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[22] viXra:2108.0164 [pdf] replaced on 2025-03-04 00:57:35
Authors: Méhdi Pascal
Comments: 12 Pages.
This paper is a correction of the last paper I published in viXra in 2021, "On the signs of Lagrange " I had made some big mistakes, and since it's really very important, so I'm correcting it.This paper contains a proof of Fermat's theorem for the first cases, a link between Fermat's theorem and a work by Lagrange, and a link between Fermat's theorem and Galois theory.
Category: Number Theory
[21] viXra:2108.0162 [pdf] submitted on 2021-08-30 22:27:17
Authors: Tae Beom Lee
Comments: 25 Pages.
Goldbach's Conjecture(GC) states that any even integer ≥ 4 can be represented by the sum of two prime numbers. This was conjectured by Christian Goldbach in 1742 and still remains unproved. In this thesis we proved GC by introducing, we called them, Goldbach Partition Model Table(GPMT) and Sieve Functions(SFs). GPMT is a 2-dimensional table of all possible pair of two numbers (x, 2n – x), whose sum can be any even number 2n. To functionally treat the sieve of Eratosthenes, we devised SFs that have sinusoidal symmetry and period properties. By using GPMT and the SFs, we could induce GC False Conditions(GCFC) that must be satisfied if GC is false. And we proved that GCFC can not be satisfied, so, GC is true.
Category: Number Theory
[20] viXra:2108.0138 [pdf] submitted on 2021-08-25 21:11:17
Authors: Ibrahima Sambegou Diallo
Comments: 16 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
In this paper, the author proposes to establish, using relatively simple means, that the quantity of twin primes is infinite, and therefore that the conjecture concerning this notion is in fact a theorem.
Category: Number Theory
[19] viXra:2108.0109 [pdf] replaced on 2021-10-16 12:27:51
Authors: Zeolla Gabriel Martín
Comments: 11 Pages.
Argentest II is born, a personal research project that develops a new exclusive probabilistic primality test for Twin prime numbers. I present a test referenced in Fermat's little theorem.
Category: Number Theory
[18] viXra:2108.0108 [pdf] submitted on 2021-08-21 11:22:13
Authors: Marko V. Jankovic
Comments: 13 Pages.
In this paper proof of the Polignac's Conjecture for gap equal to six is going to be presented. Consecutive primes with gap six are known as sexy primes. The proof represents an extension of the proof of the twin prime conjecture. It will be shown that sexy primes could be obtained through two stage sieve process, and that will be used to prove that infinitely many sexy primes exist.
Category: Number Theory
[17] viXra:2108.0106 [pdf] replaced on 2025-06-02 03:13:33
Authors: Zhang Tianshu
Comments: 16 Pages.
First, let us expound certain basic concepts relating to Collatz conjecture. Next, list the mathematical induction that proves the conjecture. Then again, prepare several judging criteria, which are solely used to determine whether each such operational result fits the conjecture. After that, we sort positive integers successively and prove directly one of certain sorts after each sorting, until the last two sorts are proved bidirectionally. The bidirectional proofs mean that for these two sorts of integers, on the one hand, start with several proven kinds of integers to expand successively the scope of proven kinds of integers, up to all kinds of integers. On the other, each unproven kind of integers is operated by the operational rule to find an integer expression that is less than the unproven kind of integers, from this, it meets a judging criterion, such that the unproven kind of integers is proved to fit the conjecture.
Category: Number Theory
[16] viXra:2108.0105 [pdf] replaced on 2023-03-16 02:00:50
Authors: Tianshu Zhang
Comments: 16 Pages. (Corrections made by viXra Admin on name and title - Please conform!)
The subject of this article is exactly to analyze Beal’s conjecture and prove it. First, we classify mathematical expressions which consist of AX, BY and CZ, according to the parity of A, B and C, then get rid of two combinations of AX, BY and CZ, for they have nothing to do with the conjecture. After that, we exemplify AX+BY=CZ under the necessary constraints, where A, B, and C have at least one common prime factor. Secondly, divide AX+BY≠CZ under the necessary constraints into two kinds, and prove one kind thereof in which any two terms have a common prime factor while another term has not it. Next, under known constraints, divide another kind of AX+BY≠CZ into four inequalities. Furthermore, we derive four conclusions from the interrelation between an even number as the symmetric center and a sum of two odd numbers. This is just a preparation for proving the first two inequalities. Then, the first two inequalities are proved by the mathematical induction, fundamental theorem of arithmetic and the binomial theorem. Then again, other two inequalities are proved by the reduction to absurdity.Finally, after comparing AX+BY=CZ and AX+BY≠CZ under the necessary constraints, we came to the conclusion that Beal's conjecture is true.
Category: Number Theory
[15] viXra:2108.0104 [pdf] replaced on 2023-02-13 10:31:06
Authors: Zhang Tianshu
Comments: 14 Pages.
In this article, we classify positive integers step by step, and use the formulation to represent a certain class therein until all classes. First, divide all integers ≥2 into 8 kinds, and formulate each of 7 kinds therein into a sum of 3 unit fractions. For the unsolved kind, again divide it into 3 genera, and formulate each of 2 genera therein into a sum of 3 unit fractions. For the unsolved genus, further divide it into 5 sorts, and formulate each of 3 sorts therein into a sum of 3 unit fractions. For two unsolved sorts i.e. 4/(49+120c) and 4/(121+120c) where c≥0, we use an unit fraction plus a proper fraction to replace each of them, then take out the unit fraction as 1/x. After that, we take out an unit fraction from the proper fraction and regard the unit fraction as 1/y, and finally, prove that the remainder can be identically converted to 1/z.
Category: Number Theory
[14] viXra:2108.0092 [pdf] submitted on 2021-08-18 09:02:17
Authors: Brian Scannell
Comments: 12 Pages.
The nulls in antenna radiation patterns show the non-trivial Riemann zeta zeros. Here we simulate antenna designs that show the zeros
Category: Number Theory
[13] viXra:2108.0080 [pdf] submitted on 2021-08-16 19:52:45
Authors: Juan Elias Millas Vera
Comments: 2 Pages.
This paper is a resume of my investigation on the Collatz conjecture. We take a look to the hypothesis of infinite systems of iterations or combined iterations.
Category: Number Theory
[12] viXra:2108.0077 [pdf] submitted on 2021-08-15 19:45:09
Authors: Shekhar Suman
Comments: 11 Pages.
In this manuscript we denote a unit disc by $\mathbb{D}=\{z\in \mathbb{C} \mid |z|<1\}$ and a semi plane as\\ $\mathbb{P}=\{s\in\mathbb{C}\mid \Re(s)>\frac{1}{2}\}$. We denote, $\mathbb{R}_{\geq 0}=\{x\in \mathbb{R}\mid x\geq 0\}$ and $\mathbb{R}_{\geq 1}=\{x\in \mathbb{R}\mid x\geq 1\}$. Considering non negative real axis as a branch cut, we define a map from slit unit disc to the slit plane as $s:\mathbb{D}\setminus \mathbb{R}_{\geq 0}\to \mathbb{P}\setminus\mathbb{R}_{\geq 1}$ defined as $s(z)=\frac{1}{1-\sqrt{z}}$ which is proved to be one-one and onto. Next, we define a function $f(z)=(s-1)\zeta(s)$ where $s=s(z)$ and both $s(z)$ and $f(z)$ are proved to be analytic in $\mathbb{D}\setminus \mathbb{R}_{\geq 0}$. Next we prove that $s=s(z)$ is a conformal map. We also show that $f$ is continuous at $0$. Using Cauchy's residue theorem to a keyhole contour and Lebesgue's dominated convergence theorem along with Schwarz reflection principle, we prove that, $$\int_{-\infty}^\infty \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt=0$$
This settles the Riemann Hypothesis because this relation is an equivalent version of Riemann Hypothesis as proved by Balazard, Saias and Yor [1].
Category: Number Theory
[11] viXra:2108.0073 [pdf] submitted on 2021-08-15 20:05:04
Authors: Andrea Berdondini
Comments: 6 Pages.
In statistics, to evaluate the significance of a result, one of the most used methods is the statistical hypothesis test. Using this theory, the fundamental problem of statistics can be expressed as follows: "A statistical data does not represent useful information, but becomes useful information only when it is shown that it was not obtained randomly". Consequently, according to this point of view, among the hypotheses that perform the same prediction, we must choose the result that has a lower probability of being produced randomly. Therefore, the fundamental aspect of this approach is to calculate correctly this probability value. This problem is addressed by redefining what is meant by hypothesis. The traditional approach considers the hypothesis as the set of rules that actively participate in the forecast. Instead, we consider as hypotheses the sum of all the hypotheses made, also considering the hypotheses preceding the one used. Therefore, each time a prediction is made, our hypothesis increases in complexity and consequently increases its ability to adapt to a random data set. In this way, the complexity of a hypothesis can be precisely determined only if all previous attempts are known. Consequently, Occam's razor principle no longer has a general value, but its application depends on the information we have on the tested hypotheses.
Category: Number Theory
[10] viXra:2108.0066 [pdf] submitted on 2021-08-14 04:08:05
Authors: Han Geurdes
Comments: 6 Pages. A new attempt. This is rewritten after comments .
With simple basic mathematics it is possible to demonstrate a conflicting result in complex number theory using Euler’s identity, simple trigonometry and deMoivre’s formula for n=2.
Category: Number Theory
[9] viXra:2108.0065 [pdf] submitted on 2021-08-14 06:43:24
Authors: Marcin Barylski
Comments: 5 Pages.
Lemoine’s conjecture (LC), still unsolved, states that all positive odd integer ≥ 7 can be expressed as the sum of a prime and an even semiprime. But do we need all primes to satisfy this conjecture? This work is devoted to selection of must-have primes and formulation of stronger version of LC with reduced set of primes.
Category: Number Theory
[8] viXra:2108.0064 [pdf] submitted on 2021-08-14 06:47:14
Authors: Marcin Barylski
Comments: 10 Pages.
Goldbach strong conjecture states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). Hypothesis still remains open and is confirmed experimentally for bigger and bigger n. This work studies different approaches to finding the first
confirmation of this conjecture in order to select the most effective confirmation method.
Category: Number Theory
[7] viXra:2108.0058 [pdf] submitted on 2021-08-12 06:56:36
Authors: Marcin Barylski
Comments: 3 Pages.
Goldbach strong conjecture, still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). Each prime p>3 can be expressed as 6k ± 1. This work is devoted to studies of 6k ± 1 primes in Goldbach partitions and enhanced Goldbach strong conjecture with the lesser of twin primes of form 6k − 1 used as a baseline.
Category: Number Theory
[6] viXra:2108.0057 [pdf] replaced on 2021-08-13 11:40:56
Authors: Marcin Barylski
Comments: 5 Pages.
Goldbach Strong Conjecture (GSC), still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). But do we need all primes to satisfy this conjecture? This work is devoted to selection of must-have primes and formulation of stronger version of GSC with reduced set of primes.
Category: Number Theory
[5] viXra:2108.0056 [pdf] replaced on 2021-08-13 11:45:14
Authors: Marcin Barylski
Comments: 4 Pages.
Goldbach strong conjecture, still unsolved, states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). We can also formulate it from the opposite perspective: from a set of prime numbers you may pick two primes, sum them, and you will be able to build every even number n>2. This work is devoted to studies on sum of two prime numbers.
Category: Number Theory
[4] viXra:2108.0055 [pdf] submitted on 2021-08-12 07:09:45
Authors: Marcin Barylski
Comments: 4 Pages.
Goldbach strong conjecture states that all even integers n>2 can be expressed as the sum of two prime numbers (Goldbach partitions of n). This work is devoted to studies on twin primes present in Goldbach partitions. Based on executed experiments original Goldbach conjecture has been extended to a form that all even integers n>4 can be expressed as the sum of twin prime and prime.
Category: Number Theory
[3] viXra:2108.0036 [pdf] replaced on 2021-11-25 17:47:00
Authors: Zeolla Gabriel Martín
Comments: 11 Pages.
This text develops and formulates the discovery of an unknown pattern for prime numbers, with amazing and calculable characteristics. Using a mechanism similar to the Collatz conjecture.
The latest version includes Python program
Category: Number Theory
[2] viXra:2108.0024 [pdf] submitted on 2021-08-08 19:39:07
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper we study the shortest addition chains of numbers of special forms. We obtain the crude inequality $$\iota(2^n-1)\leq n+1+G(n)$$ for some function $G:\mathbb{N}\longrightarrow \mathbb{N}$. In particular we obtain the weaker inequality $$\iota(2^n-1)\leq n+1+\left \lfloor \frac{n-2}{2}\right \rfloor$$ where $\iota(n)$ is the length of the shortest addition chain producing $n$.
Category: Number Theory
[1] viXra:2108.0008 [pdf] submitted on 2021-08-03 22:09:56
Authors: Gaurav Krishna
Comments: 11 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
The approach to the problem is in the reverse. Instead of moving from 3n+1, we move from the final answer and extract all the possible values of n. They key was to find out a unique way of all the possible n’s that emanate from 2y. Tried it the other way round, gets messier. So just sharing the final simpler version. The paper contains tables, which are basically equations for sets of data and rearrangement of those tables is just playing with the underlying equations. The solution: Redefining the problem → Identifying the pattern → analyzing the pattern → eliminating rouge data → testing for all the possible values of n.
Category: Number Theory
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