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[20] viXra:2602.0145 [pdf] submitted on 2026-02-24 21:55:54
Authors: Niccan Mandal
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
In this paper, we derive a formal inversion identity from the Taylor expansion of $sqrt[k]{x}$ to get $x$ as an infinite series of the function. Along with the derivation, we also give a proof of the identity by justifying some crucial mathematically rigorous statements regarding analyticity, validity of Cauchy's convolution, and the convergence, and also derive a trivial infinite series for $pi$, $e$ (Euler's constant) and a formal infinite series identity of $gamma$ (Euler-Mascheroni constant) in terms of their $k$-th roots.
Category: Number Theory
[19] viXra:2602.0144 [pdf] replaced on 2026-03-15 00:40:40
Authors: Marcin Barylski
Comments: 5 Pages. Editorial fixes + more experiments for hypothesis 1 (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
There are several interesting properties of triangular numbers and research work devoted to them. One of the categories is correlation between them and primes- there is hypothesis that between every two different triangularnumbers >1 there is always a prime number. This paper is focused on detailed examination of such difference, mainly between triangular numbers and their closest (smaller or greater) primes (this difference is called in this work delta,δTP), including its extreme values, also in spirit of finding effective test to search for prime numbers.
Category: Number Theory
[18] viXra:2602.0139 [pdf] submitted on 2026-02-23 19:56:27
Authors: P. Murugesha
Comments: 2 Pages.
In this paper i going to compare matrix determinants and Beals conjuncture.. In this paper I consider equation (az^2-by^2)x^2-(bx^2-cz^2)y^2=(ax^2-cy^2)z^2.Next taken bracket containing things in above equation as x, y, z and proved equation 5 as no solution.... So,I concluded in this paper( if we replace (az^l-by^m)x^n-(bx^n-cz^l)y^m=(ax^n-cy^m)z^l then again taken this equation az^l-by^m=x and bx^n-cz^l=y and ax^n-cy^m=z(note l, m, n are >=2). Then also get no solution (it get same as equation 5 only changes in x, y, z powers) it still holds the properties of relatively prime properties.) So we finally got answer for beal conjuncture. When powers greater than 2 has no solution.
Category: Number Theory
[17] viXra:2602.0131 [pdf] submitted on 2026-02-22 10:40:17
Authors: Timothy Jones
Comments: 2 Pages.
We use a TI-84 program to show pi's rationality implies that a radius at 90 degrees must have a defined slope, a contradiction.
Category: Number Theory
[16] viXra:2602.0109 [pdf] submitted on 2026-02-21 19:37:24
Authors: Christoper Muoki Mututu
Comments: 13 Pages. (Note by viXra Admin: For the last time, Please cite and list scientific references!)
We investigate a structural subclassification of twin prime pairs based on intersections between two modular quadruplet configurations, an admissible (2, 4, 2) prime pattern and a complimentary forbidden quadruplet pattern eliminated modulo 3. We define an overlap counting function ��(��) measuring the number of twin primes up to �� arising from such structural intersections and compare it to the total twin prime count ��(��). Computational data up to ��=3×1011 shows that the ratio ��(��)=��(��) ��(��) increases from approximately 0.4 at 103 to approximately 0.6568 at 3×1011. We prove that the structural configurations underlying the overlap occur infinitely often as arithmetic patterns and that ��(��)→∞ ���� ��→∞. We do not prove infinitude of twin primes nor do we establish a limiting value of ��(��). However, the data suggests that the overlap subclass forms a substantial and stable proportion of observed twin primes at large computational scales. This work provides an empirical decomposition of twin primes that they may compliment probabilistic models such as the Hardy-Littlewood heuristic.
Category: Number Theory
[15] viXra:2602.0108 [pdf] submitted on 2026-02-20 20:35:02
Authors: Arthur V. Shevenyonov
Comments: 4 Pages.
An introduction to L-gebra, a promising algebraic apparatus spanning areas as diverse as calculus & number theory to name but a few, and bridging the otherwise distinct if disparate operations & operators, should suffice for a potent yet succinct treatment of Fermat’s augmented last proposition & Riemann’s hypothesis.
Category: Number Theory
[14] viXra:2602.0103 [pdf] replaced on 2026-02-23 20:04:59
Authors: Theophilus Agama
Comments: 8 Pages.
Denote the minimal length of a fixed degree d>1 addition chain that leads to n by l^d(n). We introduce the concept of a strong Brauer number of rank d>1 and show that all numbers belonging to this class satisfy the inequality l^d(d^n-1)
Category: Number Theory
[13] viXra:2602.0095 [pdf] submitted on 2026-02-18 20:40:02
Authors: Francesco Aquilante
Comments: 6 Pages.
Beal Conjecture, which asserts that for $a^k + b^m = c^n$ with $k, m, n > 2$, the bases $a, b,$ and $c$ must share a common prime factor. We prove it to be true with an approach that utilizes a sequence of rational perturbations $delta={delta_i}:{delta}_{i in mathbb{N}} subset mathbb{Q}$ , $delta_i > 0$ and $lim_{i to infty} delta_i = 0$ to treat such Diophantine equation as the critical limit-state of a geometrically constrained configuration. By defining a sequence of non-degenerate triangles $mathcal{T}_delta$ with rational side lengths ${a^k, b^m, c^n - delta_i}$, we establish a continuous mapping to the moduli stack of elliptic curves $mathcal{M}_{1,1}$.We demonstrate that the requirement for {rationality of the configuration} (the existence of a rational altitude $h_delta$) induces a sequence of Frey-Hellegouarch curves $E_delta$ that converge algebraically to the limit-state $E_{Beal}$. For signatures where $min(k,m,n) geq 3$, we invoke Ribet’s Level-Lowering Theorem to show that the associated Galois representation $ho_{E,n}$ is necessitated to reside within the {em empty space} of weight-2 cuspidal modular forms $S_2(Gamma_0(2))$.Simply speaking, our proof follows the often anticipated path of reasoning by which if Beal Conjecture were trueit must ultimately stand on the foundationthat underpins the validity of Fermat's Last Theorem.Furthermore, we provide a formal textit{Parity Lemma} to delineate the bifurcation at $n=2$, explaining why the modular sieve permits coprime solutions in Fermat-Catalan and Pythagorean signatures. This topological and arithmetic framework confirms that for strictly hyperbolic signatures, a solution exists if and only if $gcd(a, b, c) > 1$.
Category: Number Theory
[12] viXra:2602.0090 [pdf] submitted on 2026-02-18 13:18:57
Authors: Óscar E. Chamizo Sánchez
Comments: 6 Pages.
An ancient conjecture, named after its discoverer as Goldbach conjecture [1][2], that is to say, every even number greater than 2 can be represented by the sum of two primes, is a simple and intractable statement that has been torturing mathematicians for more than 250 years. We wonder if the divide et impera method, so useful in programming and algorithmics, could provide some service here. The goal is simplify and separate the whole problem into three independent and fairly manegeable subproblems. An approach that, as far as I know, has not been tested before
Category: Number Theory
[11] viXra:2602.0086 [pdf] submitted on 2026-02-17 00:46:47
Authors: Ryujin Choi
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
We study the distribution of integers obtained by removing fixed residue classes moduloprimes. Using an explicit upper-bound sieve argument, we show that admissible integers cannotoccupy arbitrarily long contiguous intervals. In the case of two arithmetic progressions, thisleads to the existence of simultaneous prime values. As a consequence, Goldbach’s conjectureand the twin prime conjecture follow.
Category: Number Theory
[10] viXra:2602.0073 [pdf] submitted on 2026-02-12 19:56:03
Authors: Francesco Aquilante
Comments: 7 Pages.
We present the quantum Riemann sum ($Q$-sum) operator framework and use it to prove the irrationality of the Riemann-$zeta$ function at odd integers, the Dirichlet-$beta$ function atat all positive integers $n geq 2$, as well as that of the Euler-Mascheroni constant ($gamma$). By establishing a recursive functional hierarchy, we circumvent the classical ``parity barrier'' that has traditionally isolated even and odd zeta-type constants. We utilize the $p$-adic Newton Polygon to demonstrate that the arithmetic complexity of the operator kernel is an invariant of the functional hierarchy. Therefore, the irrationality of the transcendental anchors $zeta(2)$ and $beta(1)$ necessitates the irrationality of the entire chain. This line of reasoning can be extended to incorporate $gamma$, thereby substantiating its long-held irrationality.
Category: Number Theory
[9] viXra:2602.0063 [pdf] submitted on 2026-02-09 18:15:31
Authors: Edgar Valdebenito
Comments: 9 Pages.
In this note, we study the sum S=(1/2)+(1/10)+(1/30)+(1/68)+...
Category: Number Theory
[8] viXra:2602.0054 [pdf] submitted on 2026-02-06 21:13:45
Authors: Ammar Hamdous
Comments: 18 Pages. Creative Commons Attribution 4.0 International
In earlier work [1], we introduced a refined and more structurally representative Collatz tree, within which we identified a singularity. A subsequent preprint [2] established a methodological generalization of the Collatz sequences that preserves this singularity by extending it to a generalized singularity. In the present paper, we investigate the structure of the generalized Collatz tree—referred to as the k-Tree—arising from this transformation. Our analysis focuses on the ordering, propagation, and interaction of branch beginnings across ranks, with particular attention to the structural sets Bk and Ak. This study aims to elucidate the internal architecture of the generalized tree and to clarify the extent to which the geometric and dynamical features of the classical Collatz tree persist under the generalization.
Category: Number Theory
[7] viXra:2602.0036 [pdf] submitted on 2026-02-06 19:36:32
Authors: Silvana di Savino
Comments: 5 Pages.
The even number 2n, which is the product of two or more prime numbers with 2, is always equal to the sum of only two prime numbers equidistant from half their sum; the odd number 2n+1, which is the product of two or more prime numbers with 2 + the odd number 1, is always equal to the sum of two prime numbers equidistant from half their sum, which is an even number + a prime number, 1+2n, which is the difference between the two equidistant primes.
Category: Number Theory
[6] viXra:2602.0031 [pdf] submitted on 2026-02-05 20:18:03
Authors: Theophilus Agama
Comments: 19 Pages.
We denote the length of an addition chain with fixed degree d>2 leading to n by l^d(n). We study the counting function F_d(m,r):=#{nin [d^m,d^{m+1})~:~ l^d(n)<m+r} establishing upper and lower bounds, which generalizes previous classical investigations of De Koninck, Doyon, and Verreault.
Category: Number Theory
[5] viXra:2602.0024 [pdf] replaced on 2026-02-10 09:39:21
Authors: Sriramadesikan Jagannathan
Comments: 11 Pages. References appended
This paper presents a proof of the Riemann Hypothesis by examining the geometric and arithmetic properties of the Dirichlet eta function. By assuming the existence of zeros off the critical line, and analyzing the resulting alternating series in the complex plane, we establish a logical contradiction. The proof relies on insights into the structure of these series, demonstrating that all non-trivial zeros must possess a real part of exactly 12.
Category: Number Theory
[4] viXra:2602.0018 [pdf] submitted on 2026-02-03 20:26:41
Authors: Srihan Dutta, Subhraneel Dutta
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper investigates two specific modular exponentiation identities involving fixed integers. First, we determine the set of non-negative integers m satisfying aN ≡ am (mod N) for a fixed N > 1 and all integers a, deriving the minimum such m. Second, we analyze the minimum positive integer n such that amn ≡ an (mod x) holds for a fixed x > 1 and all integers a, m. We provide explicit formulas for these minimal exponents in terms of the prime factorization exponents and the Carmichael function λ(·).
Category: Number Theory
[3] viXra:2602.0008 [pdf] submitted on 2026-02-01 19:50:12
Authors: Steven M. Tylock
Comments: 8 Pages.
The Collatz conjecture offers a seemingly arbitrary piecewise sequencing of two separate functions (divide by two, multiply by three and add one). Attempts have been made to partially simplify the problem by combining exactly one instance of the multiplication with one instance of the division but have not previously been able to completely separate the two alternatives. To create this separation, I define a positive integer’s Least Significant Bit as the smallest power of two that is added together to create its binary representation. I then define a replacement function as three times n plus the Least Significant Bit of n. I then show that an application of the replacement function followed by division by two has an identical result to division by two followed by the original Collatz multiplication. By using the replacement function, all division can be delayed until the result is a perfect power of two. This change removes the piecewise aspect of the Collatz conjecture that has stymied a proof. In addition, the resulting graph of transformations displays a many-to-one relationship that has previously been hidden. The replacement formula’s non-piecewise and many-to-one features offer new avenues to prove the conjecture. If one can prove that the replacement function reaches a perfect power of two, one will have proved the Collatz.
Category: Number Theory
[2] viXra:2602.0007 [pdf] submitted on 2026-02-02 02:41:50
Authors: Jonipol E. Fortaliza
Comments: 18 Pages.
Throughout mathematics history, mathematicians had created triangular array of numbers. Famous among these number triangles is the Pascal’s Triangle which had marked its prominence in many areas of mathematics and even extends its usefulness in the sciences. This paper presents an inventory of number triangles known and recognized in the mathematics world and takes a look to newly-found triangular array of numbers generated by the function, and its link to the Pascal’s Triangle particularly to the Tetrahedral Numbers.
Category: Number Theory
[1] viXra:2602.0005 [pdf] submitted on 2026-02-01 01:40:16
Authors: Md. Razib Talukder
Comments: 12 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
At a specific angle, the fundamental rule for triangles, the cosine rule simplifies to the same form as the equation in one of mathematics’ most famous problems. This connection arises when the angle is a right angle, linking a basic geometric idea with the only case for which the centuries-old statement holds true.
Category: Number Theory
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