with p = 0.0658, surpassing the 0.05 threshold, while refined zeros achieve D = 0.0901 and a Cramer-von Mises (CvM) p = 0.9465, indicating exceptional GUE alignment. A persistent 22% deviation from GUE, however, suggests a deeper mechanism. We propose that prime numbers introduce cyclic patterns in quantum states, influencing coherence and challenging GUE’s classical assumptions. This novel synthesis of quantum physics and number theory, validated by 50,000 simulations, hints at a unified framework reconciling classical and quantum realms.
Category: Number Theory
[8] viXra:2503.0036 [pdf] submitted on 2025-03-06 19:32:57
Authors: Eric M. Kelleher
Comments: 24 Pages. (Note by viXra Admin: Author's nameis required on the article)
The Collatz conjecture states any number, N0, after successive computations will always yield one, initiating the recursive sequence of 1→4→2→1→4 because 1 equals itself via (3n+1) 2 for n = 1. Finding a separate recursive sequence excluding 1 would prove the conjecture false. Analysis of the Collatz conjecture with a dual coordinate system (beta tables and gamma tables) revealed only one such recursive sequence is possible. For n, x = N0, the number sequence (6n + 5) forms beta table 2, column 1 and (6n + 1) creates beta table 3, column 1 with each successive column increasing by 22x. The formula (n−1) 3 was used on each beta entry to produce the gamma tables defined by the following sequences, for n = N0: (4n + 3) = m forms gamma table 2, column 1 and (8n + 1) = m forms gamma table 3, column 1 where each successive column increases by 4m + 1. This reveals the (3n+1)2 quotients of all odd numbers connect as follows: (4n + 3) → (6n + 5) and (8n + 1) → (6n +1), accounting for all possible unique connections between odd numbers via (3n+1)2. The differences between connecting sequences are (4n +3)−(6n+5) =2m and (8n+1)−(6n+1)= −2m where the difference only equals zero for n = 0 as follows: (8 × 0 + 1) − (6 × 0 + 1) = 0, indicating the recursive loop as found with (3n+1)2 for n = 1. Using the coordinate system, I herein demonstrate the uniqueness of the only known recursive loop and prove it is the only one of its kind thus solving a major part of the Collatz Conjecture.
Category: Number Theory
[7] viXra:2503.0033 [pdf] submitted on 2025-03-05 21:21:05
Authors: Xiaochun Mei
Comments: 13 Pages.
This paper discusses the complex extensions of Riemann Zeta function and complementary formulas of Gama function. By re-writing the Zeta function equation, it is proved that the equation described a relation between the original Zeta function Z(s) and a now function Z(s)=Z(1-s) . But the domains of these two function does not the same and incompatible, so the Riemann Zeta function equation does not hold. It is also proved that the complex extension formula of the present complementary formula of Gama function is wrong. The correct formula is given by strict calculation. The condition 0
Category: Number Theory
[6] viXra:2503.0025 [pdf] submitted on 2025-03-04 21:48:44
Authors: Pradeep Pant
Comments: 12 Pages. The supplementary file (Table S1) is available at https://bennettu-my.sharepoint.com/:x:/g/personal/pradeep_pant_bennett_edu_in/EcL2EjaiGNJFqyyLEo-SqqcBqEle9TZA32Wk3IAmC0Cy0A?e=LVosYs
Mathematics is a constantly evolving field where the search for new knowledge never stops. Discovering new mathematical series has been crucial for progress, leading to breakthroughs in many areas and offering fresh insights into the world of numbers. In this paper, we describe a new series called the Pn power series. The Pn power series is a superset of many series depending on the value of n. In number theory, we hypothesize a natural number X belongs to a power series Pn if its proper positive divisors (d1 , d2 , d3 ,. . . , X) follow; X^n = d1 * d2 * d3 *. . . * X (where n = 1, 1.5, 2, 2.5, 3, etc.). To illustrate this concept, 12 is a member of the P3 power series as its proper positive divisors, namely 1, 2, 3, 4, 6, and 12, satisfy the equation: 12^3= 1 * 2 * 3 * 4 * 6 * 12 = 1728. Similarly, 196 belongs to the P4.5 power series, with its divisors 1, 2, 4, 7, 14, 28, 49, 98, and 196,following the equation: 196^4.5 = 1 * 2 * 4 * 7 * 14 * 28 * 49 * 98 * 196 = 20,661,046,784. We believe that the implications of this observation are far-reaching, extending beyond number theory into various mathematical disciplines, and have the potential to open up new avenues of research and mathematical exploration.
Category: Number Theory
[5] viXra:2503.0024 [pdf] submitted on 2025-03-04 21:45:10
Authors: Kwangsun Song
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references)
The Collatz conjecture posits that for every natural number, a specific iterative rule leads to 1 or forms a cycle. This paper introduces a simplified Collatz function, inverse Collatz, and double inverse Collatz to prove that no cycles exist beyond the known 1 → 4 → 2 → 1. By analyzing number generation through parameters a and k, we demonstrate the logical impossibility of additional cycles.
Category: Number Theory
[4] viXra:2503.0021 [pdf] submitted on 2025-03-04 21:28:05
Authors: Bahbouhi Bouchaib
Comments: 17 Pages.
Let's assume an even number E that is one unit larger than to the largest prime number we know today. Let's call this prime number Pl. Now we have 0—E/2—E and therefore Pl > E/2. For GSC to be true E must be sum of two primes P1 and P2 such that P1 < E/2 and P2 > E/2. Therefore we have to calculate E — P2 = X. If X is composite GSC is not verified; if X = P1 then GSC is verified. We then calculate E — P2 starting with Pl and all P2 till the one which is the closest to E/2. The question is: are all the Xs resulting from calculated E — P2 = X composites? Is it possible that all Xs might be composite which means non-verification of GSC? By contrast, if only one X is prime, then GSC is true. We see that GSC is much more likely to be verified in this case because a very long sequence of composite numbers is very unlikely to be continuous from E — Pl to E — P2 which is the closest to E/2. In other words there is at least one P2 prime in [E/2—E] such that E — P2 = P1 → E = P1 + P2. The small primes are those that give the most composite numbers because their multiples are the most frequent but it is unlikely that all P2 of [E/2—E] would give composite numbers when calculating E — P2. If anyone, using this procedure, is able to find a sequence of composite numbers E - P2 = X in the whole E/2 — E interval; then He will be the first one who finds the solution to Goldbach's strong conjecture because this means its final rejection, and no mathematician can cast any doubt on his result. However, let us not forget that if only one X is prime, then GSC is true. As long as we cannot find this very precious and historical counterexample of an uninterrupted sequence of composite numbers by E — P2 = X (P2 is in E/2-E interval); Goldbach's strong conjecture will remain true although unprovable. This article gives the TWO THEOREMS of CONGRUENCE-MODULO which always predetermine whether GSC is true or not when calculating E - P2 = X. These two theorems described in this article will predict whether X is prime or composite. Nevertheless, these two theorems require the use of euclidean divisions in series with the calculation of the remainders for each P2.
Category: Number Theory
[3] viXra:2503.0019 [pdf] submitted on 2025-03-03 20:51:25
Authors: Khazri Bouzdi Fethi
Comments: 1 Page.
This article explores a géometric interpretation of the Riemann zêta function as angle measures.This perspective could explain the spiral arragement of prime numbers,as observed in Ulam’s spiral (1963).
Category: Number Theory
[2] viXra:2503.0004 [pdf] submitted on 2025-03-01 22:35:10
Authors: Mustapha Kharmoudi
Comments: 10 Pages. (Note by viXra Admin: Article title should be in English; author name should be after article title; an abstract is required; and scientific references should cited and listed))
In this article, we approach Fermat's famous theorem in an original manner. To achieve this, I will use two applications, one of which is well-known, while the other seems to be unprecedented.
Category: Number Theory
[1] viXra:2503.0001 [pdf] submitted on 2025-03-01 00:25:07
Authors: José Acevedo Jiménez
Comments: 11 Pages.
This article explores the notion of artificial prime numbers in the context of specific sets of integers. A formal definition is presented, properties are discussed, and comparisons are made with the classical notion of primality. Additionally, potential applications of these numbers in number theory and cryptography are analyzed.
Category: Number Theory
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