**Previous months:**

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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)

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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(33)

2019 - 1901(12) - 1902(11) - 1903(22) - 1904(15)

Any replacements are listed farther down

[1973] **viXra:1904.0334 [pdf]**
*submitted on 2019-04-18 03:49:29*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

Although it carried out in real number 1, There is a divergence between Plot and Parametric plot.

**Category:** Number Theory

[1972] **viXra:1904.0276 [pdf]**
*submitted on 2019-04-16 05:34:06*

**Authors:** Toshiro Takami

**Comments:** 17 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 non-trivial zero values.
That is, it takes eternal number of non-trivial zeros of the positive and negative regions on the axis 0.5.
(3)^2= 2^(-2c)*n^(-2c) = n^(1 - 2c)/(4^c*(1 - 2c))
when c≠0.5
(3)^2= 2^(-2c)*n^(-2c) ≠ n^(1 - 2c)/(4^c*(1 - 2c)), and x of (3) can not take non-trivial zero.
when c=0.5
2^(-2c)*n^(-2c) = n^(1 - 2c)/(4^c*(1 - 2c))
{ n^(1 - 2c)/(4^c*(1 - 2c)) },{c=0.5} = ∞^~ = ComplexInfinity. and x of (3) can take all eternal number of non-trivial zeros.
=(3)^2
=(3)^2
When c = 0.5, x can have infinite and infinitesimal values. That is, it takes all of the infinite non-trivial zeros of the positive and negative regions on the axis 0.5.
However, when x is not 0.5, non-trivial zeros can not be taken.
The proof is completed.

**Category:** Number Theory

[1971] **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

[1970] **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

[1969] **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

[1968] **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

[1967] **viXra:1904.0179 [pdf]**
*submitted on 2019-04-09 20:52:50*

**Authors:** Toshiro Takami

**Comments:** 18 Pages.

If you enter a non-trivial zero value in x, the value of the equation is zero.
The above equation is derived from cos(π/3)=0.5 after trial and error, and as a result of many examinations, it turned out that this equation is correct.
And it turned out that the same result can be obtained by putting sin instead of cos. This also performed a number of inspections.
And, c=0.5 is best. If you use 0.499 or 0.501 instead of 0.5, it will not converge.
And when a non-trivial zero value is entered, the tendency to converge to 0 is seen, but if the non-trivial zero value is shifted even by 0.01, the tendency to converge generally tends to disappear.
If you add it to infinity instead of 10000, it will converge to 0.

**Category:** Number Theory

[1966] **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

[1965] **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

[1964] **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

[1963] **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

[1962] **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

[1961] **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

[1960] **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

[1959] **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

[1958] **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

[1957] **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

[1956] **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

[1955] **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

[1954] **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

[1953] **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

[1952] **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

[1951] **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

[1950] **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

[1949] **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

[1948] **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

[1947] **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

[1946] **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

[1945] **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

[1944] **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

[1943] **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

[1942] **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

[1941] **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

[1940] **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

[1939] **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

[1938] **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

[1937] **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

[1936] **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

[1935] **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

[1934] **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

[1933] **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

[1932] **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

[1931] **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

[1930] **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

[1929] **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

[1928] **viXra:1902.0040 [pdf]**
*submitted on 2019-02-02 16:34:38*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. A Complete proof of the abc conjecture using elementary calculus with numerical examples. Submitted to the Ramanujan Journal. Your comments are welcome.

In this paper, we consider the abc conjecture. Firstly, we give a proof of a first conjecture that c

**Category:** Number Theory

[1927] **viXra:1902.0036 [pdf]**
*submitted on 2019-02-03 00:15:01*

**Authors:** Simon Plouffe

**Comments:** 10 Pages.

Une famille de formules permettant d'obtenir une suite de longueur arbitraire de nombres premiers. Ces formules sont nettement plus efficaces que celles de Mills ou Wright. Le procédé permet de produire par exemple une suite dont la croissance est double exponentielle mais l'exposant = 101/100.

**Category:** Number Theory

[1926] **viXra:1902.0005 [pdf]**
*submitted on 2019-02-01 10:53:12*

**Authors:** James Edwin Rock

**Comments:** 2 Pages of exposition. 5 Tables. Copyright 2019 James Edwin Rock Creative Commons Attribution-ShareAlike 4.0 International License

Let P_n be the n_th prime. For twin primes P_n – P_(n-1) = 2. Let X be the number of (6j –1, 6j+1) pairs in the interval [P_n, P_n^2]. The number of twin primes (TPAn) in [P_n, P_n^2] can be approximated by the formula
(a_3 /5)(a_4 /7)(a_5 /11)…(a_n /P_n)(X) for 3 ≤ m ≤ n, a_m = P_m –2 .
We establish a lower bound for TPAn (3/5)(5/7)(7/9)…(P_n–2)/P_n)(X) = 3X/P_n < TPAn.
We exhibit a formula showing as P_n increases, the number of twin primes in the interval [P_n, P_n^2] also increases. Let P_n – P_(n-1) = c. For all n (TPAn-1)(1+(2c –2)/2P_(n-1)+(c^2–2c)/2P_(n-1)^2) < TPAn

**Category:** Number Theory

[1925] **viXra:1901.0436 [pdf]**
*submitted on 2019-01-29 10:25:44*

**Authors:** Kenneth A. Watanabe

**Comments:** 10 Pages.

Legendre's conjecture, states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that determines the number of prime numbers less than n for large values of n. Then by mathematical induction, it is proven that there is at least 1 prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture.

**Category:** Number Theory

[1924] **viXra:1901.0430 [pdf]**
*submitted on 2019-01-29 01:24:31*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we consider the $ABC$ conjecture,then we give a proof that $C<rad^2(ABC)$ that it will be the key of the proof of the $ABC$ conjecture.

**Category:** Number Theory

[1923] **viXra:1901.0300 [pdf]**
*submitted on 2019-01-19 14:41:09*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

On calculation by Euler 's formula(3), at least notice that the zero point of the real part is not on x = 0.5.
Using Euler 's formula, we found that at least the real part' s zero point is not x = 0.5 but about x = 0.115444. Moreover, the imaginary point is around i14.524.
And s=0.8355 +i39.
And s=0.1645 +i39.
And s=0.884556 +i14.524.
And s=0.115444 +i14.524
Also, replacing sin with cos, the imaginary part becomes zero.
I do not know at all whether the collapse of Riemann hypothesis or not?
In addition, books are printed as cos instead of sin.
Also, I have collected ζ on the left side.
In addition, (8) is Euler's formula found overseas, which is also a singular point in this. Also, I have collected ζ on the left side.
In this case, sin is printed instead of cos, but if sin and cos are exchanged, the zero point moves only from the real part to the imaginary part.

**Category:** Number Theory

[1922] **viXra:1901.0297 [pdf]**
*submitted on 2019-01-19 22:35:49*

**Authors:** Michael Grützmann

**Comments:** 2 Pages.

every prime number can be a sum of p=3+...+3+2 or
q=3+...+3+4, the number of '3's in both equatons always being odd.
there are the for two primes p+q the combinations
'p+q', 'p+p' and 'q+q'.
we consider case 1: p+q=3k+2+3l+4
3k must be odd, as the product of two odd numbers,
3l must be odd, for the same reason.
but the sum of two odd numbers is an even number always. also, if you add more even numbers, like 2 and 4, the result will always be even also.
So this results in an even number.
cases 'p+p' and 'q+q' analogue.

**Category:** Number Theory

[1921] **viXra:1901.0227 [pdf]**
*submitted on 2019-01-16 12:07:43*

**Authors:** James Edwin Rock

**Comments:** 5 Pages. Copyright 2018 James Edwin Rock Create Commons Attribution-ShareAlike 4.0 International License

Collatz sequences are formed by applying the Collatz algorithm to any positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. If the Collatz conjecture is true eventually you always get back to one. A connected Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[1920] **viXra:1901.0155 [pdf]**
*submitted on 2019-01-11 06:22:54*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents three trigonometric identities.

**Category:** Number Theory

[1919] **viXra:1901.0116 [pdf]**
*submitted on 2019-01-10 02:36:15*

**Authors:** Quang Nguyen Van

**Comments:** 2 Pages.

The equation a^5 + b^5 = c^2 has no solution in integer. However, related to Fermat- Catalan conjecture, the equation a^5 + b^5 = 2c^2 has a solution in integer. In this article, we give a parametric equation of the equation a^5 + b^5 = 2c^2.

**Category:** Number Theory

[1918] **viXra:1901.0108 [pdf]**
*submitted on 2019-01-08 11:13:12*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. A Proof of ABC conjecture is submitted to the Journal of Number Theory (2019). Comments Welcome.

In this paper, we assume that the ABC conjecture is true, then we give a proof that Beal conjecture is true. We consider that Beal conjecture is false then we arrive to a contradiction. We deduce that the Beal conjecture is true.

**Category:** Number Theory

[1917] **viXra:1901.0101 [pdf]**
*submitted on 2019-01-09 00:16:39*

**Authors:** Johnny E. Magee

**Comments:** 7 Pages.

We identify equivalent restatements of the Brocard-Ramanujan diophantine equation, $(n! + 1) = m^2$; and employing the properties and implications of these equivalencies, prove that for all $n > 7$, there are no values of $n$ for which $(n! + 1)$ can be a perfect square.

**Category:** Number Theory

[1916] **viXra:1901.0046 [pdf]**
*submitted on 2019-01-04 11:35:20*

**Authors:** Nazihkhelifa

**Comments:** 4 Pages.

Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
The theory of numbers is an area of mathematis hiih eals ith
the propertes of hole an ratonal numbers... In this paper I ill
intro uie relaton bet een Euler phi funiton an prime iountnn
an neneratnn formula, as ell as a ioniept of the possible
operatons e ian use ith them. There are four propositons hiih
are mentone in this paper an I have use the efnitons of these
arithmetial funitons an some Lemmas hiih refeit their
propertes, in or er to prove them

**Category:** Number Theory

[1915] **viXra:1901.0030 [pdf]**
*submitted on 2019-01-03 17:17:16*

**Authors:** Robert C. Hall

**Comments:** 24 Pages.

The Benford's Law Summation test consists of adding all numbers that begin with a particular first or first two digits and determining its distribution with respect to these first or first two digits numbers. Most people familiar with this test believe that the distribution is a uniform distribution for any distribution that conforms to Benford's law i.e. the distribution of the mantissas of the logarithm of the data set is uniform U[0,1). The summation test that results in a uniform distribution is true for an exponential function (geometric progression) but not true for a data set that conforms to a Log Normal distribution even when the Log Normal distribution itself closely approximates a Benford's Law distribution.

**Category:** Number Theory

[1914] **viXra:1901.0007 [pdf]**
*submitted on 2019-01-01 16:19:56*

**Authors:** Rachid Marsli

**Comments:** 11 Pages.

In this work, we show a sufficient and necessary condition for an integer of the form
(z^n-y^n)/(z-y)to be divisible by some perfect nth power p^n, where p is an odd prime. We also show how to construct such integers. A link between
the main result and Fermat’s last theorem is discussed. Other related ideas, examples and applications are provided.

**Category:** Number Theory

[1913] **viXra:1812.0497 [pdf]**
*submitted on 2018-12-31 12:45:35*

**Authors:** Pedro Hugo García Peláez

**Comments:** 5 Pages.

Series De Números Cuyos Factores Son La Lista De Los Números Primos Desde El Principio y Enumerando Todos Sin excepción

**Category:** Number Theory

[1912] **viXra:1812.0496 [pdf]**
*submitted on 2018-12-31 13:08:20*

**Authors:** Pedro Hugo García Peláez

**Comments:** 5 Pages.

Series of numbers whose factors are the list of prime numbers from the beginning and listing all without exception

**Category:** Number Theory

[1911] **viXra:1812.0495 [pdf]**
*submitted on 2018-12-31 14:29:57*

**Authors:** James Edwin Rock

**Comments:** Pages.

Let P_n be the n_th prime. For twin primes P_n – P_n-1 = 2. We show that in the interval (P_n, P_n^2) as P_n gets larger, there is an increasing number of twin primes.

**Category:** Number Theory

[1910] **viXra:1812.0494 [pdf]**
*submitted on 2018-12-31 09:51:52*

**Authors:** Simon Plouffe

**Comments:** 5 Pages.

We show here a new set of formulas for producing primes with a growth rate much smaller than the ones of Mills and Wright. Several examples of formulas are given. All results are empirical.

**Category:** Number Theory

[1909] **viXra:1812.0488 [pdf]**
*submitted on 2018-12-30 10:59:58*

**Authors:** Zeolla Gabriel Martín

**Comments:** 8 Pages.

This paper develops the analysis of Simple composite numbers by golden patterns. Examine how the Simple composite numbers are distributed in different combinations of multiples.

**Category:** Number Theory

[993] **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

[992] **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

[991] **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

[990] **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

[989] **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

[988] **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

[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:1902.0106 [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

[972] **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

[971] **viXra:1902.0005 [pdf]**
*replaced on 2019-03-29 12:35:08*

**Authors:** James Edwin Rock

**Comments:** 7 Pages.

Let P_n be the n_th prime. For twin primes P_n – P_(n-1) = 2. Let X be the number of (6j –1, 6j+1) pairs in the interval [P_n, P_n^2]. The number of twin primes (TPAn) in [P_n, P_n^2] can be approximated by the formula
(a_3 /5)(a_4 /7)(a_5 /11)…(a_n /P_n)(X) for 3 ≤ m ≤ n, a_m = P_m –2 .
We establish a lower bound for TPAn (3/5)(5/7)(7/9)…(P_n–2)/P_n)(X) = 3X/P_n < TPAn.
We exhibit a formula showing as P_n increases, the number of twin primes in the interval [P_n, P_n^2] also increases. Let P_n – P_(n-1) = c. For all n (TPAn-1)(1+(2c –2)/2P_(n-1)+(c^2–2c)/2P_(n-1)^2) < TPAn

**Category:** Number Theory

[970] **viXra:1902.0005 [pdf]**
*replaced on 2019-03-04 08:20:36*

**Authors:** James Edwin Rock

**Comments:** 7 Pages.

Let P_n be the n_th prime. For twin primes P_n – P_(n-1) = 2. Let X be the number of (6j –1, 6j+1) pairs in the interval [P_n, P_n^2]. The number of twin primes (TPAn) in [P_n, P_n^2] can be approximated by the formula
(a_3 /5)(a_4 /7)(a_5 /11)…(a_n /P_n)(X) for 3 ≤ m ≤ n, a_m = P_m –2 .
We establish a lower bound for TPAn (3/5)(5/7)(7/9)…(P_n–2)/P_n)(X) = 3X/P_n < TPAn.
We exhibit a formula showing as P_n increases, the number of twin primes in the interval [P_n, P_n^2] also increases. Let P_n – P_(n-1) = c. For all n (TPAn-1)(1+(2c –2)/2P_(n-1)+(c^2–2c)/2P_(n-1)^2) < TPAn

**Category:** Number Theory

[969] **viXra:1902.0005 [pdf]**
*replaced on 2019-02-19 08:32:39*

**Authors:** James Edwin Rock

**Comments:** 2 pages of exposition and 7 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

**Category:** Number Theory

[968] **viXra:1902.0005 [pdf]**
*replaced on 2019-02-16 12:01:31*

**Authors:** James Edwin Rock

**Comments:** 2 Pages of exposition and 7 pages of graphs

**Category:** Number Theory

[967] **viXra:1902.0005 [pdf]**
*replaced on 2019-02-13 08:29:18*

**Authors:** James Edwin Rock

**Comments:** 2 pages of exposition and 6 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

**Category:** Number Theory

[966] **viXra:1901.0436 [pdf]**
*replaced on 2019-02-26 16:28:14*

**Authors:** Kenneth A. Watanabe

**Comments:** 13 Pages.

Legendre's conjecture, states that there is a prime number between n^2 and(n + 1)^2 for every positive integer n. In this paper, an equation was derived that accurately determines the number of prime numbers less than n for large values of n. Then it is proven by induction that there is at least one prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture.

**Category:** Number Theory

[965] **viXra:1901.0436 [pdf]**
*replaced on 2019-02-13 08:35:02*

**Authors:** Kenneth A. Watanabe

**Comments:** 8 Pages.

Legendre's conjecture, states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that determines the number of prime numbers less than n for large values of n. Then it is proven by mathematical induction that there is at least one prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture.

**Category:** Number Theory

[964] **viXra:1901.0436 [pdf]**
*replaced on 2019-02-11 08:33:18*

**Authors:** Kenneth A. Watanabe

**Comments:** 7 Pages.

Legendre's conjecture, states that there is a prime number between n^$ and (n + 1)^2 for every positive integer n. In this paper, an equation was derived that determines the number of prime numbers less than n for large values of n. Then it is proven by mathematical induction that there is at least 1 prime number between n^2 and (n + 1)^2 for all positive integers n thus proving Legendre’s conjecture.

**Category:** Number Theory

[963] **viXra:1901.0430 [pdf]**
*replaced on 2019-01-31 12:44:22*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. Paper corrected of a mistake in the last version. Comments welcome.

In this paper, we consider the abc conjecture, then we give a proof of the conjecture c<rad^2(abc) that it will be the key to the proof of the abc conjecture.

**Category:** Number Theory

[962] **viXra:1901.0430 [pdf]**
*replaced on 2019-01-29 08:20:57*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 5 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we consider the ABC conjecture, then we give a proof that C<rad*2(ABC) that it will be the key of the proof of the ABC conjecture.

**Category:** Number Theory

[961] **viXra:1901.0227 [pdf]**
*replaced on 2019-03-18 08:58:13*

**Authors:** James Edwin Rock

**Comments:** 11 Pages.

Collatz sequences are formed by applying the Collatz algorithm to any positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. If the Collatz conjecture is true eventually you always get back to one. A connected Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[960] **viXra:1901.0227 [pdf]**
*replaced on 2019-02-25 10:42:27*

**Authors:** James Edwin Rock

**Comments:** 12 pages. Section 5 rewritten This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Collatz sequences are formed by applying the Collatz algorithm to any positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. If the Collatz conjecture is true eventually you always get back to one. A connected Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[959] **viXra:1901.0227 [pdf]**
*replaced on 2019-02-21 08:01:40*

**Authors:** James Edwin Rock

**Comments:** 12 pages. Section 5 is rewritten.This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

**Category:** Number Theory

[958] **viXra:1901.0227 [pdf]**
*replaced on 2019-01-29 11:21:03*

**Authors:** James Edwin Rock

**Comments:** 7 pages. This paper should be much easier to understand than the paper it is replacing

**Category:** Number Theory

[957] **viXra:1901.0227 [pdf]**
*replaced on 2019-01-17 08:32:12*

**Authors:** James Edwin Rock

**Comments:** 5 pages. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

**Category:** Number Theory

[956] **viXra:1812.0495 [pdf]**
*replaced on 2019-01-30 16:38:19*

**Authors:** James Edwin Rock

**Comments:** 5 Pages. This replacement paper adds a ratio TPA / 2.06 column to Table 2 and Table 1 contains a comparison of the twin prime pairs formula for [743,743^2] and [19993,19993^2]. It gives a better explanation for the twin prime pairs formula.

Let P_n be the n_th prime. For twin primes P_n – P_n-1 = 2. We exhibit formula for calculating the number of twin primes in the closed interval [P_n, P_n^2]. We prove that as P_n increases the number of twin primes in the interval [P_n, Pn^2] also increases.

**Category:** Number Theory

[955] **viXra:1812.0495 [pdf]**
*replaced on 2019-01-25 15:03:02*

**Authors:** James Edwin Rock

**Comments:** 2 pages of exposition and 3 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Let P_n be the n_th prime. For twin primes P_n – P_n-1 = 2. We exhibit formula for calculating the number of twin primes in the closed interval [P_n, P_n^2]. We prove that as P_n increases the number of twin primes in the interval [P_n, Pn^2] also increases.

**Category:** Number Theory

[954] **viXra:1812.0495 [pdf]**
*replaced on 2019-01-22 09:05:33*

**Authors:** James Edwin Rock

**Comments:** 5 Pages.

Let Pn be the n_th prime. For twin primes Pn – Pn-1 = 2. We exhibit two formulas for calculating the number of twin primes in the closed interval [Pn, Pn^2]. We show there is a lower limit for the number of twin primes in the closed interval [Pn, Pn^2].

**Category:** Number Theory

[953] **viXra:1812.0495 [pdf]**
*replaced on 2019-01-14 10:52:26*

**Authors:** James Edwin Rock

**Comments:** 2 pages of exposition and 3 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Let Pn be the n_th prime. For twin primes Pn – Pn-1 = 2. We exhibit two formulas for calculating the number of twin primes in the closed interval [Pn, Pn^2]. We show there is a lower limit for the number of twin primes in the closed interval [Pn, Pn^2].

**Category:** Number Theory

[952] **viXra:1812.0495 [pdf]**
*replaced on 2019-01-07 12:49:24*

**Authors:** James Edwin Rock

**Comments:** 1 page of exposition and 3 pages with supporting tables. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Let P_n be the n_th prime. For twin primes P_n – P_n-1 = 2. We exhibit two formulas for calculating the number of twin primes in the closed interval [P_n, P_n^2]. We show there is a lower limit for the number of twin primes in the closed interval [P_n, Pn^2].

**Category:** Number Theory

[951] **viXra:1812.0494 [pdf]**
*replaced on 2019-01-18 17:04:43*

**Authors:** Simon Plouffe

**Comments:** 10 Pages.

A set of new formulas for primes are given. These formulas have a growth rate much smaller than the ones of Mills and Wright.

**Category:** Number Theory

[950] **viXra:1812.0494 [pdf]**
*replaced on 2019-01-07 23:07:56*

**Authors:** Simon Plouffe

**Comments:** 7 Pages.

In 1947, W. H. Mills published a paper describing a formula that gives primes : if A = 1.3063778838630806904686144926… then ⌊A^(3^n ) ⌋ is always prime, here ⌊x⌋ is the integral part of x. Later in 1951, E. M. Wright published another formula, if g_0=α = 1.9287800… and g_(n+1)=2^(g_n ) then
⌊g_n ⌋= ⌊2^(…2^(2^α ) ) ⌋ is always prime.
The primes are uniquely determined by α , the prime sequence is 3, 13, 16381, …
The growth rate of these functions is very high since the fourth term of Wright formula is a 4932 digit prime and the 8’th prime of Mills formula is a 762 digit prime.
A new set of formulas is presented here, giving an arbitrary number of primes minimizing the growth rate. The first one is : if a_0=43.8046877158…and a_(n+1)= 〖〖〖(a〗_n〗^(5/4))〗^n , then if S(n) is the rounded values of a_n, S(n) = 113,367,102217,1827697,67201679,6084503671, …. Other exponents can also give primes like 11/10, or 101/100. If a_0 is well chosen then it is conjectured that any exponent > 1 can also give an arbitrary series of primes. The method for obtaining the formulas is explained. All results are empirical.

**Category:** Number Theory

[949] **viXra:1812.0494 [pdf]**
*replaced on 2018-12-31 12:31:57*

**Authors:** Simon Plouffe

**Comments:** 6 Pages.

A new set of formulas for primes is presented, they are better than the ones of Mills and Wright.

**Category:** Number Theory