**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(2) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(7) - 1203(6) - 1204(6) - 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(33) - 1408(50) - 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(13) - 1510(11) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(53) - 1605(28) - 1606(17) - 1607(17) - 1608(15) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(23) - 1703(25) - 1704(32) - 1705(25) - 1706(25) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(23) - 1712(34)

2018 - 1801(32) - 1802(20) - 1803(22) - 1804(27) - 1805(31) - 1806(16) - 1807(18) - 1808(14) - 1809(22) - 1810(17) - 1811(26) - 1812(32)

2019 - 1901(12) - 1902(11) - 1903(22) - 1904(26) - 1905(10)

Any replacements are listed farther down

[1992] **viXra:1905.0365 [pdf]**
*submitted on 2019-05-19 12:19:51*

**Authors:** Emmanuil Manousos

**Comments:** 20 Pages.

“Every natural number, with the exception of 0 and 1, can be written in a unique way as a linear combination of consecutive powers of 2, with the coefficients of the linear combination being -1 or +1”. According to this theorem we define the L/R symmetry of the natural numbers. The L/R symmetry gives the factors which determine the internal structure of natural numbers. As a consequence of this structure, we have an algorithm for determining prime numbers and for factorization of natural numbers.

**Category:** Number Theory

[1991] **viXra:1905.0269 [pdf]**
*submitted on 2019-05-17 15:12:11*

**Authors:** Wilson Torres Ovejero

**Comments:** 16 Pages.

160 years ago that in the complex analysis a hypothesis was raised, which was used in principle
to demonstrate a theory about prime numbers, but, without any proof; with the passing Over the years, this
hypothesis has become very important, since it has multiple applications to physics, to number theory, statistics,
among others In this article I present a demonstration that I consider is the one that has been dodging all this
time.

**Category:** Number Theory

[1990] **viXra:1905.0250 [pdf]**
*submitted on 2019-05-16 16:10:59*

**Authors:** Yuly Shipilevsky

**Comments:** 5 Pages.

We consider a new conjecture regarding powers of integer numbers and
more specifically, we are interesting in existence and finding pairs of integers:
n ≥ 2 and m ≥ 2, such that nm
= mn. We conjecture that n = 2, m = 4
and n = 4, m = 2 are the only integral solutions.
Next, we consider the corresponding generalizations for Hypercomplex
Integers: Gaussian and Lipschitz Integers.

**Category:** Number Theory

[1989] **viXra:1905.0210 [pdf]**
*submitted on 2019-05-14 15:29:38*

**Authors:** Arthur Shevenyonov

**Comments:** 6 Pages. trilinear

A set of minimalist demonstrations suggest how the key premises of RH may have been inspired and could be qualified, by proposing a linkage between the critical strip (0..n) and Re(s)=x-1/2 interior of candidate solutions. The solution density may be concentrated around the focal areas amid the lower and upper bound revealing rarefied or latent representations. The RH might overlook some of the ontological structure while confining search to phenomena while failing to distinguish between apparently concentrated versus seemingly non-distinct candidates.

**Category:** Number Theory

[1988] **viXra:1905.0137 [pdf]**
*submitted on 2019-05-10 01:25:34*

**Authors:** Marko Jankovic

**Comments:** 11 Pages.

In this paper a proof of the existence of an infinite number of Sophie Germain primes, is going to be presented. In order to do that, we analyse the basic formula for prime numbers and decide when this formula would produce a Sophie Germain prime, and when not. Originally very difficult problem (in observational space) has been transformed into a simpler one (in generative space) that can be solved by elementary math.

**Category:** Number Theory

[1987] **viXra:1905.0111 [pdf]**
*submitted on 2019-05-07 22:32:35*

**Authors:** Toshiro Takami

**Comments:** 74 Pages.

new version
I believe this is proof of the Riemann hypothrsis.
I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.

**Category:** Number Theory

[1986] **viXra:1905.0098 [pdf]**
*submitted on 2019-05-06 16:48:48*

**Authors:** Zeolla Gabriel Martín

**Comments:** 7 Pages.

This document develops and demonstrates the discovery of a new cubic potentiation algorithm that works absolutely with all the numbers using the formula of the cubic of a binomial.

**Category:** Number Theory

[1985] **viXra:1905.0041 [pdf]**
*submitted on 2019-05-02 12:38:19*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. Submitted to the journal Monatshefte für Mathematik. Comments welcome.

In this paper, we consider the abc conjecture in the case c=a+1. Firstly, we give the proof of the first conjecture that c

**Category:** Number Theory

[1984] **viXra:1905.0021 [pdf]**
*submitted on 2019-05-01 08:54:31*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

This is an easy approach to proving zeta(2) is irrational. The reasoning is by analogy with gym weights that are rational proportions of a unit. Sometimes the sum of such weights is expressible as a multiple of a single term in the sum and sometimes it isn't. The partials of zeta(2) are of the latter type. We use a result of real analysis and this fact to show the infinite sum has this same property and hence is irrational.

**Category:** Number Theory

[1983] **viXra:1905.0010 [pdf]**
*submitted on 2019-05-01 18:09:11*

**Authors:** Yuly Shipilevsky

**Comments:** 7 Pages.

We introduce a special class of complex numbers, wherein their
absolute values and arguments given in a polar coordinate system are integers,
which when considered within the complex plane, constitute Unicentered
Radial Lattice and similarly for quaternions.

**Category:** Number Theory

[1982] **viXra:1904.0592 [pdf]**
*submitted on 2019-04-30 08:39:11*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we recall a formula for Pi.

**Category:** Number Theory

[1981] **viXra:1904.0561 [pdf]**
*submitted on 2019-04-30 05:20:15*

**Authors:** Kouji Takaki

**Comments:** 11 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[1980] **viXra:1904.0517 [pdf]**
*submitted on 2019-04-26 09:22:45*

**Authors:** Afmika, AF. Michael

**Comments:** 4 Pages.

This is a simple proof of the Legendre's conjecture.
afmichael73@gmail.com
afmichael.san@gmail.com

**Category:** Number Theory

[1979] **viXra:1904.0507 [pdf]**
*submitted on 2019-04-27 04:27:00*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages.

This remarks proves, that Riemann zeta function has infinitesimal amount of zeros.

**Category:** Number Theory

[1978] **viXra:1904.0489 [pdf]**
*submitted on 2019-04-26 01:20:05*

**Authors:** Kunle Adegoke

**Comments:** 5 Pages.

We evaluate the sums $\sum_{j=0}^k{u_{rj+s}^{2n}\,z^j}$, $\sum_{j=0}^k{u_{rj+s}^{2n-1}\,z^j}$ and $\sum_{j=0}^k{v_{rj+s}^{n}\,z^j}$, where $r$, $s$ and $k$ are any integers, $n$ is any nonnegative integer, $z$ is arbitrary and $(u_n)$ and $(v_n)$ are the Lucas sequences of the first kind and of the second kind, respectively. As natural consequences we obtain explicit forms of the generating functions for the powers of the terms of Lucas sequences with indices in arithmetic progression. This paper therefore extends the results of P.~Sta\u nic\u a who evaluated $\sum_{j=0}^k{u_{j}^{2n}\,z^j}$ and $\sum_{j=0}^k{u_{j}^{2n-1}\,z^j}$; and those of B. S. Popov who obtained generating functions for the powers of these sequences.

**Category:** Number Theory

[1977] **viXra:1904.0454 [pdf]**
*submitted on 2019-04-23 08:38:55*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

In this note we give some formulas related with the number: alpha=0.5*arccos(0.5*arccos(0.5*arccos(0.5*...))).

**Category:** Number Theory

[1976] **viXra:1904.0446 [pdf]**
*submitted on 2019-04-23 18:28:40*

**Authors:** Zeolla Gabriel Martín

**Comments:** 7 Pages.

This document develops and demonstrates the discovery of a new square potentiation algorithm that works absolutely with all the numbers using the formula of the square of a binomial.

**Category:** Number Theory

[1975] **viXra:1904.0428 [pdf]**
*submitted on 2019-04-22 21:43:23*

**Authors:** Ralf Wüsthofen

**Comments:** 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an arithmetic antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, i.e. a set that is a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[1974] **viXra:1904.0422 [pdf]**
*submitted on 2019-04-21 06:22:45*

**Authors:** Hajime Mashima

**Comments:** 2 Pages.

The three sides of the right triangle are rational numbers, and those with natural numbers are congruent numbers.

**Category:** Number Theory

[1973] **viXra:1904.0410 [pdf]**
*submitted on 2019-04-21 15:17:55*

**Authors:** Yuly Shipilevsky

**Comments:** 3 Pages.

We consider generalized Fermat equation for hypercomplex numbers, in
order to stimulate research and development of those generalization

**Category:** Number Theory

[1972] **viXra:1904.0386 [pdf]**
*submitted on 2019-04-19 11:38:30*

**Authors:** Divyendu Priyadarshi

**Comments:** 1 Page.

In this short paper, I have tried to give a physical meaning to irrational numbers.

**Category:** Number Theory

[1971] **viXra:1904.0378 [pdf]**
*submitted on 2019-04-19 21:33:45*

**Authors:** Pedro Caceres

**Comments:** 27 Pages.

The Riemann Zeta function or Euler–Riemann Zeta function, ζ(s), is a function of a complex variable z that analytically continues the sum of the Dirichlet series:
() = ∑ ^(-z) from k=1,∞
The Riemann zeta function is a meromorphic function on the whole complex z-plane, which is holomorphic everywhere except for a simple pole at z = 1 with residue 1. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x), and also provided insights into the roots (zeros) of the zeta function, formulating a conjecture about the location of the zeros of () in the critical line Re(z)=1/2.
The Riemann Zeta function is one of the most studied and well known mathematical functions in history. In this paper, we will formulate nine new propositions to advance in the knowledge of the Riemann Zeta function

**Category:** Number Theory

[1970] **viXra:1904.0376 [pdf]**
*submitted on 2019-04-20 00:47:11*

**Authors:** Surajit Ghosh

**Comments:** 26 Pages.

Starting with proof of Riemann hypothesis, zeta values are renormalised to remove the poles of zeta function and get relationships between numbers and prime. Imaginary number i has been deﬁned such a way that it eases the complex logarithm and accounts for the scale diﬀerence between very big and very small. Other unsolved prime conjectures are also proved with the help of newly gathered information.

**Category:** Number Theory

[1969] **viXra:1904.0235 [pdf]**
*submitted on 2019-04-12 17:45:46*

**Authors:** Arthur Shevenyonov

**Comments:** 8 Pages. Trilinear, IIIVNII

A set of distinct and elementary approaches, all embarking on the Euler-Riemann equivalence representing the zeta at zero, invariably point to a consistent solution structure. The Riemann Hypothesis as regards Re=1/2 gains full support as a core solution, albeit one amounting to a special nontrivial case warranting extensions and qualifications.

**Category:** Number Theory

[1968] **viXra:1904.0227 [pdf]**
*submitted on 2019-04-11 07:40:26*

**Authors:** Elizabeth Gatton-Robey

**Comments:** 6 Pages.

I created an algorithm capable of proving Goldbach's Conjecture. This is not a claim to have proven the conjecture. The algorithm and all work contained in this document is original, so no outside sources have been used. This paper explains the algorithm then applies the algorithm with examples. The final section of the paper contains a series of proof-like reasoning to accompany my thoughts on why I believe Goldbach's Conjecture can be proven with the use of my algorithm.

**Category:** Number Theory

[1967] **viXra:1904.0219 [pdf]**
*submitted on 2019-04-11 18:49:36*

**Authors:** Yuly Shipilevsky

**Comments:** 2 Pages.

We consider a new conjecture regarding powers of integer numbers and
more specifically, we are interesting in existence and finding pairs of integers:
n ≥ 2 and m ≥ 2, such that n^m = m^n.

**Category:** Number Theory

[1966] **viXra:1904.0214 [pdf]**
*submitted on 2019-04-12 03:21:35*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. Rigorous Proof for Polignac's and Twin prime conjectures dated April 12, 2019

Prime numbers are Incompletely Predictable numbers calculated using complex algorithm Sieve of Eratosthenes. Involving proposals that prime gaps and associated sets of prime numbers are infinite in magnitude, Twin prime conjecture deals with even prime gap 2 and is a subset of Polignac's conjecture which deals with all even prime gaps 2, 4, 6, 8, 10,.... Treated as Incompletely Predictable problems, we solve these conjectures as Plus Gap 2 Composite Number Continuous Law and Plus-Minus Gap 2 Composite Number Alternating Law obtained using novel research method Information-Complexity conservation.

**Category:** Number Theory

[1965] **viXra:1904.0146 [pdf]**
*submitted on 2019-04-07 14:40:11*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we consider the $abc$ conjecture in the case $c=a+1$. Firstly, we give the proof of the first conjecture that $c

**Category:** Number Theory

[1964] **viXra:1904.0105 [pdf]**
*submitted on 2019-04-06 00:57:16*

**Authors:** Idriss Olivier Bado

**Comments:** 7 Pages.

In this paper we give a proof for Beal's conjecture . Since the discovery of the proof of Fermat's last theorem by Andre Wiles, several questions arise on the correctness of Beal's conjecture. By using a very rigorous method we come to the proof. Let $ \mathbb{G}=\{(x,y,z)\in \mathbb{N}^{3}: \min(x,y,z)\geq 3\}$
$\Omega_{n}=\{ p\in \mathbb{P}: p\mid n , p \nmid z^{y}-y^{z}\}$ ,
$$\mathbb{T}=\{(x,y,z)\in \mathbb{N}^{3}: x\geq 3,y\geq 3,z\geq 3\}$$
$\forall(x,y,z) \in \mathbb{T}$ consider the function $f_{x,y,z}$ be the function defined as :
$$\begin{array}{ccccc}
f_{x,y,z} & : \mathbb{N}^{3}& &\to & \mathbb{Z}\\
& & (X,Y,Z) & \mapsto & X^{x}+Y^{y}-Z^{z}\\
\end{array}$$
Denote by $$\mathbb{E}^{x,y,z}=\{(X,Y,Z)\in \mathbb{N}^{3}:f_{x,y,z}(X,Y,Z)=0\}$$
and $\mathbb{U}=\{(X,Y,Z)\in \mathbb{N}^{3}: \gcd(X,Y)\geq2,\gcd(X,Z)\geq2,\gcd(Y,Z)\geq2\}$
Let $ x=\min(x,y,z)$ . The obtained result show that :if $ A^{x}+B^{y}=C^{z}$ has a solution and $ \Omega_{A}\not=\emptyset$, $\forall p \in \Omega_{A}$ ,
$$ Q(B,C)=\sum_{j=1}^{x-1}[\binom{y}{j}B^{j}-\binom{z}{j}C^{j}]$$ has no solution in $(\frac{\mathbb{Z}}{p^{x}\mathbb{Z}})^{2}\setminus\{(\overline{0},\overline{0})\} $ Using this result we show that Beal's conjecture is true since $$ \bigcup_{(x,y,z)\in\mathbb{T}}\mathbb{E}^{x,y,z}\cap \mathbb{U}\not=\emptyset$$ Then $\exists (\alpha,\beta,\gamma)\in \mathbb{N}^{3}$ such that $\min(\alpha,\beta,\gamma)\leq 2$ and $\mathbb{E}^{\alpha,\beta,\gamma}\cap \mathbb{U}=\emptyset$
The novel techniques use for the proof can be use to solve the variety of Diophantine equations . We provide also the solution to Beal's equation . Our proof can provide an algorithm to generate solution to Beal's equation

**Category:** Number Theory

[1963] **viXra:1904.0070 [pdf]**
*submitted on 2019-04-03 09:55:42*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

The Polignac prime conjecture, was made by Alphonse de Polignac in 1849. Alphonse de Polignac (1826 – 1863) was a French mathematician whose father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown in1830. Polignac attended the École Polytechnique (commonly known as Polytechnique) a French public institution of higher education and research, located in Palaiseau near Paris. In 1849, the year Alphonse de Polignac was admitted to Polytechnique, he made what's known as Polignac's conjecture:
For every positive integer k, there are infinitely many prime gaps of size 2k.
Alphonse de Polignac made other significant contributions to number theory, including the de Polignac's formula, which gives the prime factorization of n!, the factorial of n, where n ≥ 1 is a positive integer.
This paper presents a complete and exhaustive proof of the Polignac Prime Conjecture. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers.

**Category:** Number Theory

[1962] **viXra:1904.0035 [pdf]**
*submitted on 2019-04-02 14:51:02*

**Authors:** Stephen Marshall

**Comments:** 10 Pages.

Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
If n is a composite number then so is 2n − 1. More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.
Mersenne primes Mp are also noteworthy due to their connection to perfect numbers.
A new Mersenne prime was found in December 2017. As of January 2018, 50 Mersenne primes are now known. The largest known prime number 277,232,917 − 1 is a Mersenne prime. Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. Ever since M521 was proven prime in 1952, the largest known prime has always been Mersenne primes, which shows that Mersenne primes become large quickly. Since the prime numbers are infinite, and since all large primes discovered since 1952 have been Mersenne primes, this seems to be evidence indicating the infinitude of Mersenne primes since there has to continually be an infinite number of large primes, even if we don’t find them. Additional evidence, is that since prime numbers are infinite, there exist an infinite number of Mersenne numbers of form 2p – 1, meaning there exist an infinite number of Mersenne numbers that are candidates for Mersenne primes. However, as with 211 – 1, we know not all Mersenne numbers of form 2p – 1 are primes. All of this evidence makes it reasonable to conjecture that there exist an infinite number of Mersenne primes. First we will provide additional evidence indicating an infinite number of Mersenne primes. Then we will provide the proof.

**Category:** Number Theory

[1961] **viXra:1904.0034 [pdf]**
*submitted on 2019-04-02 15:00:11*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

Fermat prime is a prime number that are a special case, given by the binomial number of the form:
Fn = 22n + 1, for n ≥ 0
They are named after Pierre de Fermat, a Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory.
The only known Fermat primes are:
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65,537
It has been conjectured that there are only a finite number of Fermat primes, however, we will use the same technique the author used to prove that the Mersenne primes are infinite, to prove the Fermat primes are infinite.

**Category:** Number Theory

[1960] **viXra:1904.0033 [pdf]**
*submitted on 2019-04-02 15:07:04*

**Authors:** Stephen Marshall

**Comments:** 9 Pages.

The Wagstaff prime is a prime number q of the form:
q = (2^p- 1)/3
where, p is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr. Wagstaff primes appear in the New Mersenne conjecture and have applications in cryptography.
The New Mersenne conjecture (Bateman et al. 1989) states that for any odd natural number p, if any two of the following conditions hold, then so does the third:
1. p = 2k ± 1 or p = 4k ± 3 for some natural number k.
2. 2p − 1 is prime (a Mersenne prime).
3. (2p + 1) / 3 is prime (a Wagstaff prime).
There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving which is very time consuming.
A Wagstaff prime can also be interpreted as a repunit prime of base , as
if p is odd, as it must be for the above number to be prime.
The first three Wagstaff primes are 3, 11, and 43 because
The first few Wagstaff primes are:
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, … (sequence A000979 in the OEIS)
As of October 2014, known exponents which produce Wagstaff primes or probable primes are:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, (all known Wagstaff primes)
95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531 (Wagstaff probable primes) (sequence A000978 in the OEIS)
In February 2010, Tony Reix discovered the Wagstaff probable prime:
which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.
In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:
and,
Each is a probable prime with slightly more than 4 million decimal digits. It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.
Note that when p is a Wagstaff prime, need not to be prime, the first counterexample is p = 683, and it is conjectured that if p is a Wagstaff prime and p>43, then is composite.

**Category:** Number Theory

[1959] **viXra:1904.0032 [pdf]**
*submitted on 2019-04-02 15:13:48*

**Authors:** Stephen Marshall

**Comments:** 7 Pages.

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:
1.Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2.Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3.Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
4.Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1?
We will solve Landau’s fourth problem by proving there are infinitely many primes of the form n2 + 1.

**Category:** Number Theory

[1958] **viXra:1904.0025 [pdf]**
*submitted on 2019-04-03 05:22:19*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

En ce qui concerne la conjecture forte, chaque nombre pair n, à partir de 4 peut générer plusieurs couples dont les éléments a et b < n et que parmi ces couples, qui déjà répondent à la conjecture par la sommation (n=a+b).Le nombre ou le cardinal des couples premiers sera estimé par le théorème fondamentale des nombres premiers , en démontrant que ce cardinal > 0 c'est-à-dire ∀ N pair > 3, ∃ un couplet Goldbach premier (p, p’) généré par N / N= p + p’
En établissant l’inéquation de Goldbach qui exprime autrement la conjecture
dédié à Mostafa , mon petit frère décédé d'une mort subite (R.A).

**Category:** Number Theory

[1957] **viXra:1903.0553 [pdf]**
*submitted on 2019-03-30 08:25:45*

**Authors:** Daoudi Rédoane

**Comments:** 1 Page.

Here I present one formula that produces prime numbers. There are counterexamples for this formula.

**Category:** Number Theory

[1956] **viXra:1903.0548 [pdf]**
*submitted on 2019-03-30 12:21:44*

**Authors:** Ilija Barukčić

**Comments:** 6 pages. Copyright © 2019 by Ilija Barukčić, Jever, Germany. All rights reserved. Published:

Abstract
Objectives:
The scientific knowledge appears to grow by time. However, every scientific progress involves different kind of mistakes, which may survive for a long time. Nevertheless, the abandonment of partially true or falsified theorems, theories et cetera, for positions which approach more closely to the truth, is necessary. In a critical sense, a reduction of the myth in science demands the non-ending detection of contradictions in science and the elimination the same too.
Methods:
Nullity as one aspect of the trans-real arithmetic and equally as one of today’s approaches to the solution of the problem of the division of zero by zero is re-analyzed. A systematic mathematical proof is provided to prove the logical consistency of Nullity.
Results:
There is convincing evidence that Nullity is logically inconsistent. Furthermore, the about 2000 year old rule of the addition of zero’s (0+0+…+0 = 0) is proved as logically inconsistent and refuted.
Conclusion: Nullity is self-contradictory and refuted.
Keywords: Indeterminate forms, Classical logic, Zero divided by zero

**Category:** Number Theory

[1955] **viXra:1903.0546 [pdf]**
*submitted on 2019-03-30 14:17:21*

**Authors:** Dick Hudson

**Comments:** 5 Pages.

Originated by Lothar Collatz in 1937 [1], the conjecture states: given the recursive function, y=3x+1 if x is odd, or y=x/2 if x is even, for any positive integer x, y will equal 1 after a finite number of steps. This analysis examines number form and uses a tree type graph to prove the process.

**Category:** Number Theory

[1954] **viXra:1903.0543 [pdf]**
*submitted on 2019-03-31 01:17:07*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 2 Pages.

I give definition of Beautiful Natural Numbers (BNNs) and relate it to the theorem I claimed earlier on distinct proper fractions that sum to 1.

**Category:** Number Theory

[1953] **viXra:1903.0503 [pdf]**
*submitted on 2019-03-27 07:27:20*

**Authors:** Timothy W. Jones

**Comments:** 8 Pages.

In this article we revisit Sondow geometric proof of the irrationality of e. This is done by using circles with rational sector areas. Attempting to extend the idea to the series for zeta(n), challenges are met.

**Category:** Number Theory

[1952] **viXra:1903.0483 [pdf]**
*submitted on 2019-03-28 02:01:19*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. Rigorous Proof for Riemann hypothesis dated Thursday 28 March 2019

Riemann hypothesis proposed all nontrivial zeros to be located on critical line of Riemann zeta function. Treated as Incompletely Predictable problem, we obtain the novel Dirichlet Sigma-Power Law as final proof of solving this problem. This Law is derived as equation and inequation from original Dirichlet eta function (proxy function for Riemann zeta function). Performing a parallel procedure help explain closely related Gram points.

**Category:** Number Theory

[1951] **viXra:1903.0464 [pdf]**
*submitted on 2019-03-27 01:45:05*

**Authors:** Ilija Barukčić

**Comments:** Pages.

Abstract
Objectives:
The problem of the division of zero by zero appears to be as old as science itself, and may be older. Nonetheless, the solution of this to long lasting and not ending issue in mathematics and physics is coming nearer. In point of fact, an end of discussions on the issue of the division of zero by zero is not in sight as long as the solutions of this problem proposed or published are grounded on logical contradictions. Roughly, any contradiction in a formal axiomatic system become disastrous because any theorem can be proven as true (Principle of explosion).
Methods:
A systematic mathematical proof is provided to re-analyze the logical foundations of Saitho's approach to the problem of the division of zero by zero. A direct proof (Inversion) was used to show the truth or falsehood of Saitho's published statement with respect to the division of zero by zero.
Results:
Noncontradiction implies that it cannot be both true, +1=+1 and +1=+0. There is convincing evidence that the Saitho's solution of the problem of zero divided by zero is logically inconsistent.
Conclusion: Saitho’s equality (1/0)=(0/0) is self-contradictory and refuted.
Keywords: Indeterminate forms, Classical logic, Zero divided by zero

**Category:** Number Theory

[1950] **viXra:1903.0439 [pdf]**
*submitted on 2019-03-24 07:13:24*

**Authors:** Yuly Shipilevsky

**Comments:** 7 Pages.

We introduce a special class of complex numbers, wherein their
absolute values and arguments given in a polar coordinate system are integers
and we introduce the corresponding class of the Optimization Problems:
"Polar Complex Integer Optimization

**Category:** Number Theory

[1949] **viXra:1903.0390 [pdf]**
*submitted on 2019-03-21 22:53:24*

**Authors:** Soerivhe Iriene, J. Oquibo Ihwaiuwaue

**Comments:** 6 Pages.

The paper "Proof of the Polignac Prime Conjecture and other Conjectures", (although listed under the title "Elementary Proof of the Goldbach Conjecture") first published in 2017 claimed to have proven Polignac's conjecture, and in doing so also the twin prime conjecture. The said paper had several problems, not least of which was a catastrophic basic error that completely invalidated the proof. Polignac's conjecture remains unproven, as does the twin primes conjecture.

**Category:** Number Theory

[1948] **viXra:1903.0387 [pdf]**
*submitted on 2019-03-22 04:31:14*

**Authors:** Juan Moreno Borrallo

**Comments:** 6 Pages. Spanish language

En este breve artículo se propone y demuestra una curiosa identidad de la función zeta, equivalente a la suma de las progresiones geométricas de los recíprocos de todos los enteros positivos que no son potencias, con numeradores cuyo valor es la función divisor del exponente de cada término de la progresión.

**Category:** Number Theory

[1947] **viXra:1903.0353 [pdf]**
*submitted on 2019-03-19 14:19:09*

**Authors:** Sally Myers Moite

**Comments:** 9 Pages.

Numbers of form 6N – 1 and 6N + 1 factor into numbers of the same form. This observation provides elimination sieves for numbers N that lead to primes and prime pairs. The sieves do not explicitly reference primes.

**Category:** Number Theory

[1946] **viXra:1903.0333 [pdf]**
*submitted on 2019-03-18 18:07:39*

**Authors:** Toshiro Takami

**Comments:** 180 Pages.

I also found a zero point which seems to deviate from 0.5.
I thought that the zero point outside 0.5 can not be found very easily in the area
which can not be shown in the figure, but this area can not be represented in the
figure but can be found one after another.
It is completely unknown whether this axis is distorted in the 0.5 axis or just by
coincidence.
The number of zero points in the area that can not be shown in the figure is now 43.
No matter how you looked it was not found in other areas.
It seemed that there is no other way to interpret this axis as 0.5 axis is distorted in
this area.
Somewhere on the net there is a memory that reads the mathematician's view that
"there are countless zero points in the vicinity of 0.5 on high area".
We are reporting that the zero point search of the high-value area of the imaginary
part which was giving up as it is no longer possible with the supercomputer is no
longer possible, is reported.
43 zero-point searches in the high-value area of the imaginary part are thus
successful.
This means that the zero point search in the high-value area of the imaginary part
has succeeded in the 43.
We will also write 43 zero point searches of the successful high-value area of the
imaginary part.
There are many counterexamples far beyond 0.5, which is far beyond the limit, but
the computer can not calculate it.
Moreover, I believe that it can only be confirmed on supercomputer whether this is
really counterexample. In addition, it is necessary to make corrections in the
supercomputer.

**Category:** Number Theory

[1945] **viXra:1903.0296 [pdf]**
*submitted on 2019-03-15 19:04:45*

**Authors:** Masashi Furuta

**Comments:** 21 Pages.

We define the "Div sequence" that sets up the number of times divided by 2 in the Collatz operation.
Using this and the "infinite descent", we prove the Collatz conjecture.

**Category:** Number Theory

[1944] **viXra:1903.0295 [pdf]**
*submitted on 2019-03-15 22:14:20*

**Authors:** Aaron Chau

**Comments:** 2 Pages.

也因为多与少，即填得满与填不满的视觉凭证是零点空格，所以，零点空格证明黎猜不成立。

**Category:** Number Theory

[1943] **viXra:1903.0209 [pdf]**
*submitted on 2019-03-11 18:46:22*

**Authors:** Bambore Dawit Geinamo

**Comments:** 2 Pages. For more improvement comments and corrections are expected

This paper magically shows very interesting and simple proof of Fermata Last Theorem. The proof describes sufficient conditions of that the equation holds and contradictions on them to satisfy the theorem. If Fermat had proof most probably his proof may be similar with this one.

**Category:** Number Theory

[1942] **viXra:1903.0200 [pdf]**
*submitted on 2019-03-12 06:40:54*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Which Cube is sum of six cubes?

**Category:** Number Theory

[1941] **viXra:1903.0167 [pdf]**
*submitted on 2019-03-09 10:51:21*

**Authors:** Zeolla Gabriel Martín

**Comments:** 11 Pages. Idioma Español

Este documento desarrolla y demuestra el descubrimiento de un nuevo algoritmo de multiplicación que funciona absolutamente con todos los números.

**Category:** Number Theory

[1940] **viXra:1903.0157 [pdf]**
*submitted on 2019-03-10 00:49:01*

**Authors:** Toshiro Takami

**Comments:** 20 Pages.

I also found a zero point which seems to deviate from 0.5.
I thought that the zero point outside 0.5 can not be found very easily in the area which can not be shown in the figure, but this area can not be represented in the figure but can be found one after another.
It is completely unknown whether this axis is distorted in the 0.5 axis or just by coincidence.
The number of zero points in the area that can not be shown in the figure is now 43.
No matter how you looked it was not found in other areas.
It seemed that there is no other way to interpret this axis as 0.5 axis is distorted in this area.
Somewhere on the net there is a memory that reads the mathematician's view that "there are countless zero points in the vicinity of 0.5 on high area".
We are reporting that the zero point search of the high-value area of the imaginary part which was giving up as it is no longer possible with the supercomputer is no longer possible, is reported.
43 zero-point searches in the high-value area of the imaginary part are thus successful.
This means that the zero point search in the high-value area of the imaginary part has succeeded in the 43.
We will also write 43 zero point searches of the successful high-value area of the imaginary part.
There are many counterexamples far beyond 0.5, which is far beyond the limit, but the computer can not calculate it.
Moreover, I believe that it can only be confirmed on supercomputer whether this is really counterexample. In addition, it is necessary to make corrections in the supercomputer.

**Category:** Number Theory

[1939] **viXra:1903.0144 [pdf]**
*submitted on 2019-03-08 12:57:37*

**Authors:** Emmanuil Manousos

**Comments:** 10 Pages.

In this article, we define a pair of sequences (α, β). By using the properties of the pair (α, β), we establish a method for determining large prime numbers.

**Category:** Number Theory

[1938] **viXra:1903.0059 [pdf]**
*submitted on 2019-03-05 05:22:37*

**Authors:** Espen Gaarder Haug

**Comments:** 4 Pages.

In this paper, we point out an interesting asymmetry in the rules of fundamental mathematics between positive and negative numbers. Further, we show that there exists an alternative numerical system that is basically identical to today’s system, but where positive numbers dominate over negative numbers. This is like a mirror symmetry of the existing number system. The asymmetry in both of these systems leads to imaginary and complex numbers.
We suggest an alternative number system with perfectly symmetric rules – that is, where there is no dominance of negative numbers over positive numbers, or vice versa, and where imaginary and complex numbers are no longer needed. This number system seems to be superior to other number systems, as it brings simplicity and logic back to areas that have been dominated by complex rules for much of the history of mathematics. We also briefly discuss how the Riemann hypothesis may be linked to the asymmetry in the current number system.

**Category:** Number Theory

[1937] **viXra:1903.0031 [pdf]**
*submitted on 2019-03-02 16:28:58*

**Authors:** Ahmad Telfah

**Comments:** 10 pages

this paper carrying a method to introduce the distribution of the densities of the prime numbers and the composite numbers along in natural numbers, the method basically depends on the direct deduction of the composite numbers in a specified intervals that also with using some corrections and modifications to reach maximum and minimum values of the composites and primes densities, this allowed us to detect some special conjectures related to the primes density.

**Category:** Number Theory

[1936] **viXra:1903.0030 [pdf]**
*submitted on 2019-03-02 16:40:03*

**Authors:** Ahmad Telfah

**Comments:** 5 pages

this paper carrying a method to calculate an approximation to the number of the prime numbers in the natural numbers interval I={1,2,3,4,……,P_n,P_n+1,P_n+2,……,P_n^2 } by using the primes (2,3,5,…,P_n ) to specify the primes density in the sub intervals I(P_n ) as
I(P_n )={P_n^2 〖,P〗_n^2+1,P_n^2+2,P_n^2+3,……,P_(n+1)^2-1} has primes density of (d(P_n ))= ( ∏_(i=1)^(i=n)▒〖( 1- 1/P_i 〗 ).

**Category:** Number Theory

[1935] **viXra:1902.0406 [pdf]**
*submitted on 2019-02-25 03:49:20*

**Authors:** Dariusz Gołofit

**Comments:** 8 Pages.

If we elimate an ordered subset from the ordered set, we will receive a subset od orderly character.

**Category:** Number Theory

[1934] **viXra:1902.0405 [pdf]**
*submitted on 2019-02-25 03:54:25*

**Authors:** Dariusz Gołofit

**Comments:** 12 Pages.

If we elimate an ordered subset from the ordered set, we will receive a subset od orderly character.

**Category:** Number Theory

[1933] **viXra:1902.0395 [pdf]**
*submitted on 2019-02-23 20:22:29*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

This proof identifies the three solutions to the three ABC-conjecture formulations. Given that the ABC-conjecture’s relevance to a slew of unsolved problems, other equations will be proven by inspection. Aside from the ABC conjecture, this proof will solve for a hypothetical Moore graph of diameter 2, girth 5, degree 57 and order 3250 (degree-diameter problem); the Collatz conjecture; and the Beal conjecture. Discussion and conclusion will review a unifying solution by spectral graph theory.

**Category:** Number Theory

[1932] **viXra:1902.0390 [pdf]**
*submitted on 2019-02-24 03:18:22*

**Authors:** ZhangAik, Leet_Noob

**Comments:** 4 Pages.

The elementary proof to the twin conjecture.

**Category:** Number Theory

[1931] **viXra:1902.0235 [pdf]**
*submitted on 2019-02-13 05:23:19*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. Submitted to the The Ramanujan Journal. Comments welcome.

In this paper, we consider the abc conjecture in the case c=a+1. Firstly, we give the proof of the first conjecture that c

**Category:** Number Theory

[1930] **viXra:1902.0200 [pdf]**
*submitted on 2019-02-11 06:24:07*

**Authors:** Faisal Amin Yassein Abdelmohssin

**Comments:** 2 Pages.

I claim that the sum of following distinct proper fractions [(1/2),(1/3),(1/6)] is the only triple of distinct proper fraction that sum to 1 {i.e. [(1/2)+(1/3)+(1/6)]=1}.

**Category:** Number Theory

[1929] **viXra:1902.0147 [pdf]**
*submitted on 2019-02-08 09:11:21*

**Authors:** Kenneth A. Watanabe

**Comments:** 13 Pages.

The Near-Square Prime conjecture, states that there are an infinite number of prime numbers of the form x^2 + 1. In this paper, a function was derived that determines the number of prime numbers of the form x^2 + 1 that are less than n^2 + 1 for large values of n. Then by mathematical induction, it is proven that as the value of n goes to infinity, the function goes to infinity, thus proving the Near-Square Prime conjecture.

**Category:** Number Theory

[1928] **viXra:1902.0106 [pdf]**
*submitted on 2019-02-06 07:50:18*

**Authors:** Algirdas Anatano Maknickas

**Comments:** 2 Pages.

This remark gives analytical solution of Last Fermat's Theorem

**Category:** Number Theory

[1018] **viXra:1905.0111 [pdf]**
*replaced on 2019-05-11 18:48:26*

**Authors:** Toshiro Takami

**Comments:** 62 Pages.

new version
I believe this is proof of the Riemann hypothrsis.
I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.

**Category:** Number Theory

[1017] **viXra:1905.0111 [pdf]**
*replaced on 2019-05-11 02:29:10*

**Authors:** Toshiro Takami

**Comments:** 56 Pages.

new version
I believe this is proof of the Riemann hypothrsis.
I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.

**Category:** Number Theory

[1016] **viXra:1905.0111 [pdf]**
*replaced on 2019-05-09 23:18:25*

**Authors:** Toshiro Takami

**Comments:** 65 Pages.

**Category:** Number Theory

[1015] **viXra:1904.0561 [pdf]**
*replaced on 2019-05-14 07:19:32*

**Authors:** Kouji Takaki

**Comments:** 10 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[1014] **viXra:1904.0561 [pdf]**
*replaced on 2019-05-09 05:38:18*

**Authors:** Kouji Takaki

**Comments:** 10 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[1013] **viXra:1904.0561 [pdf]**
*replaced on 2019-05-06 01:37:04*

**Authors:** Kouji Takaki

**Comments:** 11 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[1012] **viXra:1904.0561 [pdf]**
*replaced on 2019-05-05 03:58:42*

**Authors:** Kouji Takaki

**Comments:** 10 Pages.

We have obtained the conclusion that there are no odd perfect numbers.

**Category:** Number Theory

[1011] **viXra:1904.0507 [pdf]**
*replaced on 2019-05-02 07:08:03*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages.

This remarks prove, that Riemann zeta function has infinitesimal amount of zeros.

**Category:** Number Theory

[1010] **viXra:1904.0507 [pdf]**
*replaced on 2019-04-29 01:28:22*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages.

This remarks proves, that Riemann zeta function has infinitesimal amount of zeros.

**Category:** Number Theory

[1009] **viXra:1904.0428 [pdf]**
*replaced on 2019-05-03 18:29:29*

**Authors:** Ralf Wüsthofen

**Comments:** 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, i.e. a set that is a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[1008] **viXra:1904.0428 [pdf]**
*replaced on 2019-04-23 09:33:49*

**Authors:** Ralf Wüsthofen

**Comments:** 2 Pages. Proof of the Goldbach conjecture on http://vixra.org/abs/1702.0300

Based on a strengthened form of the strong Goldbach conjecture, this paper presents an antinomy within the Peano arithmetic (PA). We derive two contradictory statements by using the same main instrument as in the proof of the conjecture, i.e. a set that is a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[1007] **viXra:1904.0423 [pdf]**
*replaced on 2019-05-07 06:48:06*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.
The formula is (1).
Although x is treated as a real number, x is a nontrivial zero values.
That is, it takes eternal number of nontrivial zeros of the positive and negative regions on the axis 0.5.

**Category:** Number Theory

[1006] **viXra:1904.0423 [pdf]**
*replaced on 2019-05-05 21:21:08*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.
The formula is (1).
Although x is treated as a real number, x is a nontrivial zero values.
That is, it takes eternal number of nontrivial zeros of the positive and negative regions on the axis 0.5.

**Category:** Number Theory

[1005] **viXra:1904.0423 [pdf]**
*replaced on 2019-05-04 04:02:44*

**Authors:** Toshiro Takami

**Comments:** 75 Pages.

I could give a complete proof by the number theory method to Riemann hypothesis.
I found the following number law. This proved that Riemann hypothesis is correct.
The formula is (1).
Although x is treated as a real number, x is a nontrivial zero values.
That is, it takes eternal number of nontrivial zeros of the positive and negative regions on the axis 0.5.

**Category:** Number Theory

[1004] **viXra:1904.0423 [pdf]**
*replaced on 2019-05-02 22:33:21*

**Authors:** Toshiro Takami

**Comments:** 88 Pages.

**Category:** Number Theory

[1003] **viXra:1904.0423 [pdf]**
*replaced on 2019-05-01 07:20:42*

**Authors:** Toshiro Takami

**Comments:** 85 Pages.

**Category:** Number Theory

[1002] **viXra:1904.0422 [pdf]**
*replaced on 2019-05-02 04:00:00*

**Authors:** Hajime Mashima

**Comments:** 2 Pages.

The three sides of the right triangle are rational numbers, and those
with natural numbers are congruent numbers.

**Category:** Number Theory

[1001] **viXra:1904.0422 [pdf]**
*replaced on 2019-04-26 07:50:06*

**Authors:** Hajime Mashima

**Comments:** 2 Pages.

The three sides of the right triangle are rational numbers, and those
with natural numbers are congruent numbers.

**Category:** Number Theory

[1000] **viXra:1904.0422 [pdf]**
*replaced on 2019-04-23 08:41:55*

**Authors:** Hajime Mashima

**Comments:** 1 Page.

**Category:** Number Theory

[999] **viXra:1904.0376 [pdf]**
*replaced on 2019-05-14 02:42:38*

**Authors:** Surajit Ghosh

**Comments:** 32 Pages.

Based on Euler ’s formula a concept of duality unit or dunit circle is discovered. Continuing with Riemann hypothesis is proved from diﬀerent angles, zeta values are renormalised to remove the poles of zeta function and discover relationships between numbers and primes. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be deﬁned such a way that it eases the complex logarithm and accounts for the scale diﬀerence between very big and very small. Pi can also be a base to natural logarithm and complement the scale gap. 96 complex constants derived from complex logarithm can explain everything in the universe.

**Category:** Number Theory

[998] **viXra:1904.0376 [pdf]**
*replaced on 2019-05-10 08:35:55*

**Authors:** Surajit Ghosh

**Comments:** 31 Pages.

Based on Euler ’s formula a concept of duality unit or dunit circle is discovered. Continuing with Riemann hypothesis is proved from diﬀerent angles, zeta values are renormalised to remove the poles of zeta function and discover relationships between numbers and primes. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory. Imaginary number i can be deﬁned such a way that it eases the complex logarithm and accounts for the scale diﬀerence between very big and very small. Pi can also be a base to natural logarithm and complement the scale gap. 96 complex constants derived from complex logarithm can explain everything in the universe.

**Category:** Number Theory

[997] **viXra:1904.0376 [pdf]**
*replaced on 2019-04-30 08:16:21*

**Authors:** Surajit Ghosh

**Comments:** 31 Pages.

Based on Euler ’s formula a concept of duality unit or dunit circle is discovered. Continuing with Riemann hypothesis is proved from diﬀerent angles, zeta values are renormalised to remove the poles of zeta function and discover relationships between numbers and primes. Imaginary number i can be deﬁned such a way that it eases the complex logarithm and accounts for the scale diﬀerence between very big and very small. Pi can also be a base to natural logarithm and complement the scale gap. Other unsolved prime conjectures are also proved with the help of theorems of numbers and number theory.

**Category:** Number Theory

[996] **viXra:1904.0214 [pdf]**
*replaced on 2019-05-13 19:48:03*

**Authors:** John Yuk Ching Ting

**Comments:** 18 Pages. Rigorous proofs for Polignac's and Twin prime conjectures.

Prime numbers are Incompletely Predictable numbers calculated using complex algorithm Sieve of Eratosthenes. Involving proposals that prime gaps and associated sets of prime numbers are infinite in magnitude, Twin prime conjecture deals with even prime gap 2 and is a subset of Polignac's conjecture which deals with all even prime gaps 2, 4, 6, 8, 10,.... Treated as Incompletely Predictable problems, we solve these conjectures with research method Information-Complexity conservation to get Plus Gap 2 Composite Number Continuous Law and Plus-Minus Gap 2 Composite Number Alternating Law.

**Category:** Number Theory

[995] **viXra:1904.0025 [pdf]**
*replaced on 2019-05-09 08:03:40*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

en ce qui concerne la conjecture forte, chaque nombre pair n, à partir de 4 peut générer plusieurs couples dont les éléments a et b < n et que parmi ces couples, qui déjà répondent à la conjecture par la sommation (n=a+b).Le nombre ou le cardinal des couples premiers sera estimé par le théorème fondamentale des nombres premiers , en démontrant que ce cardinal > 0 c'est-à-dire ∀ N pair > 3, ∃ un couplet Goldbach premier (p, p’) généré par N / N= p + p’ En établissant l’inéquation de Goldbach qui exprime autrement la conjecture dédié à Mostafa , mon petit frère décédé d'une mort subite (R.A).

**Category:** Number Theory

[994] **viXra:1904.0025 [pdf]**
*replaced on 2019-04-05 06:14:33*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

en ce qui concerne la conjecture forte, chaque nombre pair n, à partir de 4 peut générer plusieurs couples dont les éléments a et b < n et que parmi ces couples, qui déjà répondent à la conjecture par la sommation (n=a+b).Le nombre ou le cardinal des couples premiers sera estimé par le théorème fondamentale des nombres premiers , en démontrant que ce cardinal > 0 c'est-à-dire ∀ N pair > 3, ∃ un couplet Goldbach premier (p, p’) généré par N / N= p + p’
En établissant l’inéquation de Goldbach qui exprime autrement la conjecture dédié à Mostafa , mon petit frère décédé d'une mort subite .

**Category:** Number Theory

[993] **viXra:1903.0548 [pdf]**
*replaced on 2019-04-15 08:36:19*

**Authors:** Ilija Barukčić

**Comments:** 18 Pages.

Abstract
Objectives:
The scientific knowledge appears to grow by time. However, every scientific progress involves different kind of mistakes, which may survive for a long time. Nevertheless, the abandonment of partially true or falsified theorems, theories et cetera, for positions which approach more closely to the truth, is necessary. In a critical sense, a reduction of the myth in science demands the non-ending detection of contradictions in science and the elimination the same too.
Methods:
Nullity as one aspect of the trans-real arithmetic and equally as one of today’s approaches to the solution of the problem of the division of zero by zero is re-analyzed. A systematic mathematical proof is provided to prove the logical consistency of Nullity.
Results:
There is convincing evidence that Nullity is logically inconsistent. Furthermore, the about 2000 year old rule of the addition of zero’s (0+0+…+0 = 0) is proved as logically inconsistent and refuted.
Conclusion: Nullity is self-contradictory and refuted.
Keywords: Indeterminate forms, Classical logic, Zero divided by zero

**Category:** Number Theory

[992] **viXra:1903.0548 [pdf]**
*replaced on 2019-04-01 14:56:16*

**Authors:** Ilija Barukčić

**Comments:** 10 Pages.

Abstract
Objectives:
The scientific knowledge appears to grow by time. However, every scientific progress involves different kind of mistakes, which may survive for a long time. Nevertheless, the abandonment of partially true or falsified theorems, theories et cetera, for positions which approach more closely to the truth, is necessary. In a critical sense, a reduction of the myth in science demands the non-ending detection of contradictions in science and the elimination the same too.
Methods:
Nullity as one aspect of the trans-real arithmetic and equally as one of today’s approaches to the solution of the problem of the division of zero by zero is re-analyzed. A systematic mathematical proof is provided to prove the logical consistency of Nullity.
Results:
There is convincing evidence that Nullity is logically inconsistent. Furthermore, the about 2000 year old rule of the addition of zero’s (0+0+…+0 = 0) is proved as logically inconsistent and refuted.
Conclusion: Nullity is self-contradictory and refuted.
Keywords: Indeterminate forms, Classical logic, Zero divided by zero

**Category:** Number Theory

[991] **viXra:1903.0503 [pdf]**
*replaced on 2019-04-17 03:51:24*

**Authors:** Timothy W. Jones

**Comments:** 16 Pages. A new section that shows with greater clarity the extension of Sondow has been added.

We modify Sondow's geometric proof of the irrationality of e. The modification uses sector areas on circles, rather than closed intervals. Using this circular version of Sondow's proof, we see a way to understand the irrationality of a series. We evolve the idea of proving all possible rational value convergence points of a series are excluded because all partials are not expressible as fractions with the denominators of their terms. If such fractions cover the rationals, then the series should be irrational. Both the irrationality of e and that of zeta(n>=2) are proven using these criteria: the terms cover the rationals and the partials escape the terms.

**Category:** Number Theory

[990] **viXra:1903.0503 [pdf]**
*replaced on 2019-04-15 05:15:24*

**Authors:** Timothy W. Jones

**Comments:** 15 Pages. Substantially reorganized with more examples and theory.

We modify Sondow's geometric proof of the irrationality of e. The modification uses sector areas on circles, rather than closed intervals. Using this circular version of Sondow's proof, we see a way to understand the irrationality of a series. We evolve the idea of proving all possible rational value convergence points of a series are excluded because all partials are not expressible as fractions with the denominators of their terms. If such fractions cover the rationals, then the series should be irrational. Both the irrationality of e and that of zeta(n>=2) are proven using these criteria: the terms cover the rationals and the partials escape the terms.

**Category:** Number Theory

[989] **viXra:1903.0503 [pdf]**
*replaced on 2019-04-05 09:59:59*

**Authors:** Timothy W. Jones

**Comments:** 11 Pages.

In this article we revisit Sondow geometric proof of the irrationality of e. This is done by using circles with rational sector areas. Attempting to extend the idea to the series for zeta(n), challenges are met.

**Category:** Number Theory

[988] **viXra:1903.0483 [pdf]**
*replaced on 2019-05-13 19:43:22*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. Rigorous proof for Riemann hypothesis and explaining Gram points.

Riemann hypothesis proposed all nontrivial zeros to be located on critical line of Riemann zeta function. Treated as Incompletely Predictable problem, we obtain Dirichlet Sigma-Power Law as final proof of solving this problem. This Law is derived as equation and inequation from original Dirichlet eta function (proxy function for Riemann zeta function). Performing a parallel procedure help explain closely related Gram points.

**Category:** Number Theory

[987] **viXra:1903.0483 [pdf]**
*replaced on 2019-04-12 03:15:49*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. Rigorous proof of Riemann hypothesis and explanation of Gram points.

Riemann hypothesis proposed all nontrivial zeros to be located on critical line of Riemann zeta function. Treated as Incompletely Predictable problem, we obtain the novel Dirichlet Sigma-Power Law as final proof of solving this problem. This Law is derived as equation and inequation from original Dirichlet eta function (proxy function for Riemann zeta function). Performing a parallel procedure help explain closely related Gram points.

**Category:** Number Theory

[986] **viXra:1903.0483 [pdf]**
*replaced on 2019-04-07 15:46:17*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. Rigorous proof of Riemann hypothesis and explanation of Gram points.

Riemann hypothesis proposed all nontrivial zeros to be located on critical line of Riemann zeta function. Treated as Incompletely Predictable problem, we obtain the novel Dirichlet Sigma-Power Law as final proof of solving this problem. This Law is derived as equation and inequation from original Dirichlet eta function (proxy function for Riemann zeta function). Performing a parallel procedure help explain closely related Gram points.

**Category:** Number Theory

[985] **viXra:1903.0483 [pdf]**
*replaced on 2019-03-29 21:36:31*

**Authors:** John Yuk Ching Ting

**Comments:** 20 Pages. Rigorous proof of Riemann hypothesis and explanation of Gram points.

**Category:** Number Theory

[984] **viXra:1903.0464 [pdf]**
*replaced on 2019-03-28 08:32:43*

**Authors:** Ilija Barukčić

**Comments:** 5 Pages.

Saitho’s equality (1/0)=(0/0) is self-contradictory and refuted.

**Category:** Number Theory

[983] **viXra:1903.0387 [pdf]**
*replaced on 2019-03-22 08:40:24*

**Authors:** Juan Moreno Borrallo

**Comments:** 7 Pages. Spanish Language

At this brief paper, it is proposed and demonstrated a curious identity of Zeta Function, equivalent to the sum of the geometric progression of reciprocals of all the positive integers which are not perfect powers, having as numerators the number of divisors of the exponent of each term of the progression.

**Category:** Number Theory

[982] **viXra:1903.0353 [pdf]**
*replaced on 2019-04-18 21:47:07*

**Authors:** Sally Myers Moite

**Comments:** 9 Pages.

Numbers of form 6N – 1 and 6N + 1 factor into numbers of the same form. This observation provides elimination sieves for numbers N that lead to primes and prime pairs. The sieves do not explicitly reference primes.

**Category:** Number Theory

[981] **viXra:1903.0333 [pdf]**
*replaced on 2019-04-17 17:28:23*

**Authors:** Toshiro Takami

**Comments:** 175 Pages.

I also found a zero point which seems to deviate from 0.5.
I thought that the zero point outside 0.5 can not be found very easily in the area which can not be shown in the figure, but this area can not be represented in the figure but can be found one after another.
It is completely unknown whether this axis is distorted in the 0.5 axis or just by coincidence.
The number of zero points in the area that can not be shown in the figure is now 43.
No matter how you looked it was not found in other areas.
It seemed that there is no other way to interpret this axis as 0.5 axis is distorted in this area.
Somewhere on the net there is a memory that reads the mathematician's view that
"there are countless zero points in the vicinity of 0.5 on high area".
We are reporting that the zero point search of the high-value area of the imaginary part which was giving up as it is no longer possible with the supercomputer is no longer possible, is reported.
43 zero-point searches in the high-value area of the imaginary part are thus successful.
This means that the zero point search in the high-value area of the imaginary part has succeeded in the 43.
We will also write 43 zero point searches of the successful high-value area of the imaginary part.
There are many counterexamples far beyond 0.5, which is far beyond the limit, but the computer can not calculate it.
Moreover, I believe that it can only be confirmed on supercomputer whether this is really counterexample.
In addition, it is necessary to make corrections in the supercomputer.

**Category:** Number Theory

[980] **viXra:1903.0200 [pdf]**
*replaced on 2019-03-13 09:22:44*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Two cubes are a sum of nine cubes

**Category:** Number Theory

[979] **viXra:1903.0200 [pdf]**
*replaced on 2019-03-13 04:09:18*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

Which Cube is sum of six cubes?

**Category:** Number Theory

[978] **viXra:1903.0157 [pdf]**
*replaced on 2019-03-11 16:26:03*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

**Category:** Number Theory

[977] **viXra:1902.0390 [pdf]**
*replaced on 2019-03-16 21:21:15*

**Authors:** ZhangAik, Leet_Noob

**Comments:** 2 Pages.

The elementary proof to the twin conjecture.

**Category:** Number Theory

[976] **viXra:1902.0390 [pdf]**
*replaced on 2019-03-15 08:50:15*

**Authors:** ZhangAik, Leet_Noob

**Comments:** 2 Pages.

The elementary proof to the twin conjecture.

**Category:** Number Theory

[975] **viXra:1902.0390 [pdf]**
*replaced on 2019-03-09 15:54:30*

**Authors:** ZhangAik, Leet_Noob

**Comments:** 3 Pages.

The elementary proof to the twin conjecture.

**Category:** Number Theory

[974] **viXra:1902.0390 [pdf]**
*replaced on 2019-03-08 17:57:51*

**Authors:** ZhangAik, Leet_Noob

**Comments:** 3 Pages.

The elementary proof to the twin conjecture.

**Category:** Number Theory

[973] **viXra:1902.0147 [pdf]**
*replaced on 2019-05-03 15:42:36*

**Authors:** Kenneth A. Watanabe

**Comments:** 9 Pages.

The Near-Square Prime conjecture, states that there are an infinite number of prime numbers of the form x^2 + 1. In this paper, a function was derived that determines the number of prime numbers of the form x^2 + 1 that are less than n^2 + 1 for large values of n. Then by mathematical induction, it is proven that as the value of n goes to infinity, the function goes to infinity, thus proving the Near-Square Prime conjecture.

**Category:** Number Theory

[972] **viXra:1902.0147 [pdf]**
*replaced on 2019-04-29 08:58:59*

**Authors:** Kenneth A. Watanabe

**Comments:** 9 Pages.

The Near-Square Prime conjecture, states that there are an infinite number of prime numbers of the form x^2 + 1. In this paper, a function was derived that determines the number of prime numbers of the form x^2 + 1 that are less than n^2 + 1 for large values of n. Then by mathematical induction, it is proven that as the value of n goes to infinity, the function goes to infinity, thus proving the Near-Square Prime conjecture.

**Category:** Number Theory

[971] **viXra:1902.0106 [pdf]**
*replaced on 2019-02-10 10:53:42*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages.

This remark gives analytical solution of Last Fermat's Theorem

**Category:** Number Theory

[970] **viXra:1902.0106 [pdf]**
*replaced on 2019-02-09 02:11:25*

**Authors:** Algirdas Antano Maknickas

**Comments:** 2 Pages. In previous version was mistake in middle name.

This remark gives analytical solution of Last Fermat's Theorem

**Category:** Number Theory