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[31] viXra:2501.0175 [pdf] replaced on 2025-02-06 22:18:37
Authors: Samuel Bonaya Buya
Comments: 8 Pages.
The Riemann zeta function is a function of a complex variable s = q +it where q and t as real real numbers. In this research we will examine the different ways in which this complex variable can be constructed. We will examine one possible case of violation of the Riemann hypothesis. We willlook at functions that caprure distribution of primes through their zeroes.
Category: Number Theory
[30] viXra:2501.0171 [pdf] submitted on 2025-01-31 21:09:47
Authors: B. N. Pathak
Comments: 9 Pages. (Note by viXra Admin: An abstract in the article is required and scientific references should be cited and listed)
Proof to Fermat's Last Theorem [is presented in this paper].
Category: Number Theory
[29] viXra:2501.0170 [pdf] submitted on 2025-01-30 22:27:42
Authors: Prajay Sajjan Thul
Comments: 81 Pages.
There are different versions of the abc Conjecture available in books and on internet, each claiming that there are only finitely many triples of coprime positive integers $a$, $b$, and $c$ such that the sum of the first two is the third one, and then the third one satisfies a certain specific condition (relating the sum with the product of distinct primes in the prime factorization of each integer.) This paper deals with all versions, proving the claim and explaining the concept. This paper also deals with some abc triples numerical examples by verifying the claimed and proven results.
Category: Number Theory
[28] viXra:2501.0168 [pdf] submitted on 2025-01-30 22:28:30
Authors: Samuel Bonaya Buya
Comments: 4 Pages.
In this research a general algebraic relationship is established between a pair of primes. This relationship shown to be a useful tool to prove the Binary Goldbach conjecture and other conjectures involving prime gap.
Category: Number Theory
[27] viXra:2501.0164 [pdf] submitted on 2025-01-29 22:03:28
Authors: Samuel Bonaya Buya
Comments: 6 Pages.
In this research a theorem for primes to qualify for Goldbach partition composite even numbers is presented. In the paper reference [1] a neccessary and sufficient condition for proof of Goldbach partition of a composite number was es-tablished and proved. In the paper it was proved that the square of every integer greater than 1 is equal to the sum of the square of an integer greater or equal to zero and a Goldbach partitition semiprime. The proved theorem ef-fectively means that every composite even number has a composite even number. A Goldbach partition semiprime is a semiprime that is a product of two Goldbach partition primes. For a prime to be a Goldbach partition prime it hasto have a Goldbach partition partner for a the specific composite composite even number under consideration. In the paper reference [1] it was proved that every composite even number has at least 1 Goldbach partition semiprime for itsGoldbach parition. In this paper we shall closely examine the primes constituting the Goldbach partition semiprime.
Category: Number Theory
[26] viXra:2501.0161 [pdf] submitted on 2025-01-29 03:17:34
Authors: Samuel Bonaya Buya
Comments: 23 Pages.
In this paper an identity is established connecting to consecutive primes. Bertrand’s postulate is used together with the identity to establish a quadratic inequality that can be used to establish minimum intervals containing at least three primes in between its limits. A generalization of the quadratic inequalityis introduced to establish the minimum interval containining at least one pair of primes for Goldbach partitition. The concepts of Goldbach partition deviation and Goldbach partition interval are introduced by which it is shown that the minimum number of Goldbach partitions of a composite even number is 1.
Category: Number Theory
[25] viXra:2501.0157 [pdf] replaced on 2025-10-27 19:25:00
Authors: Tim Samshuijzen
Comments: 24 Pages.
In the symmetries in the numbers that are coprime with the primorial we find proof of the Hardy-Littlewood K-tuple Conjecture and, consequently, the Twin Prime Conjecture. Using a primorial-based sieve, called the bitstring sieve, we prove that the number of prime k-tuples of size k between p(n)^2 and p(n+1)^2, where p(n) is the n-th prime, increases on average with increasing n.
Category: Number Theory
[24] viXra:2501.0149 [pdf] submitted on 2025-01-28 00:59:09
Authors: Ahcene Ait Saadi
Comments: 9 Pages. (Note by viXra Admin: Pleease cite and list scientific references)
In this paper, I give formulas on the combinatory, this using an equality that I discovered about 40 years ago.(I don't know if this equality already exists, but to my knowledge, no).I use a kind of derivative calculation that I have titled: pseudo derivative calculation. This allows me to find infinity of formulas on combinatorial analysis. Here are some of them. I hope that the young researchers will deepen this work.
Category: Number Theory
[23] viXra:2501.0133 [pdf] replaced on 2025-01-29 16:11:30
Authors: Bahbouhi Bouchaib
Comments: 14 Pages. The paper sheds new light on Goldbach's strong conjecture with new data.
For the first time, this article introduces the notion of natural equidistant-equiranked prime numbers (NEEP) which are the only ones to verify the strong Goldbach conjecture naturally in the set of natural integers. If E is an even ≥ 4 and E = p + q such that q > p, NEEP are the equidistant primes which also have the same ranking for p between 0 and E/2 and for q between E and E/2. Primes are counted from 0 to E/2 on one hand, and inversely from E to E/2 on the other hand. Therefore, primes having the same ranking face each other on a same line and if equidistant relatively to E/2 then their sum = E. From the NEEP, we calculate the deducible equidistant prime numbers (DEP) and it is only from NEEP + DEP that we obtain all the possible sums of two prime numbers of a given even number. No current algorithm for converting even numbers to the sum of two prime numbers distinguishes NEEP from DEP. There are evens like 30 or 90 which don't have NEEP and therefore not satisfying naturally Goldbach's strong conjecture (GSC) unless DEP are deduced by calculation. This is a new matter of thinking : should GSC be refuted because there are evens not having NEEP ? Is this conjecture only deducible by calculation ? Normally one expects GSC to be true with NEEP before getting to DEP. Should the fact that some even numbers not having NEEP be interpreted as a rejection of the Goldbach' Strong Conjecture? The natural presence of NEEP has been exploited here to set up for the first time a system of coding and deciphering even numbers which allows a calculator to deduce all their possible sums of two prime numbers. This article then has two originalities not published before which will certainly be subject to debate
Category: Number Theory
[22] viXra:2501.0124 [pdf] submitted on 2025-01-23 18:25:36
Authors: Johan Noldus
Comments: 2 Pages. (Note by viXra Admin: An abstract in the article is required and scientific references should be cited listed - Please conform)
We provide for ultrashort proofs of all those statements.
Category: Number Theory
[21] viXra:2501.0123 [pdf] replaced on 2026-05-12 09:53:35
Authors: A. A. Frempong
Comments: 10 Pages. Copyright © by A. A. Frempong
By applying basic mathematical principles, the author surely, and instructionally, proves, directly, the Beal conjecture which states that if A^(x+2) + B^(y+2) = C^(z+2) with x > 1 or x = 1, and where A, B, C, x, y, z are positive integers, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^(x+2) + B^(y+2) = C^(z+2) becomes (Dr)^(x+2) + (Es)^(y+2) = (Ft)^(z+2). The proof would be complete after showing that the equalities, r^(x+2) = t^(x+2), s^(y+2) = t^(y+2) and r = s = t, are true. More formally, the conjectured equality, r^(x+2) = t^(x+2) would be true if and only if (r^(x+2))/(t^(x+2)) =1; and the conjectured equality s^(y+2) = t^(y+2) would be true if and only if (s^(y+2))/(t^(y+2)) = 1. These conjectures would be proved in the Beal conjecture proof. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.
Category: Number Theory
[20] viXra:2501.0117 [pdf] replaced on 2025-01-24 15:42:38
Authors: Bahbouhi Bouchaib
Comments: 6 Pages.
The aim of this short paper is to define the real mathematical problem of Goldbach's strong conjecture (GSC), which remains officially unsolved. It attempts to propose a method based on the analysis of remainders of Euclidean division to explain why two equidistant primes add up to an even number. Let's not forget that in mathematics, posing the problem correctly is the surest way to the solution.
Category: Number Theory
[19] viXra:2501.0114 [pdf] submitted on 2025-01-21 21:22:08
Authors: Marko V. Jankovic
Comments: 6 Pages.
In this paper, a piece of advice from Nikola Tesla is going to be used, in order to create a digital electronic circuit that detects/generates prime numbers. Consequently, a formula for the n-th prime is going to be presented.
Category: Number Theory
[18] viXra:2501.0109 [pdf] submitted on 2025-01-20 20:16:29
Authors: Silvio Gabbianelli
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references)
Semiprime numbers are natural integers obtained from the multiplication of two prime numbers. In attempting to solve the factorization of a semiprime number, I considered the inverse procedure of the long multiplication method we learned in elementary school. After careful reflection, I found it to be both useful and feasible. The only known element in this procedure is the result (which must be an odd integer semiprime). We need to find the two sets [n1,n2,...nk][n1, n2, ... nk] and [m1,m2,...mk][m1, m2, ... mk] that will represent the two unique multiplicands that produce the known result. These sets are reconstructed through a triangular matrix, where the rows represent units, tens, hundreds, and so on, and the columns grow and then shrink in size. This matrix allows us to identify the necessary multiplicative pairs by ensuring the result satisfies the condition of being an odd integer that doesn't end in 5.
Category: Number Theory
[17] viXra:2501.0108 [pdf] submitted on 2025-01-20 20:15:18
Authors: Ahcene Ait Saadi
Comments: 9 Pages.
In this paper I solve equations of degrees 5,7 and 9 using an equation of degrees 3. I have the method (the counter that allows me to calculate the different values of the parameters: a, b, c, α,βu2026.etc.) to solve an infinity of equations of degrees (2n+1 ) using equations of lower degrees..My wish is to share it with the young researchers for deepening [knowledge].
Category: Number Theory
[16] viXra:2501.0106 [pdf] replaced on 2025-09-23 17:12:25
Authors: Marco Ripà
Comments: 5 Pages. Revised version (v2): refined presentation and style, clarified proofs, results unchanged.
We present a collection of five nontrivial exercises in number theory (Questions 1—4) and graph theory (Question 5). These problems can be efficiently solved using insights and shortcuts derived from the author's previously published papers. This preprint invites readers to test their expertise in these fields and assess their ability to independently solve the proposed exercises. Detailed solutions are included.
Category: Number Theory
[15] viXra:2501.0097 [pdf] submitted on 2025-01-17 21:24:43
Authors: Thomas Clapies
Comments: 2 Pages. (Correction made by viXra Admin to conform with schoarly norm - Please conform!)
This article reminds us that certain fundamental constants are associated with functions, in particular the base of the exponential function. We use certain remarkable identities associated with the exponential function and Lambert's function W (also called Product log) to obtain a new representation of the Euler constant e = exp(1) using a function.
Category: Number Theory
[14] viXra:2501.0093 [pdf] submitted on 2025-01-16 20:35:58
Authors: Wiroj Homsup, Nathawut Homsup
Comments: 6 Pages.
The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even, or (3n+1)/2 , if n is odd. The conjecture states that for all starting positive integers n the sequence eventually reaches the trivial cycle 1, 2, 1, 2u2026u2026The inverted Collatz sequences can be represented as a Collatz tree with 1 as its root node. In order to prove the Collatz conjecture, one must demonstrate that a Collatz tree covers all positive integers. In this paper, we construct a Collatz tree with 1 as its root node by rearranging the perfect binary tree. We prove that a Collatz tree is a connected tree and covers all positive integers.
Category: Number Theory
[13] viXra:2501.0090 [pdf] replaced on 2025-04-03 03:52:30
Authors: David Adam, Laurent Denis
Comments: 19 Pages. Réagencement de l'article. Les résultats sont identiques.
In this article, we prove the algebraic independence of values special features of the Carlitz-Goss zeta function in a finite place. We characterize also the algebraic independence of values u200bu200bof Carlitz polylogarithms in onefinished place. To do this, we overcome the restrictions in the method of Mahler developed by the second author within the framework of the finished characteristic. This puts us in a position to show results of algebraic independence in finite characteristics in finite places that Papanikolas' method does not yet allow to obtain.
Category: Number Theory
[12] viXra:2501.0075 [pdf] replaced on 2025-01-19 14:03:52
Authors: Samuel Bonaya Buya
Comments: 18 Pages. Some typos were edited. Some more details were added
This research paper is in the form of lesson notes. In it an identity is established connecting to consecutive primes. Bertrand's postulate is used together with the identity to establish a quadratic inequality that can be used to establish minimum intervals containing at least three primes in between its limits. A generalization of the quadratic inequality is introduced to establish the minimum interval containining at least one pair of primes for Goldbach partitition. The concepts of Goldbach partition deviation and Goldbach partition interval are introduced by which it is shown that the minimum number of Goldbach partitions of a composite even number is 1.
Category: Number Theory
[11] viXra:2501.0066 [pdf] replaced on 2025-01-26 09:27:30
Authors: Bahbouhi Bouchaib
Comments: 50 Pages. The paper sheds new light on Goldbach's strong conjecture with new data.
This article emphasizes the most fundamental rules to verify Goldbach's strong conjecture that an even number is the sum of two primes. One rule states that for an even number E to split into two primes there must be two equidistant prime numbers p and p' such that E/2 - p = p' - E/2. The strong conjecture also applies to biprime numbers that are x2 — y2. Two prime numbers equidistant with respect to an integer n have a specific property of Modulo when divided by the gap that separates them from n. The paper further proposes methods to convert even and odd numbers into sums of two and three prime numbers by the equation M ± 1 such that M is prime or multiple of primes except 2 and 3 knowing that there are two types of prime numbers 6x - 1 and 6x + 1. The data also show a strong correlation coefficient between close equidisant primes indicating they are likely to happen in a regular fashion. Finally, the paper describes new rules that explain how a prime numbers gives another one and this is where the truth of Goldbach's conjecture lies and show congruence rules between the two additive primes. These rules allow to demonstrate how an even ends up to be a sum of two primes and proves Goldbach's strong conjecture. This article can have new applications in computing and sheds new lights on the Goldbach's strong and weak conjectures.
Category: Number Theory
[10] viXra:2501.0065 [pdf] replaced on 2025-11-18 23:08:16
Authors: Masashi Furuta
Comments: 7 Pages.
This paper is positioned as a sequel edition of [1]. First, as in [1], define "division sequence", "complete division sequence", and "star conversion". Next, we consider loops and divergences in the Collatz conjecture, respectively. Through elementary arguments, the probability of there being a counterexample to the Collatz conjecture is almost 0. Also, Theorem Proving is not used in this paper.
Category: Number Theory
[9] viXra:2501.0060 [pdf] submitted on 2025-01-10 19:27:32
Authors: Theophilus Agama
Comments: 7 Pages. (Note by viXra Admin: Furher abstract speculations may not be accepted)
Determining the natural textit{density} of Ulam numbers remains an open question. We denote the sequence of all Ulam numbers by $U$. In this paper, we show for the textit{logarithmic} density of Ulam numbers $$mathcal{D}_{log}(U):=lim limits_{nlongrightarrow infty}frac{1}{log x}sum limits_{substack{nleq xin U}}frac{1}{n}=0.$$
Category: Number Theory
[8] viXra:2501.0053 [pdf] submitted on 2025-01-09 21:09:43
Authors: Theophilus Agama
Comments: 6 Pages.
In this note, we study the distribution of the product of consecutive terms in an addition chain of a given length. If $1,2,ldots,s_{delta(n)-1},s_{delta(n)}=n$ is an addition chain producing $n$ and of length $delta(n)$, with associated sequence of generators begin{align}1+1,s_2=a_2+r_2,ldots,s_{delta(n)-1}=a_{delta(n)-1}+r_{delta(n)-1},s_{delta(n)}=a_{delta(n)}+r_{delta(n)}=nonumberend{align} then $$sum limits_{l=1}^{delta(n)}log s_l=delta(n)log n-O(delta(n)).$$ It follows in particular that $$prod limits_{l=1}^{delta(n)}s_lsim n^{delta(n)}.$$
Category: Number Theory
[7] viXra:2501.0045 [pdf] submitted on 2025-01-08 21:34:38
Authors: Algoni Mohamed
Comments: 18 Pages.
We propose a method to solve the modular equation ax ≡ c[b] where a, b and c are natural numbers such that a ≡ ±(10/p) [b] with p ∈ {1, 2, 5, 10}. The method is based on the analysis units of the products pb and pc, which we designate respectively by u and uu2032. This approach allows you to simplify and accelerate the solution process by focusing on these units.
Category: Number Theory
[6] viXra:2501.0042 [pdf] submitted on 2025-01-08 21:32:08
Authors: Algoni Mohamed
Comments: 6 Pages.
We propose a method to solve the modular equation system:x ≡ a_1[b_1]αx ≡ a_2[b_2]with α, a_1, a_2, b_1 and b_2 ∈ N∗ such that a_2 ≥ αa_1 and αb_1 ≡ ±(10/p)[b_2] with p ∈ {1, 2, 5, 10}This method is based on our latest work "On the Diophantine equation ax+by=c with a ≡ ±(10/p)[b] and p ∈ {1, 2, 5, 10}". It is therefore based on the analysis of the units of products pb_2 and p(a_2 − αa_1), which we designate respectively by u and uu2032 .
Category: Number Theory
[5] viXra:2501.0037 [pdf] submitted on 2025-01-07 22:05:08
Authors: Kamal Barghout
Comments: 4 Pages.
The 6 variable general equation of Beal’s conjecture equation〖 x〗^a+y^b=z^c, where x, y, z, a, b, and c are positive integers, and a,b,c≥3, is identified as an identity made by expansion of powers of binomials of integers x and y; where x, y and z have common prime factor. Here, a proof of the conjecture is presented in two folds. First, powers of binomials of integers x and y expand to all integer solutions of Beal’s equation if they have common prime factor. Second, powers of binomials of coprime positive integers x and y expand to two terms such that if one of them is a perfect power the other one is not a perfect power.
Category: Number Theory
[4] viXra:2501.0035 [pdf] submitted on 2025-01-07 21:58:31
Authors: Theophilus Agama
Comments: 4 Pages.
In this note, we study the harmonic distribution of addition chains of a given length. If $1,2,ldots,s_{delta(n)-1},s_{delta(n)}=n$ is an addition chain producing $n$ and of length $delta(n)$, with associated sequence of generators begin{align}1+1,s_2=a_2+r_2,ldots,s_{delta(n)-1}=a_{delta(n)-1}+r_{delta(n)-1},s_{delta(n)}=a_{delta(n)}+r_{delta(n)}=nonumberend{align}then $$sum limits_{l=1}^{delta(n)}frac{1}{s_l}=frac{3}{2}+frac{delta(n)}{n+1}+sum limits_{l=3}^{delta(n)}sum limits_{v=1}^{infty}frac{1}{(n+1)^{v+1}}bigg(sum limits_{j=l}^{delta(n)}r_jbigg)^{v}+O(frac{1}{n})$$ where $sum limits_{j=l}^{delta(n)}r_j<n-1$ is the sum of the regulators in the generator of the chain for each $3leq lleq delta(n)$.
Category: Number Theory
[3] viXra:2501.0013 [pdf] submitted on 2025-01-04 14:22:10
Authors: Alister John Munday
Comments: 6 Pages.
This paper introduces the Fundamental Circularity Theorem (FCT), establishing that certain mathematical behaviours are inherently unprovable because they emerge directly from fundamental properties that cannot themselves be proven without circular reasoning. Using the Collatz conjecture as our primary example, we demonstrate how mathematical behaviours that arise purely from the interaction of fundamental properties resist formal proof not due to complexity or logical paradox, but because any proof would require proving unprovable fundamentals. This insight offers a new understanding of mathematical unprovability distinct from Gödel's incompleteness theorems or independence results.
Category: Number Theory
[2] viXra:2501.0009 [pdf] submitted on 2025-01-02 21:25:01
Authors: Andrey V. Voron
Comments: 7 Pages. (Note by viXra Admin: For the last time, the article heading should be in the following order: Article title, author name and abstract - Future non-compliant submission will not be accepted)
Formulas are defined for calculating the sum of a series of numbers — the so—called constant - that make up a square and a cube of a certain order. The central symmetry of magic squares from the 2nd to the 8th order, as well as magic cubes from the 3rd to the 5th order, is analyzed. It is revealed that the magic squares of the 2nd, 3rd, 4th, 5th, 7th, and 8th order have a "homogeneous" central symmetry (relative to the diagonals), and the magic square of the 6th order has a "mixed" central symmetry. The character of symmetry is determined in the same way for magic cubes of the 3rd, 4th, and 5th orders. "Homogeneous" symmetry is characteristic of the magic cube of the 3rd order and the 5th order, and "heterogeneous" - for the magic cube of the 4th order. Based on the logic of constructing magic squares and cubes, two similar magical objects are constructed — a cube in a cube and cubes in a cube. The first one is based on a magic square of the 4th order (Albrecht Dürer, 1514), and the second one is based on a magic square of the 8th order. These magical figures have a "mixed" symmetry.
Category: Number Theory
[1] viXra:2501.0006 [pdf] replaced on 2025-01-05 22:24:41
Authors: Juan Elias Millas Vera
Comments: 1 Page.
Proof by contradiction of non existence of odd perfect numbers by parity comparasion.
Category: Number Theory
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