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[18] viXra:2511.0148 [pdf] submitted on 2025-11-30 02:24:38
Authors: Durga Shankar Akodia
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We 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 >= 2, x must be odd, and k must be an odd prime. Furthermore, we establish the strict congruence k = 7 (mod 8) for all solutions with n >= 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 <= 31, revealing 16 distinct non-trivial solutions. We propose the Akodia Conjecture concerning the infinitude of such non-trivial solutions
Category: Number Theory
[17] viXra:2511.0136 [pdf] submitted on 2025-11-26 11:16:02
Authors: Dmitri Martila
Comments: 2 Pages.
I am writing a shortest proof of the Riemann Hypothesis using Wang's paper as a starting point.
Category: Number Theory
[16] viXra:2511.0133 [pdf] replaced on 2025-12-09 01:08:30
Authors: Ryan Hackbarth
Comments: 5 Pages. (Note by viXra Admin: An abstract is required in the article)
In this paper, I utilize Euler's derivation of the Product Formula the Zeta function to produce a similar Product Formula for the Dirichlet Eta Function. I then examine how this formula relates to the critical line and the zeros of the Riemann Zeta Function.
Category: Number Theory
[15] viXra:2511.0130 [pdf] submitted on 2025-11-26 01:26:58
Authors: Ahcene Ait Saadi
Comments: 7 Pages. (Note by viXra Admin: Further repetition may not be accepted; for the last time, please cite and list scientific references!)
This document deals with enigmas related to the numbers 17 and 19, by presenting equations involving squares of natural integers. It is composed of two parts. 1-Enigmas of the numbers 17 and 19[;] 2-Enigma of the number 17
Category: Number Theory
[14] viXra:2511.0124 [pdf] submitted on 2025-11-24 22:07:52
Authors: Cesar A. P. Correa
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper presents a heuristic approach to the validity of the Riemann Hypothesis, utilizing techniques from Harmonic Analysis and Partial Differential Equations. By decomposing the Dirichlet series using integer-part functions, an oscillatory discrepancy term generated by the discrete nature of the summands is isolated. Modeling this term via Fourier series and subjecting it to the Laplacian operator in the complex plane, it is demonstrated that the condition for the annihilation of the function $zeta(s)$ requires an equilibrium of magnitudes in the second-order partial derivatives. Analytical results indicate that such equilibrium is unstable for $text{Re}(s) eq 1/2$, providing strong theoretical evidence in favor of Riemann's original conjecture.
Category: Number Theory
[13] viXra:2511.0118 [pdf] submitted on 2025-11-24 01:49:22
Authors: Quency Nixon
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)
This paper presents a deterministic, structural proof of the Twin Prime Conjecture, moving beyond traditional probabilistic models. We introduce the "Midpoint Generator" (Mn = Pn/2), a geometric center of gravity that necessitates the existence of Twin Prime pairs at every scale of the number line. We define the "Recursive Binary Descent," an algorithm that mechanically links candidate pairs at infinite scales to verified Twin Primes in a finite "Safe Haven." Finally, using a Proof by Contradiction, we demonstrate that the assumption of a finite number of Twin Prime pairs creates a structural paradox, thereby proving that the set of TwinPrime pairs must be infinite.
Category: Number Theory
[12] viXra:2511.0114 [pdf] submitted on 2025-11-23 00:25:22
Authors: Michael Spencer
Comments: 63 Pages. (Note by viXra Admin: Please cite and list scientific references of other authorities besides self-citations and submit article written with AI assistance to ai.viXra.org)
This paper presents a complete arithmetic resolution of the Collatz Conjecture by separating its structure into two complementary components: a finite residue and phase transition system, and a global affine counting framework. The reverse Collatz step is shown to act only on the two live odd residue classes, and every valid reverse exponent produces a predictable affine expansion whose inverse matches the expected dyadic frequency. From this, every odd integer belongs to exactly one uniquely defined dyadic slice, forming a disjoint partition of all odd numbers. Independently, a zero-state index reveals that every live odd number also generates a unique sequence obtained by repeatedly applying the transformation four times the number plus one, and these sequences likewise partition the odd integers without overlap. The two partitions are proven to be identical, establishing a single global organizational structure for the entire Collatz map. Because the forward and reverse processes are locked to each other and the residue-phase system is finite, every forward trajectory follows one non-branching path that must ultimately return to one. This eliminates the possibility of infinite growth or nontrivial cycles. All structural components, classifications, and counting methods are original to this work and together provide a fully closed arithmetic description of the Collatz dynamics.
Category: Number Theory
[11] viXra:2511.0111 [pdf] submitted on 2025-11-22 01:10:10
Authors: F. F. Martinez Gamo
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We present computational evidence for a novel spectral characterization of prime numbers through the Laplacian eigenvalues of Paley-type graphs. For integers n ≡ 1 (mod 4), we demonstrate that the second smallest Laplacian eigenvalue λu2082 of the graph constructed from quadratic residues modulo n satisfies λu2082(G) = (n - √n)/2 if and only if n is prime, with numerical precision limited only by floating-point accuracy (~10u207b¹u2075). Composite numbers exhibit substantial deviation from this formula, with gaps ranging from 3 to over 60 for n < 300. Statistical analysis over 29 primes and 30 composites shows a separation ratio exceeding 10¹u2075 between prime and composite gap magnitudes. This result establishes a connection between number-theoretic primality and graph spectral properties, with implications for understanding the algebraic structure of finite fields versus rings with zero divisors.
Category: Number Theory
[10] viXra:2511.0103 [pdf] submitted on 2025-11-21 00:04:18
Authors: Shinsuke Hamaji
Comments: 12 Pages. [Submitted to a journal]; Zenodo: https://doi.org/10.5281/zen (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
Abstract: This paper identifies the fundamental difficulty of the long-unresolved Collatz Conjecture as stemming from an unconscious self-limitation in conventional mathematics—specifically, a structural ``lack of collaboration'' between the linear definition of natural numbers (Peano Axioms) and non-linear structural analysis (Cantor's Set Theory). We propose that, to solve a non-linear Halting Problem like the Collatz Conjecture, the proof must prioritize the structural fundamental of ``symmetry of the start and stop point ($1$)'' instead of adhering to linear methods. Specifically, we resolve the ``bijective definition deficit'' of the mapping existing between specific sub-patterns (e.g., $4n+1 leftrightarrow 3n+1$) derived from the Collatz operation, by using Cantor's dimensional expansion pairing function. This method reconstructs the Collatz operation as a closed, bijective structure centered at $1$, structurally and completely excluding the possibility of cycles other than $1$ and divergence to infinity. This represents a structural solution that, by integrating the Peano Axioms and Cantor's Set Theory, rigorously guarantees the global stability of the Collatz infinite tree for the first time.Keywords: Collatz, Tree Equivalence Theorem, Peano's Successor Function, Cantor's Pairing Function.MSC 2020: 03D50, 11B83.
Category: Number Theory
[9] viXra:2511.0096 [pdf] submitted on 2025-11-20 00:16:58
Authors: Christoper Muoki Mututu
Comments: 10 Pages. (Note by viXra Admin: Please cite and list scientific references)
Prime numbers have fascinated mathematicians for centuries due to their inherent unpredictability and fundamental role in number theory. Despite extensive research into their distribution and patterns, primes continue to surprise and challenge scholars. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, is a prime example of this unpredictability. This paper introduces a newly discovered class of prime numbers which under two distinct alternating gap sequences — 2,4,2 and 12,14,12 - predict the occurrence of six additional primes, either forward or in reverse, a phenomenon previously unknown in the study of prime numbers. This work offers not only a novel approach to prime generation but also introduces the idea that primes themselves can act as building blocks for other primes, leading to new methods of understanding prime distribution and its profound implications for both theoretical and applied mathematics.
Category: Number Theory
[8] viXra:2511.0093 [pdf] replaced on 2025-12-31 02:21:34
Authors: Immense Raj Subedi
Comments: 13 Pages.
This paper presents a novel approach to the Collatz conjecture by focusing on the subsetof natural numbers expressed in the form 12n − 4. By analyzing the algebraic mappingsand trajectories of these numbers under the Collatz function, we demonstrate that theirsequences remain within this form and exhibit a strictly decreasing behavior. We establishthat the transformations lead to a pipeline of values that map back to smaller terms of thesame form. Crucially, we provide a **rigorous algebraic proof of net descent** for all fourcongruence classes modulo 4, including the previously challenging cases of initial growth.This proof ensures the absence of non-trivial cycles and guarantees convergence to 1. Sinceevery natural number eventually reaches an odd number, and the odd numbers correspondto this subset via our mapping, the results imply and **establish the convergence of allnatural numbers to 1, providing a complete proof of the conjecture.**
Category: Number Theory
[7] viXra:2511.0068 [pdf] submitted on 2025-11-14 22:21:36
Authors: Muhammad Razzaq Aman Wattoo
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
This paper proposes a unified theoretical framework integrating the Uu2011Water Continuum Theory with the Golden Ratio (ϕ) and the Fibonacci Sequence. While Uu2011Water posits the universe as a continuous medium rather than discrete particles or relativistic distortions, the Golden Ratio represents external proportional expansion, and Fibonacci numbers represent internal developmental sequences governing growth. We formalize a mathematical fusion showing how Fibonacci structures propagate through the Uu2011Water continuum to generate ϕ-based expansions: "Flow"_n=F_n⋅ϕ^n,F_(n+2)=F_(n+1)+F_n,ϕ=(1+√5)/2 [1-3] Advanced modeling includes polar Fibonacci spirals, continuum-based differential equations, and generating functions, establishing a triadic system: internal progression (Fibonacci), external proportionality (ϕ), and medium-based continuity (Uu2011Water).
Category: Number Theory
[6] viXra:2511.0050 [pdf] submitted on 2025-11-11 21:17:49
Authors: Manikandan Karunanidhi
Comments: 4 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
This work introduces a new family of natural numbers for which the sum of two successive powers, beginning with an even exponent, results in a perfect square. The identity is derived from the condition where the base plus one equals a perfect square. A formal proof is presented, along with a recurrence relation and supporting computational evidence in tabular form. This study contributes to the field of number theory by highlighting an unexpected relationship between exponential expressions and perfect squares.
Category: Number Theory
[5] viXra:2511.0040 [pdf] submitted on 2025-11-10 20:59:22
Authors: J. Kuzmanis
Comments: 23 Pages.
Primary abc-triples, formed by the set of roots for the generalized Pell’s equations x^2 -Dy^2 = +/-N (with N [larger than] 2), induce formation of secondary abc-triples in the set of roots for equations x^2 - Dy^2 = N^2.
Category: Number Theory
[4] viXra:2511.0035 [pdf] submitted on 2025-11-08 16:14:33
Authors: Felipe Wescoup
Comments: 9 Pages.
This paper presents a method for generating lists of prime numbers. The algorithm presentedcalculates the set of all composite (non-prime) numbers up to a given limit and the set of primes is subsequently defined as its complement. The method is demonstrated through a JavaScript implementation, which is evolved over four iterative levels increasing optimization. Level 1 presents the core mathematical concept. Levels 2 and 3 include logical assumptions thatimprove efficiency. Level 4 is capable of calculating all primes up to 20 million in seconds within a standard web browser. This paper is supplemented by a public GitHub repository containing the complete operational code.
Category: Number Theory
[3] viXra:2511.0025 [pdf] submitted on 2025-11-07 01:30:00
Authors: Colm Gallagher
Comments: 11 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org)
A deterministic arithmetic reformulation of the mod 6 lattice revealing geometric symmetry in the distribution of primes. Rotational symmetry when applied to these residuals visits all of the non-prime numbers stepwise, which we refer to as "hops", that generates a deterministic arithmetic framework that reproduces the sequence of primes and their gaps. We present a framework for understanding the distribution of primes using modular arithmetic and iterative hop sequences. Visual patterns such as the Ulam spiral are shownto arise naturally from rotational symmetries within this framework. We provide both anintuitive explanation and a formal arithmetic treatment that reproduces the sequence ofprimes and their gaps. As first noted by Ulam and popularized by Gardner, the arrangement of integersin a spiral lattice reveals that prime numbers tend to cluster along diagonal lines. However,this observation alone does not explain the *mechanism* of the clustering. The hop-based interpretation offers a natural explanation: primes occupy loci defined by arithmetic propagation rather than arbitrary geometric coincidence.
Category: Number Theory
[2] viXra:2511.0024 [pdf] replaced on 2025-11-23 17:36:46
Authors: Satya Das
Comments: 22 Pages. The content is changed as part of an improvement
We establish a structural correspondence between the Collatz map and the signedJacobsthal numbers, providing an arithmetic reformulation of the 3x + 1 problem. Byrepresenting Collatz iterations through powers of signed Jacobsthal numbers, we derivenecessary and sucient conditions for the existence of cycles and for the validity ofthe coecient stopping time conjecture. This formulation translates the combinatorialdynamics of the Collatz map into explicit number-theoretic identities, revealing anunderlying algebraic framework that connects iteration, recurrence, and integrality.The results suggest a pathway toward analyzing the conj
Category: Number Theory
[1] viXra:2511.0016 [pdf] submitted on 2025-11-06 02:27:33
Authors: WenBin Hu
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper proposes a set of logical operation rules for sequences and formulates a generation rule for difference-free sequences that satisfy the operations. The "difference-free sequence" in this paper refers to a sequence where the difference between any two arbitrary numbers within the sequence is not equal to any number in other sequences.Common generation rules for difference-free sequences include:The new term of the sequence satisfies a_(n+1)>2a_nComputer-generated sequences based on the greedy algorithm
Category: Number Theory
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