# Number Theory

Previous months:
2007 - 0703(3) - 0706(2)
2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)
2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)
2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)
2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)
2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)
2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)
2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(51) - 1409(47) - 1410(16) - 1411(16) - 1412(18)
2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(15) - 1508(14) - 1509(14) - 1510(12) - 1511(9) - 1512(25)
2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(19) - 1608(16) - 1609(22) - 1610(23) - 1611(12) - 1612(20)
2017 - 1701(19) - 1702(24) - 1703(29) - 1704(32) - 1705(25) - 1706(26) - 1707(21) - 1708(26) - 1709(12)

## Recent submissions

Any replacements are listed further down

[1573] viXra:1709.0375 [pdf] submitted on 2017-09-24 18:23:32

### An Essay on the Zeroes of an L-Function

Authors: Matanari Shimoinuda

Category: Number Theory

[1572] viXra:1709.0312 [pdf] submitted on 2017-09-22 01:36:01

### The Distribution of Primes

Authors: Ihsan Raja Muda Nasution

In this paper, we analyze the behavior of prime numbers.
Category: Number Theory

[1571] viXra:1709.0295 [pdf] submitted on 2017-09-20 06:46:03

### Fermat’s Zero Theorem

Authors: Faisal Amin Yassein Abdelmohssin

Fermat’s zero theorem is stated as follows: It is impossible to separate a square of a difference of two natural numbers into two squares of differences, or a cube power of a difference into two cube powers of differences, or a fourth power of a difference into two fourth powers, or in general, any power higher than the first, into two like powers of differences.
Category: Number Theory

[1570] viXra:1709.0288 [pdf] submitted on 2017-09-19 04:55:48

### Special Rule for Certain Prime Numbers

Authors: Ranganath G. Kulkarni

A quadratic equation for prime numbers is assumed to be true that satisfy the following four rules. Some prime numbers violate these rules. Whereas some non prime numbers satisfy the four rules. They are not prime, therefore to make them violate the fourth rule we need to study how to choose the value of m and n so as make the quadratic equation as the primes generating formula.
Category: Number Theory

[1569] viXra:1709.0258 [pdf] submitted on 2017-09-17 13:33:28

### Fermat's Theorem. Proof by 2 Operations

Authors: Victor Sorokine

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
Category: Number Theory

[1568] viXra:1709.0257 [pdf] submitted on 2017-09-17 13:35:16

### Fermat's Theorem. Proof by 2 Operations French

Authors: Victor Sorokine

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
L'égalité de Fermat est contradictoire entre les deuxièmes chiffres des facteurs du nombre A.
Category: Number Theory

[1567] viXra:1709.0256 [pdf] submitted on 2017-09-17 13:36:32

### Fermat's Theorem. Proof by 2 Operations Russian

Authors: Victor Sorokine

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
Суть противоречия. Равенство Ферма противоречиво по вторым цифрам сомножителей числа А.
Category: Number Theory

[1566] viXra:1709.0227 [pdf] submitted on 2017-09-15 05:23:56

### A Minor Theorem Related with the Fermat Conjecture

Authors: José Francisco García Juliá

It is obtained a minor theorem related with the Fermat conjecture.
Category: Number Theory

[1565] viXra:1709.0128 [pdf] submitted on 2017-09-11 05:10:30

### Theorems on Pythagorean Triples and Prime Numbers

Authors: Faisal Amin Yassein Abdelmohssin

Relationships among natural numbers constituting a Pythagorean triple (PT) and between these natural numbers constituting the Pythagorean triples (PTs) and Prime Numbers (PNs) have been found. These relationships are formulated as theorems; first theorem is that the natural numbers constituting a Pythagorean triple (PT) satisfy a certain equation related to sum of their differences; second theorem is that differences of sum of the natural numbers constituting a Pythagorean triple (PT) are prime numbers.
Category: Number Theory

[1564] viXra:1709.0092 [pdf] submitted on 2017-09-08 12:19:26

### Question 383 : Nonlinear Equation , Euler Numbers , Number Pi

Authors: Edgar Valdebenito

This note presents some formulas for pi.
Category: Number Theory

[1563] viXra:1709.0039 [pdf] submitted on 2017-09-04 11:17:48

### RSA Cryptography over Polynomials

Authors: Antoine Balan
Comments: 2 pages, written in french

Here is defined, following the RSA cryptosystem, a method of cryptography for polynomials over finite rings.
Category: Number Theory

[1562] viXra:1709.0013 [pdf] submitted on 2017-09-02 03:44:43

### Proving Grimm’s Conjecture by Step-by- Step Forming Consecutive Composite Numbers’ Points at the Number Axis(Chinese)

Authors: Zhang Tianshu

Let us regard positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments which have same permutations of c kinds of integers’ points, where c≥1. In this article we shall prove Grimm’s conjecture by the method which changes stepwise symbols of each kind of composite numbers’ points at the original number axis, so as to form consecutive composite numbers’ points inside the limited field of proven Legendre- Zhang conjecture as the true.
Category: Number Theory

[1561] viXra:1709.0003 [pdf] submitted on 2017-09-01 07:17:32

### Aﬃrmative Resolve of Conway’s Problem

Authors: T.Nakashima

In this paper, we prove Conway's problem.
Category: Number Theory

[1560] viXra:1708.0421 [pdf] submitted on 2017-08-28 08:10:40

### Double Integrals and Series for Some Classical Constants

Authors: Edgar Valdebenito

This note presents a collection of double integrals for some classical constants.
Category: Number Theory

[1559] viXra:1708.0411 [pdf] submitted on 2017-08-28 10:09:24

### Implementation of a Core(c) Number Sieve.

Authors: Preininger Helmut

In this paper we give an implementation of a Core(c) Number Sieve (for a given c=1,2,3,.. we sift out numbers that have in there factorization a prime with a power >= c). For c=2 we have a squarefree number sieve. (Note, that, for c=1, our implementation compute the usual prime number sieve.) Our goal is to use only one codebase and avoid extra algorithms for every c. We use some well known algorithms and adopt it for our purpose.
Category: Number Theory

[1558] viXra:1708.0400 [pdf] submitted on 2017-08-28 00:19:33

### Units and Class Numbers Extracted from Regulators of Small Number Fields

Authors: Lulu Karami

This submission demonstrates how to use the analytic class number formula to express certain quotients of Dedekind's Eta function as a unit raised to the power of a quoteint of class numbers, for particular number fields. It includes a loose derivation for some special cases of reciprocity laws and the Fourier series of particular Eisenstein series.
Category: Number Theory

[1557] viXra:1708.0380 [pdf] submitted on 2017-08-27 09:31:10

### Division by Zero and the Arrival of Ada

Authors: I Gede Putra Yasa ''Gus Satya''

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.
Category: Number Theory

[1556] viXra:1708.0255 [pdf] submitted on 2017-08-21 21:17:24

### New Idea of the Goldbach Conjecture

Authors: Jun Chen

A new idea of the Goldbach conjecture has been studied, it is that the even number is more bigger, the average form of the sum of two primes are more larger too. And then, we prove that every sufficiently large even number is the sum of two primes.
Category: Number Theory

[1555] viXra:1708.0234 [pdf] submitted on 2017-08-19 11:06:29

### Fermat's Last Theorem

Authors: Ramaswamy Krishnan

This proof is based on an assumption that value of an infinite series cannot be obtained from a finite number of terms of the series. For all possible factors of (x + y -z) which are not factors of x or y or z, 3 infinite series can be developed, 2 convergent and 1 divergent. In all the 3 cases, the value of the infinite series can be obtained by considering only a finite number of terms. This gives the value for (x + y -z) = p to to the power of alpha * (p1) (p2) (p3). Thus proving Fermat's last theorem.
Category: Number Theory

[1554] viXra:1708.0231 [pdf] submitted on 2017-08-19 11:51:51

### A Relation for Q-Pochhammer Symbol, Q-Bracket, Q-Factorial and Q-Binomial Coefficient.

Authors: Edigles Guedes, Cícera Guedes

In this paper, we construct a relation involving q-Pochhammer symbol, q-bracket, q-factorial and q-binomial coefficient among other things.
Category: Number Theory

[1553] viXra:1708.0230 [pdf] submitted on 2017-08-19 11:56:45

### An Identity for a Q-Hypergeometric Series

Authors: Edigles Guedes, Cícera Guedes

In this paper, we construct a identity for a q-hypergeometric series.
Category: Number Theory

[1552] viXra:1708.0221 [pdf] submitted on 2017-08-19 03:46:20

### Generalization of a Ramanujan Formula

Authors: Mendzina Essomba Francois

I found a formula due to Ramanjan, I have given a generalization in this article
Category: Number Theory

[1551] viXra:1708.0220 [pdf] submitted on 2017-08-19 03:47:39

### Equation for Distribution of Prime Numbers

Authors: Ranganath G. Kulkarni

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann's formula is approximate one. We need to study the variation of prime numbers with given number x and new variable r.
Category: Number Theory

[1550] viXra:1708.0206 [pdf] submitted on 2017-08-17 05:19:11

### 2 to the Power of (p-1) is not Congruent to 1 Mod P Cubed for Any Prime 'p'

Authors: Ramaswamy Krishnan
Comments: 3 Pages. Title has been changed a little bit. Instead of mod p, it should be mod p cubed

Category: Number Theory

[1549] viXra:1708.0204 [pdf] submitted on 2017-08-17 05:25:20

### 3 to the Power of (p-1) is not Congruent to 1 Mod P Cubed if P is Congruent to 1 Mod 6

Authors: Ramaswamy Krishnan

Category: Number Theory

[1548] viXra:1708.0187 [pdf] submitted on 2017-08-16 12:56:13

### Question 273: Nonlinear Equation,Bernoulli Numbers,Number Pi

Authors: Edgar Valdebenito

This note presents some formulas for pi.
Category: Number Theory

[1547] viXra:1708.0181 [pdf] submitted on 2017-08-16 06:53:04

### The Proof of the Lonely Runner Conjecture

Authors: Kurmet Sultan
Comments: 29 pages, Written in Russian

The article presents the proof of the lonely runner conjecture.
Category: Number Theory

[1546] viXra:1708.0177 [pdf] submitted on 2017-08-16 00:48:13

### The Proof of the Collatz Conjecture

Authors: Kurmet Sultan

The article provides with the evidence of the Collatz conjecture.
Category: Number Theory

[1545] viXra:1708.0158 [pdf] submitted on 2017-08-14 08:00:35

### On the Pythagorean Triples (12,y,z)

Authors: Faisal Amin Yassein Abdelmohssin

I found a pattern in the Pythagorean triples formed of the natural number12;{(12,5,13), (12,9,15), (12,16, 20), (12,35,37)}. The pattern is the decreasing value of the difference z - y for the triples such that the differences form a sequence of the even numbers {8,6,4,2} in that order. The existence of such sequence for other natural numbers transforms the Pythagorean equation into a linear equation in y .
Category: Number Theory

[1544] viXra:1708.0141 [pdf] submitted on 2017-08-13 04:27:47

### Yet Another Proof that Zeta(2)=pi^2/6

Authors: Hervé G.

Yet another proof that zeta(2)=Pi^2/6
Category: Number Theory

[1543] viXra:1708.0114 [pdf] submitted on 2017-08-10 21:43:45

### The Binary Goldbach Conjecture: A Proof For the Existence of Prime Sums For All 2N

Authors: Joseph Dise

A minimum number of paired composite sums is shown for all 2N. By logical extension, it proves the existence of paired prime sums for all 2N.
Category: Number Theory

[1542] viXra:1708.0108 [pdf] submitted on 2017-08-10 09:44:07

### The P-alic Numbers

Authors: Antoine Balan
Comments: 2 pages, written in french

The P-alic numbers are defined and we show a formula giving the decomposition of a polynomial according the valuations.
Category: Number Theory

[1541] viXra:1708.0103 [pdf] submitted on 2017-08-09 13:24:14

### The Infinity of Twin Primes

Authors: Joseph Dise

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.
Category: Number Theory

[1540] viXra:1708.0082 [pdf] submitted on 2017-08-08 05:09:33

### New Formulas for Transcendent Functions

Authors: François Mendzina Essomba

I propose new formulas for the transcendent functions that I discovered.
Category: Number Theory

[1539] viXra:1708.0076 [pdf] submitted on 2017-08-08 07:03:16

### Classify Positive Integers to Prove Collatz Conjecture by Mathematical Induction

Authors: Zhang Tianshu

Positive integers which are able to be operated to 1 by set operational rule of the Collatz conjecture and positive integers got via operations by the operational rule versus the set operational rule are one-to-one the same, thus we refer to converse operational routes, apply the mathematical induction, next classify positive integers to prove the Collatz conjecture by substeps according to beforehand prepared two theorems concerned.
Category: Number Theory

[1538] viXra:1708.0063 [pdf] submitted on 2017-08-06 18:22:13

### A Strange Natural Logarithmm Formula

Authors: Mendzina Essomba Francois

A strange formula for the calculation of the natural logarithm of any real number which I guessed almost effortlessly. This formula has the advantage of being very efficient for the calculation of the logarithm of large numbers
Category: Number Theory

[1537] viXra:1708.0048 [pdf] submitted on 2017-08-04 15:49:56

### François Mendzina Essomba Phy Formulae

Authors: Mendzina Essomba Francois

I present some formulas found for the calculation of phy. Some of these formulas accelerate the calculation of decimals and others are written as a function of e and pi by continuous fractions
Category: Number Theory

[1536] viXra:1708.0046 [pdf] submitted on 2017-08-05 03:08:52

### François Mendzina Essomba pi-3.14 Formulae

Authors: Mendzina Essomba Francois

some of my many pi formulae
Category: Number Theory

[1535] viXra:1708.0044 [pdf] submitted on 2017-08-04 11:37:00

### Numbers

Authors: Antoine Balan
Comments: 6 pages, written in french

Foundations of numbers theory are reviewed and some basic definitions are studied.
Category: Number Theory

[1534] viXra:1707.0410 [pdf] submitted on 2017-07-31 10:45:34

### The Proof of Fermat's Last Theorem for the Base Case

Authors: Victor Sorokine

The essence of the proof: From the known properties of the Fermat’s equality An+Bn=Cn follows: If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /stay/ infinitely large.
Category: Number Theory

[1533] viXra:1707.0395 [pdf] submitted on 2017-07-29 16:17:02

Authors: Ramón Ruiz
Comments: 36 Pages. This document is written in Spanish

Goldbach's Conjecture: “Every even integer greater than 2 can be expressed as the sum of two primes”. In this document I used the prime numbers theorem enunciated by Carl Friedrich Gauss and the prime numbers theorem in arithmetic progressions. These two theorems applied to a combination of two arithmetic progressions of module 30 and that contain prime numbers, allows us to develop a nonprobability general formula to calculate, approximately, the number of prime pairs that adding up an even number x. This research is based on a approach designed solely to demonstrate the Binary Goldbach Conjecture and the Twin Prime Conjecture.
Category: Number Theory

[1532] viXra:1707.0392 [pdf] submitted on 2017-07-30 03:32:36

### Integer Difference of Powers

Authors: L. Castillo

I explore a method to characterize all the real numbers a,b such that all of $a - b, a^2 - b^2,...,a^n - b^n$ are integers for a given n and paying particular attention to the special case when neither of a and b are integers themselves.
Category: Number Theory

[1531] viXra:1707.0335 [pdf] submitted on 2017-07-26 03:43:24

### Even Gap and Polignac Conjecture

Comments: In 4 pages i give the proof to that conjecture

In this paper we give the proof of even gap conjecture whose can be expressed by it exists infinitely prime p such that p+n is prime for an even integer n and we deduce Polignac conjecture
Category: Number Theory

[1530] viXra:1707.0279 [pdf] submitted on 2017-07-20 13:09:45

### The Internal Structure of Natural Numbers and the Sets of Odd Numbers that Are not Primes and Even Numbers that Are not Powers of Two.

Authors: Emmanuil Manousos

Natural numbers have a strictly defined internal structure that is being revealed in the present article. This structure is inherent of the natural numbers and is not derived through the introduction of any axioms for the set of natural numbers. In the present article, we prove the fundamental theorems that determine this structure. As a consequence of this structure, a mathematical expression for the set of odd numbers that are not primes is derived. Given the set of odd numbers, we can identify the set of prime numbers. Additionally, a new method for expressing odd composite numbers as the product of powers of prime numbers is derived.
Category: Number Theory

[1529] viXra:1707.0258 [pdf] submitted on 2017-07-18 15:08:59

### A New Result About Prime Numbers: Lim N→+∞ N/(p(n) − N(ln N + ln ln N − 1)) = +∞

Authors: Rédoane Daoudi

In this short paper we propose a new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞ .
Category: Number Theory

[1528] viXra:1707.0241 [pdf] submitted on 2017-07-17 13:26:16

### Ramanujan Trigonometric Formula

Authors: Edgar Valdebenito

This note presents formulas and fractals related with Ramanujan's trigonometric formula.
Category: Number Theory

[1527] viXra:1707.0240 [pdf] submitted on 2017-07-17 14:34:07

### François Mendzina Essomba pi-3.14 Formulas

Authors: François Mendzina Essomba

I present in this article some of my many formulas discovered on pi
Category: Number Theory

[1526] viXra:1707.0237 [pdf] submitted on 2017-07-17 22:43:49

### Upper Bound of Prime Gaps, Lengendre's Conjecture Was Verified (I)

Authors: Quang Nguyen Van

We have found the possible max- difference between two successive prime numbers, and by them, Lengendre's conjecture is verified.
Category: Number Theory

[1525] viXra:1707.0217 [pdf] submitted on 2017-07-15 15:40:09

### How to Prove that an Integer Number is Prime with the Factoriels.

Authors: Mendzina Essomba Francois

How to prove that an integer number is prime with the factoriels. We give in this article which is not complete a property of the facoral which allows in an interval of given length to verify if the number is prime
Category: Number Theory

[1524] viXra:1707.0176 [pdf] submitted on 2017-07-13 03:54:39

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[1523] viXra:1707.0174 [pdf] submitted on 2017-07-12 07:32:30

### A Simple Proof of the Last Fermat Theorem (RUSSIAN Version)

Authors: Victor Sorokine

The proof of Fermat's last theorem for the base case / Доказательство ВТФ для базового случая ABSTRACT The essence of the proof: From the known properties of the Fermat’s equality An+Bn=Cn follows: If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become /remain/ infinitely large. Суть доказательства: Из базового равенства Ферма An+Bn=Cn следует: Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля, то новые уменьшенные числа А°, В°, С° становятся /остаются/ бесконечно большими. (See also http://vixra.org/abs/1707.0092)
Category: Number Theory

[1522] viXra:1707.0168 [pdf] submitted on 2017-07-11 17:00:37

### The Q-Universe: A Set Theoretic Construction

Authors: Wes Hansen

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.
Category: Number Theory

[1521] viXra:1707.0167 [pdf] submitted on 2017-07-11 18:24:29

### The Surprising Proofs

Authors: Leszek Włodzimierz Guła
Comments: 7 Pages. In this work we have a new deductions.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture.
Category: Number Theory

[1520] viXra:1707.0152 [pdf] submitted on 2017-07-10 13:09:21

### A Conjecture About Prime Numbers Assuming the Riemann Hypothesis

Authors: Rédoane Daoudi

In this paper we propose a conjecture about prime numbers. Based on the result of Pierre Dusart stating that the n th prime number is smaller than n(ln n + ln ln n − 0.9484) for n ≥ 39017 we propose that the n th prime number is smaller than n(ln n + ln ln n − 1+) when n → +∞.
Category: Number Theory

[1519] viXra:1707.0092 [pdf] submitted on 2017-07-06 04:02:34

### A Simple Proof of the Last Fermat Theorem

Authors: Victor Sorokine
Comments: 6 Pages. The text is in French

English. The essence of the proof From the known properties of the Fermat’s equality A n +B n =C n follows: If the second digits of all the prime factors of the numbers A, B, and C are reduced to zero, then the new reduced numbers A°, B°, C° become infinitely large. From which follows the truth of the FLT? Français. L'essence de la preuve : A partir des propriétés connues de l'égalité de Fermat A n +B n =C n il suit: Si les deuxièmes chiffres de tous les facteurs premiers des nombres A, B, C, réduit à zéro, alors les nombres de nouveaux A°, B°, C°, devenir infiniment grand. Ce qui implique la vérité du DTF ? Русский. Суть доказательства: Из базового равенства Ферма A n +B n =C n следует: Если вторые цифры всех простых сомножителей чисел А, В, С УМЕНЬШИТЬ до нуля, то новые уменьшенные числа А°, В°, С° становятся бесконечно большими. Из чего следует истинность ВТФ?
Category: Number Theory

[1518] viXra:1707.0086 [pdf] submitted on 2017-07-05 14:31:20

### Zeta Function and Infinite Sums

Authors: François Mendzina Essomba

I have come to the conclusion, after finishing a first reection on infinite sums, that all the functions which are written in the form of an infinite sum are written according to the famous Zeta function, this statement is explicitly presented in this article.
Category: Number Theory

[1517] viXra:1707.0048 [pdf] submitted on 2017-07-05 03:01:29

### More Insight in Sequence of Happy Cube Numbers

Authors: Muneer Jebreel Karama

A positive integer is called Fixed Happy Cube Numbers (FHCN) in case, if you are cubing its digits and adding them together one time you got the same number. For example the number 153 is happy cube because; 153= 1^3+5^3+3^3, in fact this paper will address new propriety of this extraordinary happy cube number .
Category: Number Theory

[1516] viXra:1707.0020 [pdf] submitted on 2017-07-02 05:41:19

### One, Zero, Ada, and Cyclic Number System

Authors: I Gede Putra Yasa Gus Satya''

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However those solution methods give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that the methods produce actual or exact results. The solution methods are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.
Category: Number Theory

[1515] viXra:1706.0543 [pdf] submitted on 2017-06-28 23:55:17

### The Max Nature Number

Authors: Liu Ran

Thank ancient philosopher Zeno, who brought such an interesting and meaningful paradox. It imply that the limit is reachable. Then we can deduct the infinity is about 618724203×10^26,
Category: Number Theory

[1514] viXra:1706.0531 [pdf] submitted on 2017-06-29 05:39:25

### The Recursive Future And Past Equation Based On The Ananda-Damayanthi Normalized Similarity Measure Considered To Exhaustion {File Closing Version+1}

In this research investigation, the author has presented a Recursive Past Equation and a Recursive Future Equation based on the Ananda-Damayanthi Normalized Similarity Measure considered to Exhaustion [1] (please see the addendum of [1] as well).
Category: Number Theory

[1513] viXra:1706.0507 [pdf] submitted on 2017-06-26 13:13:35

### Confirmation of Riemann Hypothesis Allows the Calculation of the 19th and 21st Mills' Primes

Authors: Thomas Pierre Nicolas Jean Brouard

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta fuction is 1 . Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.
Category: Number Theory

[1512] viXra:1706.0479 [pdf] submitted on 2017-06-25 18:25:43

### Factored Closed-Form Expressions for the Sums of Fibonacci and Lucas Numbers

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of a certain form.
Category: Number Theory

[1511] viXra:1706.0457 [pdf] submitted on 2017-06-23 13:24:49

### Question 935 : An Integral for Pi

Authors: Edgar Valdebenito

In this note we present an integral for the constant pi:pi=3.1415926535...
Category: Number Theory

[1510] viXra:1706.0414 [pdf] submitted on 2017-06-21 05:00:47

### Forty Inedited Sequences of Integers Involving Primes and Fermat Pseudoprimes

Authors: Marius Coman

A selection of forty sequences regarding primes and Fermat pseudoprimes from my yet unpublished papers, presented in "OEIS style", with definition of the terms of a sequence, examples, few first terms, notes and conjectures.
Category: Number Theory

[1509] viXra:1706.0410 [pdf] submitted on 2017-06-21 00:28:19

### Primes Obtained Concatenating the Numbers S(p)-D(k), Where S(p) is the Sum of Digits of a Prime P and D(1),...,d(k) the Digits of P

Authors: Marius Coman

In this paper I make the following conjecture: There exist an infinity of primes p having the property that concatenating s(p) – d(1) with s(p) – d(2) and repeatedly up to s(p) – d(k), where s(p) is the sum of digits of p and d(1),...,d(k) are the digits of p, is obtained a prime q. Example: such prime p is 127 because concatenating 9 (= 10 – 1) with 8 (= 10 – 2) and with 3 (= 10 – 7) is obtained a prime q = 983.
Category: Number Theory

[1508] viXra:1706.0408 [pdf] submitted on 2017-06-21 02:28:55

### A New Sufficient Condition by Euler Function for Riemann Hypothesis

Authors: Choe Ryong Gil

The aim of this paper is to show a new sufficient condition (NSC) by the Euler function for the Riemann hypothesis and its possibility. We build the NSC for any natural numbers ≥ 2 from well-known Robin theorem, and prove that the NSC holds for all odd and some even numbers while, the NSC holds for any even numbers under a certain condition, which would be called the condition (d).
Category: Number Theory

[1507] viXra:1706.0407 [pdf] submitted on 2017-06-21 02:30:07

### An Upper Bound for Error Term of Mertens' Formula

Authors: Choe Ryong Gil

In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.
Category: Number Theory

[1506] viXra:1706.0381 [pdf] submitted on 2017-06-18 22:43:41

### Diophantine Equation Bicar

Authors: Lahcen Aghray

We obtain a parameterization of a Diophantine equation of degree 4
Category: Number Theory

[1505] viXra:1706.0380 [pdf] submitted on 2017-06-18 23:13:05

### Diophantine Equation Third Degre

Authors: Lahcen Aghray

The resolution of a Diophantine equation by calculating the intersection of a curve of degree 3 with a line
Category: Number Theory

[1504] viXra:1706.0288 [pdf] submitted on 2017-06-15 07:46:58

### On Fermat's Last Theorem - An Elementary Approach

Authors: Gang Li

An attempt of using elementary approach to prove Fermat's last theorem (FLT) is given. For infinitely many prime numbers, Case I of the FLT can be proved using this approach. Furthermore, if a conjecture proposed in this paper is true (k-3 conjecture), then case I of the FLT is proved for all prime numbers. For case II of the FLT, a constraint for possible solutions is obtained.
Category: Number Theory

[1503] viXra:1706.0206 [pdf] submitted on 2017-06-13 13:41:27

### Complex Continued Fractions , Numbers , Sequences , Number Pi

Authors: Edgar Valdebenito

In this note we recall some formulas related with continued fractions , numbers , sequences and the constant pi.
Category: Number Theory

[1502] viXra:1706.0205 [pdf] submitted on 2017-06-13 13:45:55

### The Quintic :Z^5+z^4-1=0

Authors: Edgar Valdebenito

In this note we briefly explore the equation: z^5+z^4-1=0
Category: Number Theory

[1501] viXra:1706.0197 [pdf] submitted on 2017-06-14 09:53:22

### Why Are Odd Bernoulli Numbers Equal to Zero?

Authors: Ryan Zielinski

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.
Category: Number Theory

[1500] viXra:1706.0196 [pdf] submitted on 2017-06-14 15:11:32

### Elegant Formulas for the Natural Logarithm

Authors: Mendzina Essomba Francois

J present two algorithms for calculating the natural logarithm of any real number. The first is an algorithm obtained by the method of Archimedes for the calculation of pi and the second the product of a succession of rad icals.
Category: Number Theory

[1499] viXra:1706.0192 [pdf] submitted on 2017-06-15 02:06:54

### The End of FLT

Authors: Leszek W. Guła
Comments: 5 Pages. Certainly no scientist was working under such conditions. Nobody will ever announce to the world my creative proposals.

1. The truly marvellous proof of The Fermat's Last Theorem (FLT). 2. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2.
Category: Number Theory

[1498] viXra:1706.0134 [pdf] submitted on 2017-06-09 07:24:09

### Series Representation Ofpower Function

Authors: Kolosov Petro
Comments: 12 pages, arXiv:1603.02468, MSC 2010: 40C15, 32A05

This paper describes a method of natural exponented power's $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, based on induction, except above described theorems. Keywords: power, power function, monomial, polynomial, power series, third power, series, finite difference, divided difference, high order finite difference, derivative, binomial coefficient, binomial theorem, Newton's binomial theorem, binomial expansion, n-th difference of n-th power, number theory, cubic number, cube, Euler number, exponential function, Pascal triangle, Pascal’s triangle, mathematics, math, maths, algebra, science, arxiv, preprint, series representation, series expansion, open scicence, calculus
Category: Number Theory

[1497] viXra:1706.0112 [pdf] submitted on 2017-06-07 14:51:48

### On the Quantum Differentiation of Smooth Real-Valued Functions

Authors: Kolosov Petro
Comments: 12 pages, 6 figures, arXiv:1705.02516

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown. Keywords: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint
Category: Number Theory

[1496] viXra:1706.0111 [pdf] submitted on 2017-06-07 19:48:40

### On the Link Between Finite Difference and Derivative of Polynomials

Authors: Kolosov Petro
Comments: 12 pages, 1 figure, arXiv:1608.00801

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials. Keywords: finite difference, divided difference, high order finite difference, derivative, ode, pde, partial derivative, partial difference, power, power function, polynomial, monomial, power series, high order derivative, mathematics, differential calculus, math, maths, science, arxiv, preprint, algebra, calculus, open science, differential equations
Category: Number Theory

[1495] viXra:1706.0102 [pdf] submitted on 2017-06-06 11:10:53

### Primes Obtained Concatenating P∙q∙r-P with P∙q∙r-Q with P∙q∙r-R Then with P∙q∙r Where p, q, R Primes of the Form 6k+1

Authors: Marius Coman

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 3-Poulet numbers of the form (6k + 1)*(6h + 1)*(6j + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of pairs of primes [q, r], of the form 6h + 1 and 6j + 1, such that the number obtained concatenating p*q*r – p with p*q*r – q with p*q*r – r then with p*q*r is prime.
Category: Number Theory

[1494] viXra:1706.0097 [pdf] submitted on 2017-06-06 04:10:22

### Primes Obtained Concatenating P∙q-P with P∙q-Q Then with P∙q Where p, Q Primes of the Form 6k+1

Authors: Marius Coman

This paper is inspired by one of my previous papers, namely “Large primes obtained concatenating the numbers P - d(k) where d(k) are the prime factors of the Poulet number P”, where I conjectured that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Because some of these Poulet numbers are 2-Poulet numbers of the form (6k + 1)*(6h + 1) I extend in this paper that idea conjecturing that for any prime p of the form 6k + 1 there exist an infinity of primes q of the form 6h + 1 such that the number obtained concatenating p*q – p with p*q – q then with p*q is prime.
Category: Number Theory

[1493] viXra:1706.0037 [pdf] submitted on 2017-06-05 05:55:08

### Large Primes Obtained Concatenating the Numbers P-D(k) Where D(k) Are the Prime Factors of the Poulet Number P

Authors: Marius Coman

In this paper I conjecture that there are an infinity of primes which can be obtained concatenating the numbers P - d(1); P - d(2); ...; P – d(k); P, where d(1), ..., d(k) are the prime factors of the Poulet number P. Example: using the sign “//” with the meaning “concatenated to”, for the Poulet number 129921 (= 3*11*31*127), the number (129921 – 3)//(129921 – 11)//(129921 – 31)//(129921 – 127)//129921 = 129918129910129890129794129921 is prime. Note that such primes are obtained for 10 from the first 90 Poulet numbers!
Category: Number Theory

[1492] viXra:1706.0033 [pdf] submitted on 2017-06-04 11:53:02

### Primes Obtained Concatenating the Numbers 30-D(k) Where D(1),...,d(k) Are the Digits of a Square of a Prime

Authors: Marius Coman

In this paper I make the following observation: for many squares of primes (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(k), where d(1),..., d(k) are the digits of a square of a prime, are primes. Example: for 1369 (= 37^2) the number obtained concatenating 29 = 30 – 1 with 27 = 30 – 3 with 24 = 30 – 6 with 21 = 30 – 9, i.e. the number 29272421, is prime. Note that for 35 from the first 200 squares of primes the numbers obtained this way are primes!
Category: Number Theory

[1491] viXra:1706.0032 [pdf] submitted on 2017-06-04 12:51:12

### Primes Obtained Concatenating the Numbers 30-D(n) Where D(1),...,d(n) Are the Digits of a Poulet Number

Authors: Marius Coman

In this paper I make the following observation: for many Poulet numbers (I conjecture that for an infinity of them) the numbers obtained concatenating 30 – d(1), 30 – d(2),..., 30 – d(n), where d(1),..., d(n) are the digits of a n-digits Poulet number, are primes. Example: for 8911 the number obtained concatenating 22 = 30 – 8 with 21 = 30 – 9 with 29 = 30 – 1 with 29 = 30 – 1, i.e. the number 22212929, is prime.
Category: Number Theory

[1490] viXra:1706.0029 [pdf] submitted on 2017-06-04 02:09:22

### Trigonometric Functions Extension and Mathematic Constances

Authors: Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

Introduction of new trigonometric functions and mathematical constants. The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.
Category: Number Theory

[1489] viXra:1705.0461 [pdf] submitted on 2017-05-29 12:33:16

### The Cubic :X^3+x^2+1=0

Authors: Edgar Valdebenito

This note presents some formulas and fractals related with the equation : x^3+x^2+1=0.
Category: Number Theory

[1488] viXra:1705.0460 [pdf] submitted on 2017-05-29 07:06:59

### Virtual Container Research Method: Solving Riemann Hypothesis, Polignac's Conjecture and Twin Prime Conjecture

Authors: John Yuk Ching Ting
Comments: 66 Pages. Rigorous proofs for Riemann hypothesis, Polignac's conjecture and Twin prime conjecture

L-functions form an integral part of the 'L-functions and Modular Forms Database' which is associated with far-reaching applications and implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by German mathematician Bernhard Riemann whereby all nontrivial zeros of Riemann zeta function are conjectured to be located on the critical line. This proposal is equivalently stated in this research paper as all nontrivial zeros are conjectured to exactly match the 'Origin' intercepts of Riemann zeta function. Deeply entrenched in number theory, prime number theorem involves analysis of the prime counting function for prime numbers. Solving Riemann hypothesis would result in a crucial primary by-product whereby absolute and full delineation of this important prime number theorem will occur. Involving the study of prime numbers [which are Incompletely Predictable entities], Twin prime conjecture involves the analysis of prime gap = 2 [representing all twin primes] and is thus a subset of Polignac's conjecture which involves the analysis of all even number prime gaps = 2, 4, 6,... [representing prime numbers in totality except for the first prime number '2']. Nontrivial zeros of Riemann zeta function are also Incompletely Predictable entities. With the common presence of Incompletely Predictable entities and with this helpful presence considered a major asset; the task to solve the above mentioned intractable open problems of Riemann hypothesis, Polignac's and Twin prime conjectures is conveniently amalgamated together in this paper. We employ our novel Virtual Container Research Method which acts essentially as foundation for the mathematical framework enabling successful completion of this monumental task.
Category: Number Theory

[1487] viXra:1705.0395 [pdf] submitted on 2017-05-28 03:10:36

### Unknown Algorithms for Finding Prime Numbers Among Odd Numbers

Authors: Oleg Cherepanov

The discovered algorithm for extracting prime numbers from the natural series is alternative to both the Eratosthenes lattice and Sundaram and Atkin's sentences. The distribution of prime numbers does not have a formula, but if the number is one less than the prime number is an exponent of the integers, then there are no two scalar scalars whose sum is equal to the third integer in the same degree. This is the sound of P. Fermat's Great Theorem, the proof of which he could begin by using the Minor theorem known to him. The first part of the proof is here restored. But how did P. Fermat finish it?
Category: Number Theory

[1486] viXra:1705.0393 [pdf] submitted on 2017-05-27 18:13:53

### On Gormaund Numbers and Gormaund's Theorem

Authors: Caitherine Gormaund

Herein we introduce the subject of the Gormaund numbers, and prove a fundamental property thereof.
Category: Number Theory

[1485] viXra:1705.0390 [pdf] submitted on 2017-05-27 07:41:40

### A Recreative Method to Obtain from a Given Prime Larger Primes Based on the Powers of 3

Authors: Marius Coman

In this paper I present a method to obtain from a given prime p1 larger primes, namely inserting before of a digit of p1 a power of 3, and, once a prime p2 is obtained, repeating the operation on p2 and so on. By this method I obtained from a prime with 9 digits a prime with 36 digits (the steps are showed in this paper) using just the numbers 3, 9(3^2), 27(3^3) and 243(3^5).
Category: Number Theory

[1484] viXra:1705.0379 [pdf] submitted on 2017-05-26 06:15:45

### Open Number Theory Problem

Authors: Ricardo.gil
Comments: 1 Page. Email solutions or suggestions to Ricardo.gil@sbcglobal.net

In math the the 7 Clay Math unsolved problems? Another problem is the question if there is a God(s)? In my paper the purpose is to explain that in the end we all meet our maker and that man does not have the power to cheat death. Like the Riemann Zeta function that remains unsolved and when solved will give insight to distribution of the Primes, giving or solving this open-end problem will help me solve a problem. This is the only problem I have not been able to solve and I am open sourcing it.
Category: Number Theory

[1483] viXra:1705.0360 [pdf] submitted on 2017-05-25 05:19:12

### y^n FLT Identity Proof

Authors: Maik Becker-Sievert
Comments: 1 Page. Identity Proof FLT

This Identity proofs direct Fermats Last Theorem
Category: Number Theory

[1482] viXra:1705.0343 [pdf] submitted on 2017-05-22 13:32:15

### Some Formulas Related with Theta Functions and pi Constant

Authors: Edgar Valdebenito

This note presents some formulas for pi constant
Category: Number Theory

[1481] viXra:1705.0342 [pdf] submitted on 2017-05-22 13:36:50

### The Number Z=lambertw(sqrt(-1))

Authors: Edgar Valdebenito

This note presents some formulas related with the number z=LambertW(i),where LambertW(x) is the Lambert function.
Category: Number Theory

[1480] viXra:1705.0293 [pdf] submitted on 2017-05-19 11:18:33

### If P is Any Odd Prime Number and C is Any Odd Number Less Than p, Then there Must Exist a Positive Number C’ Less Than p, Such that Cc’= 2modp

Authors: Prashanth Rao

If p is any odd prime number and c is any odd number less than p, then there must exist a positive number c’ less than p, such that cc’= -2modp
Category: Number Theory

[1479] viXra:1705.0289 [pdf] submitted on 2017-05-19 07:56:37

### Distribution of the Residues and Cycle Counting

Authors: Helmut Preininger

In this paper we take a closer look to the distribution of the residues of squarefree natural numbers and explain an algorithm to compute those distributions. We also give some conjectures about the minimal number of cycles in the squarefree arithmetic progression and explain an algorithm to compute this minimal numbers.
Category: Number Theory

[1478] viXra:1705.0277 [pdf] submitted on 2017-05-19 01:01:55

### Analytic Demonstrations on the Fourfold Root Topics of Primes

Authors: Shaban A. Omondi Aura

This paper is concerned with formulation and demonstration of new versions of equations that can help us resolve problems concerning maximal gaps between consecutive prime numbers, the number of prime numbers at a given magnitude and the location of nth prime number. There is also a mathematical argument on why prime numbers as elementary identities on their own respect behave the way they do. Given that the equations have already been formulated, there are worked out examples on numbers that represent different cohorts. This paper has therefore attempted to formulate an equation that approximates the number of prime numbers at a given magnitude, from N=3 to N=〖10〗^25. Concerning the location of an nth prime number, the paper has devised a method that can help us locate a given prime number within specified bounds. Nonetheless, the paper has formulated an equation that can help us determine extremely bounded gaps. Lastly, using trans-algebraic number theory method, the paper has shown that unpredictable behaviors of prime numbers are due to their identity nature.
Category: Number Theory

[1477] viXra:1705.0242 [pdf] submitted on 2017-05-16 03:47:59

### Conjecture on the Pairs of Primes Obtained Inserting N with Digit Sum 12 After the First Digit of Twin Primes

Authors: Marius Coman

In this paper I conjecture that for any pair of twin primes [p, q], p ≥ 11, there exist a number n having the sum of its digits equal to 12 such that inserting n after the first digit of p respectively q are obtained two primes (almost always twins, as in the case [1481, 1483] where n = 48 is inserted in [11, 13], beside the case that the first digit of twins is different, as in the case [5669, 6661] where n = 66 is inserted in [59, 61]).
Category: Number Theory

[1476] viXra:1705.0224 [pdf] submitted on 2017-05-15 02:29:14

### A Recreative Conjecture on Primes Obtained Inserting N with Digit Sum 12 Before the Last Digit of a Prime

Authors: Marius Coman

In this paper I conjecture that for any prime p, p ≥ 7, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 before the last digit of p.
Category: Number Theory

[1475] viXra:1705.0221 [pdf] submitted on 2017-05-15 03:41:04

### A Recreative Conjecture on Primes Obtained Inserting N with Digit Sum 12 After the First Digit of a Prime

Authors: Marius Coman

In this paper I conjecture that for any prime p, p ≥ 5, there exist a prime q obtained inserting a number n with the sum of digits equal to 12 after the first digit of p.
Category: Number Theory

## Replacements of recent Submissions

[692] viXra:1709.0312 [pdf] replaced on 2017-09-24 03:38:04

### The Distribution of Primes

Authors: Ihsan Raja Muda Nasution

In this paper, we find the axiomatic pattern of prime numbers.
Category: Number Theory

[691] viXra:1708.0220 [pdf] replaced on 2017-08-23 05:44:34

### Equation for Distribution of Prime Numbers

Authors: Ranganath G. Kulkarni

An equation for distribution of prime numbers is found that agree well with actual values of prime numbers in the range x. We find that Riemann hypothesis may be wrong. We need to study the variation of new variable r with the given number x.
Category: Number Theory

[690] viXra:1708.0108 [pdf] replaced on 2017-08-12 18:30:40

### The P-alic Numbers

Authors: Antoine Balan
Comments: 2 pages, written in french

We define here a p-adic like field, called the P-alic field. We replace in fact the ring Z by Q[X] and we show a formula of decomposition of a polynomial.
Category: Number Theory

[689] viXra:1708.0108 [pdf] replaced on 2017-08-11 13:40:29

### The P-alic Numbers

Authors: Antoine Balan
Comments: 2 pages, written in french

Following the definition of p-adic numbers, we apply the definitions to the ring Z[X] instead of Z, we show a formula for a polynomial.
Category: Number Theory

[688] viXra:1708.0108 [pdf] replaced on 2017-08-10 18:38:28

### The P-alic Numbers

Authors: Antoine Balan
Comments: 2 pages, written in french

The P-alic numbers are defined following the definition of p-adic numbers. We show a formula for a polynomial.
Category: Number Theory

[687] viXra:1708.0103 [pdf] replaced on 2017-09-24 12:48:05

### The Infinity of Twin Primes

Authors: Joseph Dise

6x±1 are twin primes where x=6nm±(n±m) has no solution for positive integers x, n, and m. This paper follows that definition to its conclusion.
Category: Number Theory

[686] viXra:1707.0335 [pdf] replaced on 2017-07-26 07:25:36

### Even Gap and Polignac Conjecture Proof

Comments: In 4 pages i give the proof

In this paper i give the proof of Polignac conjecture and even gap cobjecture by using Chebotarev Artin theorem
Category: Number Theory

[685] viXra:1707.0176 [pdf] replaced on 2017-09-08 04:10:25

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[684] viXra:1707.0176 [pdf] replaced on 2017-07-21 06:58:15

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[683] viXra:1707.0176 [pdf] replaced on 2017-07-15 02:12:22

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[682] viXra:1707.0176 [pdf] replaced on 2017-07-14 06:25:46

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[681] viXra:1707.0176 [pdf] replaced on 2017-07-13 07:41:09

### Uncertainty and the Lonely Runner Conjecture

Authors: John Atwell Moody

By convolving the distribution of one of the non-chosen runners with a step function (to introduce some uncertainty in its start time) we arrange that the mutual expectation reverts to the continuous extension of its value in the transcendental case.
Category: Number Theory

[680] viXra:1707.0168 [pdf] replaced on 2017-08-05 15:40:50

### The Q-Universe: A Set Theoretic Construction

Authors: Wes Hansen

In an earlier paper, “Q-Naturals: A Counter-Example to Tennenbaum’s Theorem,” we developed a set of non-standard naturals called q-naturals and demonstrated a counter-example to Tennenbaum’s Theorem. In this paper we extend the q-naturals to the Q-Universe and explore the properties of the various subsets along the way. In the process of this development, we realize that the standard Universe and the Q-Universe are simply the zeroth-order and first-order Universes, respectively, in a countable subsumption hierarchy of recursive Universes; there exist countably many counter-examples to Tennenbaum’s Theorem.
Category: Number Theory

[679] viXra:1707.0167 [pdf] replaced on 2017-08-03 18:41:24

### The Surprising Proofs

Authors: Leszek Włodzimierz Guła
Comments: 10 Pages. This is the new variant of my work.

The proof of the Fermat’s Last Theorem. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2. The proof of the Goldbach’s Conjecture. The proof of the Beal’s Conjecture.
Category: Number Theory

[678] viXra:1707.0023 [pdf] replaced on 2017-07-18 03:50:04

### An Interval Unifying Theorem About Primes

Authors: Juan Moreno Borrallo

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.
Category: Number Theory

[677] viXra:1707.0023 [pdf] replaced on 2017-07-11 07:05:28

### An Interval Unifying Theorem About Primes

Authors: Juan Moreno Borrallo

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.
Category: Number Theory

[676] viXra:1707.0023 [pdf] replaced on 2017-07-05 03:11:26

### An Interval Unifying Theorem About Primes

Authors: Juan Moreno Borrallo

In this paper it is proved the existence of a prime number in the interval between the square of any natural number greater than one, and the number resulting from adding or subtracting this natural number to its square (Oppermann’s Conjecture). As corollaries of this proof, they are proved three classical prime number’s conjectures: Legendre’s, Brocard’s, and Andrica’s. It is also defined a new maximum interval between any natural number and the nearest prime number. Finally, it is stated as corollary the existence of infinite prime numbers equal to the square of a natural number, plus a natural number inferior to that natural number, and minus a natural number inferior to that natural number.
Category: Number Theory

[675] viXra:1707.0020 [pdf] replaced on 2017-08-04 01:55:55

### One, Zero, Ada, and Cyclic Number System

Authors: I Gede Putra Yasa Gus Satya''

Division by 0 is not defined in mathematics. Mathematics suggests solutions by work around methods. However they give only approximate, not the actual or exact, results. Through this paper we propose methods to solve those problems. One characteristic of our solution methods is that they produce actual or exact results. They are also in conformity with, and supported by, physical or empirical facts. Other characteristic is their simplicity. We can do computations easily based on basic arithmetic or algebra or other computation methods we already familiar with.
Category: Number Theory

[674] viXra:1707.0010 [pdf] replaced on 2017-07-30 11:06:38

### Nine Notes on Modular Forms

Authors: John Atwell Moody

Contents:

Analytic primality testing 1
Lefshetz numbers of modular curves 23
Grothendieck sections and rational points of modular curves 29
Rational points of modular curves 33
Outline geometric proof of Mordell’s conjecture 48
Example:the Fermat curves 63
The residue calculation 69
The meaning of positive and negative 81

Category: Number Theory

[673] viXra:1706.0507 [pdf] replaced on 2017-06-27 12:34:17

### Confirmation of Riemann Hypothesis Allows the Calculation of the 19th and 21st Mills' Primes

Authors: Thomas Pierre Nicolas Jean Brouard

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta function is 1/2. Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.
Category: Number Theory

[672] viXra:1706.0507 [pdf] replaced on 2017-06-27 08:43:31

### Add to Mendeley Add to Citeulike Add to Facebook Confirmation of Riemann Hypothesis Allows the Calculation of the 19th and 21st Mills' Primes

Authors: Thomas Pierre Nicolas Jean Brouard

Using the η function, we show that the real part of the non-trivial zeros of the Riemann zeta fuction is 1/2. Then, we calculate two big primes using the Riemann hypothesis as true. These two big primes have respectively more than one hundred millions digits and more than one billion digits.
Category: Number Theory

[671] viXra:1706.0197 [pdf] replaced on 2017-06-28 09:57:27

### Why Are Odd Bernoulli Numbers Equal to Zero?

Authors: Ryan Zielinski
Comments: 4 Pages. Version 2 is an extended version of the original paper. Both works are licensed under the CC BY 4.0, a Creative Commons Attribution License.

In this note we will use Faulhaber's Formula to explain why the odd Bernoulli numbers are equal to zero.
Category: Number Theory

[670] viXra:1706.0192 [pdf] replaced on 2017-07-04 17:51:05

### The End of FLT

Authors: Leszek W. Guła

1. The proper proof of The Fermat's Last Theorem (FLT). 2. The proof of the theorem - For all n∈{3,5,7,…} and for all z∈{3,7,11,…} and for all natural numbers u,υ: z^n≠u^2+υ^2.
Category: Number Theory

[669] viXra:1706.0134 [pdf] replaced on 2017-09-07 17:27:09

### Series Representation of Power Function

Authors: Kolosov Petro
Comments: 13 pages, 5 figures, arXiv:1603.02468, Keywords: Power function, Monomial, Polynomial, Power series, Finite difference, Derivative, Differential calculus, Differentiation, Binomial coefficient, Newton's Binomial Theorem, Exponential function

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown. In Application 3 generalized calculus of finite differences, based on expression (1.9) is shown.
Category: Number Theory

[668] viXra:1706.0134 [pdf] replaced on 2017-08-01 15:52:40

### Series Representation of Power Function

Authors: Petro Kolosov
Comments: 11 pages, 5 figures, arXiv:1603.02468, MSC 2010: 40C15, 32A05.

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Binomial sum. Additionally, in section 4 exponential function’s $e^x$ representation is shown.
Category: Number Theory

[667] viXra:1705.0460 [pdf] replaced on 2017-08-09 04:23:37

### Solving Incompletely Predictable Problems: Riemann Hypothesis, Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 69 Pages. Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros of this function are conjectured to lie on the critical line. This proposal is equivalently stated in this research paper as all nontrivial zeros are conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem involves analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities which allow us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.
Category: Number Theory

[666] viXra:1705.0460 [pdf] replaced on 2017-07-23 07:26:29

### Solving Incompletely Predictable Problems: Riemann Hypothesis, Polignac's and Twin Prime Conjectures

Authors: John Yuk Ching Ting
Comments: 68 Pages. Rigorous proofs for Riemann hypothesis, Polignac's and Twin prime conjectures

L-functions form an integral part of the 'L-functions and Modular Forms Database' with far-reaching implications. In perspective, Riemann zeta function is the simplest example of an L-function. Riemann hypothesis refers to the 1859 proposal by Bernhard Riemann whereby all nontrivial zeros of this function are conjectured to lie on the critical line. This proposal is equivalently stated in this research paper as all nontrivial zeros are conjectured to exactly match the 'Origin' intercepts of this function. Deeply entrenched in number theory, prime number theorem involves analysis of prime counting function for prime numbers. Solving Riemann hypothesis would enable complete delineation of this important theorem. Involving proposals on the magnitude of prime gaps and their associated sets of prime numbers, Twin prime conjecture deals with prime gap = 2 (representing twin primes) and is thus a subset of Polignac's conjecture which deals with all even number prime gaps = 2, 4, 6,... (representing prime numbers in totality except for the first prime number '2'). Both nontrivial zeros and prime numbers are Incompletely Predictable entities which allow us to employ our novel Virtual Container Research Method to solve the associated hypothesis and conjectures.
Category: Number Theory

[665] viXra:1705.0360 [pdf] replaced on 2017-05-25 07:46:44

### y^n FLT Identity Proof

Authors: Maik Becker-Sievert