**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(18) - 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(31) - 1802(20) - 1803(22) - 1804(25) - 1805(31) - 1806(16) - 1807(18) - 1808(14) - 1809(22) - 1810(17) - 1811(26) - 1812(29)

2019 - 1901(12) - 1902(11) - 1903(21) - 1904(25) - 1905(23) - 1906(43) - 1907(44) - 1908(13)

Any replacements are listed farther down

[2094] **viXra:1908.0416 [pdf]**
*submitted on 2019-08-19 09:50:51*

**Authors:** Johannes Abdus Salam

**Comments:** 1 Page.

I discovered an evidence of the existence of God as the mathematically beautiful equality of the Euler product.

**Category:** Number Theory

[2093] **viXra:1908.0381 [pdf]**
*submitted on 2019-08-19 02:42:39*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I treat Riemann hypothesis as a series and proved it.
Up to now, I have tried to expand this equation and prove Riemann hypothesis with the equation of cos, sin, but the proof was impossible.
However, I realized that a simple formula before expansion can prove it.

**Category:** Number Theory

[2092] **viXra:1908.0307 [pdf]**
*submitted on 2019-08-14 10:11:12*

**Authors:** Bing He

**Comments:** 16 Pages. All comments are welcome

In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Pi_{q}-identities. As a consequence, a q-identity involving Pi_{q} and Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper.

**Category:** Number Theory

[2091] **viXra:1908.0302 [pdf]**
*submitted on 2019-08-14 14:14:42*

**Authors:** Kouider Mohammed Ridha

**Comments:** 3 Pages.

We give explicit formulas to compute the Josephus-numbers where is positive integer . Furthermore we present a new fast algorithm to calculate . We also offer prosperities , and we generalized it for all positive real number non-existent, Finally we give .the proof of properties.

**Category:** Number Theory

[2090] **viXra:1908.0208 [pdf]**
*submitted on 2019-08-11 10:14:19*

**Authors:** Radomir Majkic

**Comments:** 3 Pages.

There are countable many rational distance squares, one square for each rational
trigonometric Pythagorean pair (s; c) : s^2+c^2=1 and a rational number r:

**Category:** Number Theory

[2089] **viXra:1908.0191 [pdf]**
*submitted on 2019-08-11 00:25:46*

**Authors:** Toshiro Takami

**Comments:** 23 Pages.

When we calculate by the sum method of (1) we found that the non-trivial zero point will never converge to zero.
Calculating ζ(2), ζ(3), ζ(4), ζ(5) etc. by the sum method of (1) gives the correct calculation result.
It was thought that the above equation could possibly be an expression that can be composed only of real numbers.
It seems to have not been noticed before (old) because there was no computer.
Thus, Riemann hypothesis is fundamentally wrong, and it is natural that it cannot be tried to prove it.

**Category:** Number Theory

[2088] **viXra:1908.0186 [pdf]**
*submitted on 2019-08-08 23:17:34*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist between successive primes, so that we can be sure that there is atleast one prime number between their squares.

**Category:** Number Theory

[2087] **viXra:1908.0145 [pdf]**
*submitted on 2019-08-08 08:33:33*

**Authors:** Sitangsu Maitra

**Comments:** 4 Pages.

Proof of Goldbach's strong conjecture in an unusual way

**Category:** Number Theory

[2086] **viXra:1908.0142 [pdf]**
*submitted on 2019-08-07 08:32:16*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some formulas involving Catalan's constant G=0.915965...

**Category:** Number Theory

[2085] **viXra:1908.0140 [pdf]**
*submitted on 2019-08-07 08:41:38*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents two Elementary integrals.

**Category:** Number Theory

[2084] **viXra:1908.0139 [pdf]**
*submitted on 2019-08-07 08:44:46*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some remarks on Ramanujan's integral: int(f(x),x=0..infinite)=(2/3)sqrt(pi).

**Category:** Number Theory

[2083] **viXra:1908.0115 [pdf]**
*submitted on 2019-08-08 03:28:39*

**Authors:** Andrea Berdondini

**Comments:** 4 Pages.

ABSTRACT: The following paradox is based on the consideration that the value of a statistical datum does not represent a useful information, but becomes a useful information only when it is possible to proof that it was not obtained in a random way. In practice, the probability of obtaining the same result randomly must be very low in order to consider the result useful. It follows that the value of a statistical datum is something absolute but its evaluation in order to understand whether it is useful or not is something of relative depending on the actions that have been performed. So two people who analyze the same event, under the same conditions, performing two different procedures obviously find the same value, regarding a statistical parameter, but the evaluation on the importance of the data obtained will be different because it depends on the procedure used. This condition can create a situation like the one described in this paradox, where in one case it is practically certain that the statistical datum is useful, instead in the other case the statistical datum turns out to be completely devoid of value. This paradox wants to bring attention to the importance of the procedure used to extract statistical information; in fact the way in which we act affects the probability of obtaining the same result in a random way and consequently on the evaluation of the statistical parameter.

**Category:** Number Theory

[2082] **viXra:1908.0072 [pdf]**
*submitted on 2019-08-05 02:01:30*

**Authors:** Victor Sorokine

**Comments:** 4 Pages. English version

IN THE FIRST CASE every number (A) is replaced by the sum (A'+A°n) of the last digit and the remainder. After binomial expansion of the Fermat's equality, all the members are combined in two terms: E=A'^n+B'^n-C'^n with the third digit E''', which in one of the n-1 equivalent Fermat's equalities is equal to 2, and the remainder D with the third digit D''', which is equal either to 0, or to n-1, and therefore the third digit of the number A^n+B^n-C^n is different from 0.

IN THE SECOND CASE (for example A=A°n^k, but (BС)'≠0), after having transformed the 3kn-digit ending of the number B into 1 and having left only the last siginificant digits of the numbers A, В, С, simple calculations show that the (3kn-2)-th digit of the number A^n+B^n-C^n is not 0 and does not change after the restoration of all other digits in the numbers A, B, C, because it depends only on the last digit of the number A°.

[2081] **viXra:1907.0593 [pdf]**
*submitted on 2019-07-29 06:31:31*

**Authors:** Leonid Vakhov

**Comments:** 4 Pages.

The constellation of zeros of Dirichlet eta function is similar to constellation of zeros of important subclass of L-functions (like Dirichlet series etc.). The hereby proposed simplified research can help in researching this important subclass of L-functions.

**Category:** Number Theory

[2080] **viXra:1907.0589 [pdf]**
*submitted on 2019-07-29 09:12:42*

**Authors:** Zeolla Gabriel Martin

**Comments:** 24 Pages.

: This article develops an old and well-known expression to obtain prime numbers, composite numbers and twin prime numbers. The conditioning (n) will be the key to make the formula work and the conditioning of the letter (z) will be important for the formula to be efficient.

**Category:** Number Theory

[2079] **viXra:1907.0580 [pdf]**
*submitted on 2019-07-29 22:01:51*

**Authors:** Jose R. Sousa

**Comments:** 16 Pages. I think this finding may have interesting applications in the study of the Riemann Hypothesis

This article discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of a neat power series for the prime counting function, $\pi(x)$. Among its main findings, we can cite the inversion theorem for Dirichlet series (given $F_a(s)$, we can tell what its associated function, $a(n)$, is), which enabled the creation of a formula for $\pi(x)$ in the first place, and the realization that sums of divisors and the M\"{o}bius function are particular cases of a more general concept. Another conclusion we draw is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.

**Category:** Number Theory

[2078] **viXra:1907.0579 [pdf]**
*submitted on 2019-07-29 22:06:22*

**Authors:** Jose R. Sousa

**Comments:** 7 Pages. Understanding this paper requires a reading of some of the previous papers

This is the fourth paper I'm releasing on the topic of harmonic progressions. Here we address a more complicated problem, namely, the determination of the limiting function of a generalized harmonic progression. It underscores the utility of the formula we derived for $\sum_{j=1}^{n}1/(a\ii j+b)^k$ in $\textit{Complex Harmonic Progression}$ and of results we presented in $\textit{Generalized Harmonic Numbers Revisited}$. Our objective is to create a generating function for $\sum_{k=2}^{\infty}x^k\sum_{j=1}^{\infty}1/(j+b)^k$, with complex $x$ and $b$, whose derivatives at 0 give us the limit of the harmonic progressions (of order 2 and higher) as $n$ approaches infinity.

**Category:** Number Theory

[2077] **viXra:1907.0578 [pdf]**
*submitted on 2019-07-29 22:08:45*

**Authors:** Jose R. Sousa

**Comments:** 8 Pages. This paper derives a formula that holds for nearly all generalized harmonic progressions

In $\textit{Generalized Harmonic Progression}$, we showed how to create formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, that is, $\sum_{j}1/(a j+b)^k$. Those formulae were more general than the ones we created in $\textit{Generalized Harmonic Numbers Revisited}$. In this new paper we make those formulae even more general by removing the restriction that $a$ and $b$ be integers, in other words, here we address $\sum_{j}1/(a\ii j+b)^k$, where $a$ and $b$ are complex numbers and $\ii$ is the imaginary unity. These new relatively simple formulae always hold, except when $\ii b/a\in \mathbb{Z}$. This paper employs a slightly modified version of the reasoning used previously. Nonetheless, we make another brief exposition of the principle used to derive such formulae.

**Category:** Number Theory

[2076] **viXra:1907.0577 [pdf]**
*submitted on 2019-07-29 22:10:51*

**Authors:** Jose R. Sousa

**Comments:** 8 Pages.

This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, more precisely, $\sum_{j=1}^{n}1/(a j+b)^k$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(a,b,n)$ and $S^m_{k}(a,b,n)$ (here, the term $``$harmonic progression$"$ is used loosely, as for some parameter choices, $a$ and $b$, the result may not be a harmonic progression). We provide a generalization of the formulae we created in $\textit{Generalized Harmonic Numbers Revisited}$, which was achieved by using an extension of the reasoning employed before.

**Category:** Number Theory

[2075] **viXra:1907.0558 [pdf]**
*submitted on 2019-07-28 14:39:32*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.

**Category:** Number Theory

[2074] **viXra:1907.0539 [pdf]**
*submitted on 2019-07-28 04:00:31*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

I discovered “mirror effect" of the Riemann zeta function on the critical line.
I investigated with many non-trivial zeros, the Riemann zeta function has a mirror image of about axis 1/2.
I named it, “mirror effect".
On non-trivial zero imaginary values, it is possible that one zero on the real value 1/2 or two with the real value 1/2 symmetrical may exist, but there is no possibility of the latter. Because “mirror effect" consist.
That is, on the non-trivial zero imaginary values, the real value 1/2 becomes the lowest absolute value which equal to zero, and the absolute real and imaginary value larger as it gets farther from the real value 1/2.
So, on non-trivial zero imaginary values, two with the real value 1/2 symmetrical can not exist, I conclude.

**Category:** Number Theory

[2073] **viXra:1907.0533 [pdf]**
*submitted on 2019-07-26 08:33:19*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

We give some identities for Pi.

**Category:** Number Theory

[2072] **viXra:1907.0511 [pdf]**
*submitted on 2019-07-27 04:21:14*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.

**Category:** Number Theory

[2071] **viXra:1907.0463 [pdf]**
*submitted on 2019-07-25 00:46:10*

**Authors:** Ayal Sharon

**Comments:** Pages.

The Dirichlet series of the Zeta function was long ago proven to be divergent throughout half-plane Re(s) =< 1. If also Riemann's proposition is true, that there exists an "expression" of the Zeta function that is convergent at all values of s (except at s = 1), then the Zeta function is both divergent and convergent throughout half-plane Re(s) =< 1 (except at s = 1). This result violates all three of Aristotle's "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non-Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion": Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann's analytic continuation of the Zeta function violates Cauchy's integral theorem, providing another proof of the invalidity of analytic continuation of the Zeta function. Also, Riemann's Zeta function is one of the L-functions, which are all invalid, because they are generalizations of the invalid analytic continuation of the Zeta function. This result renders unsound all theorems (e.g. Modularity, Fermat's last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an L-function (e.g. Riemann's Zeta function) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar's 3VL, Priest's LP).

**Category:** Number Theory

[2070] **viXra:1907.0437 [pdf]**
*submitted on 2019-07-23 20:48:54*

**Authors:** Hiroshi Okumura, Saburou Saitoh

**Comments:** 12 Pages. In this paper, we will give the values of the Riemann zeta function for any positive integers by means of the division by zero calculus.

In this paper, we will give the values of the Riemann zeta function for any positive integers by means of the division by zero calculus.
Zero, division by zero, division by zero calculus, $0/0=1/0=z/0=\tan(\pi/2) = \log 0 =0 $, Laurent expansion, Riemann zeta function, Gamma function, Psi function, Digamma function.

**Category:** Number Theory

[2069] **viXra:1907.0414 [pdf]**
*submitted on 2019-07-23 02:25:12*

**Authors:** Aaron chau

**Comments:** 2 Pages.

左边图有二个表示：孪生质数猜想成立。黎曼假设被推翻。右边图表示哥猜是一场没完没了的澄清运动。

**Category:** Number Theory

[2068] **viXra:1907.0400 [pdf]**
*submitted on 2019-07-21 13:24:37*

**Authors:** Jian-ping Gu

**Comments:** 1 Page.

This paper suggests extending the studies of number theory to non-decimal number systems.

**Category:** Number Theory

[2067] **viXra:1907.0393 [pdf]**
*submitted on 2019-07-21 00:21:22*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

First, ±∞ is constant at any observation point (position).

**Category:** Number Theory

[2066] **viXra:1907.0387 [pdf]**
*submitted on 2019-07-19 07:19:39*

**Authors:** Horacio useche losada

**Comments:** 29 Pages. On how to calculate the digits of Pi

El cálculo de los dı́gitos de π ha sido siempre una de las tareas más deseadas
por los matemáticos de todos los tiempos, siendo la más antigua de todas.
El número π se viene calculando desde la edad de hierro, sin exagerar, y en
este documento podrá encontrar un resumen de todos esos esfuerzos con más
de 5000 años de historia.
Actualmente el record pertenece al fı́sico de partı́culas suizo Peter Trueb,
que en noviembre de 2016, encontró 22 459 157 718 361 números decimales de
π, completamente verificados. Estos son 2.2 billones de decimales, una can-
tidad tan abrumadora que alcanzarı́a para dar 1.2 vueltas al planeta tierra,
por el ecuador, y suponiendo cada decimal del tamaño de las letras que ahora
lee.
Muchos lectores se preguntaran para que sirve calcular tantos dı́gitos de-
cimales si para calcular la circuferencia del universo con un error no superior
al radio atómico, bastarı́a una precisión de 32 decimales. La respuesta es la
misma por la cual el ser humano se empeña en reducir el tiempo de recorrido
para los 100 metros planos. Es un sı́mbolo de prepotencia y progreso, del
cual, el ser humano, no se puede desprender. Una auténtica demostración de
cerebro y máquina que presume del alcance de la especie humana.
Para realizar este tipo de esfuerzos, se deben tomar una serie de decisiones
concernientes con los algoritmos a usar, esto es, los criterios matemáticos,
además de seleccionar las herramientas de software para programar dichos
criterios y por último el hardware, o computadores fı́sicos. Todo ello junto,
conforma el arsenal de batalla para llevar a cabo hazañas como las de con-
quistar nuevos records.
Ya se trate de aficionados o matemáticos profesionales, este documento le
entrega una revista incremental, desde rústicos y antiguos criterios, hasta los
más modernos y sofisticados, usados en la ambiciosa conquista de los dı́gitos
de π, que sin duda, le darán lustre a su saber y habilidad.
Aquı́, por lo pronto, nos conformamos con llevar al lı́mite de lo posible, las
herramientas de hardware casero, con las cuales el lector podrá hacer uso de
las mejores teorı́as matemáticas para tener una idea muy fresca y fiel, de las
tormentas que se desatan en las cumbres borrascosas de la alta matemática.

**Category:** Number Theory

[2065] **viXra:1907.0378 [pdf]**
*submitted on 2019-07-19 14:15:53*

**Authors:** Horacio useche losada

**Comments:** 33 Pages. The Goldbach's strong conjeture has been proved

Abstract
The proof of Goldbach’s strong conjecture is presented, built on the
foundations of the theory of gap, which, when combined with certain
criteria about the existence of prime numbers in successions, gives us
the evidence cited. In reality, We have proof a more general statement
in relation to that attributed to Goldbach. As result, it is proved how
a even number is the sum of two odd primes, of infinite ways and as
a corollary, the conjecture about of the twin primes is also proof.

**Category:** Number Theory

[2064] **viXra:1907.0358 [pdf]**
*submitted on 2019-07-18 16:35:58*

**Authors:** Harry K. Hahn

**Comments:** 5 pages, 1 drawing

All natural numbers ( 1, 2, 3,…) can be calculated only by using constant Phi (ϕ) and 1.
I have found a way to express all natural numbers and their square roots with simple algebraic terms, which are only based on Phi (ϕ) and 1.
Further I have found a rule to calculate all natural numbers >10 and their square roots with the help of a general algebraic term.
The constant Pi (π) can also be expressed only by using constant Phi and 1 !
It seems that the irrationality of Pi (π) is fundamentally based on the constant Phi and 1, in the same way as the irrationality of all irrational square roots, and all natural numbers seems to be based on constant Phi & 1 !
This is an interesting discovery because it allows to describe many basic geometrical objects like the Platonic Solids only with Phi & 1 !
The result of this discovery may lead to a new base of number theory. Not numbers like 1, 2, 3,… and constants like Pi (π) are the base of number theory ! It seems that only the constant Phi and the base unit 1 ( which shouldn’t be considered as a number ! ) form the base of mathematics and geometry. And constant Phi and the base unit 1 must be considered as the fundamental „space structure constants“ of the real physical world !

**Category:** Number Theory

[2063] **viXra:1907.0357 [pdf]**
*submitted on 2019-07-18 16:41:14*

**Authors:** Harry K. Hahn

**Comments:** 35 pages, 17 figures, 3 tables

Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral.

**Category:** Number Theory

[2062] **viXra:1907.0356 [pdf]**
*submitted on 2019-07-18 16:44:28*

**Authors:** Harry K. Hahn

**Comments:** 44 pages, 26 figures, 7 tables

Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -second difference- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.

**Category:** Number Theory

[2061] **viXra:1907.0355 [pdf]**
*submitted on 2019-07-18 16:47:59*

**Authors:** Harry K. Hahn

**Comments:** 29 pages, 10 figures, 6 tables

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1. All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3. Riemanns Zeta Function also seems to indicate, that there is a logical connection between the mentioned number sequences and the distribution of prime numbers. This connection is indicated by lines in the diagram of the Zeta Function, which are formed by the points s where the Zeta Function is real. Another key role in the distribution of the prime numbers plays the number 5 and its periodic occurrence in the two number sequences SQ1 and SQ2. All non-prime numbers in SQ1 and SQ2 are caused by recurrences of these two number sequences with increasing wave-lengths in themselves, in a similar fashion as Overtones (harmonics) or Undertones derive from a fundamental frequency. On the contrary prime numbers represent spots in these two basic Number Sequences SQ1 and SQ2 where there is no interference caused by these recurring number sequences. The distribution of the non-prime numbers and prime numbers can be described in a graphical way with a -Wave Model- (or Interference Model) -- see Table 2.

**Category:** Number Theory

[2060] **viXra:1907.0354 [pdf]**
*submitted on 2019-07-18 16:53:39*

**Authors:** Harry K. Hahn

**Comments:** 12 pages, 6 figures

The natural numbers divisible by the Prime Factors 2, 3, 5, 11, 13 and 17 lie on defined spiral graphs, which run through the Square Root Spiral. A mathematical analysis shows, that these spiral graphs are defined by specific quadratic polynomials. Basically all natural number which are divisible by the same prime factor lie on such spiral graphs. And these spiral graphs can be assigned to a certain number of Spiral Graph Systems, which have a defined spatial orientation to each other. This document represents a supplementation to my detailed introduction study to the Square Root Spiral, and it contains the missing diagrams and analyses, showing the distribution of the natural numbers divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral. My introduction study to the Square Root Spiral can also be found in this archive. The title of this study : The ordered distribution of the natural numbers on the Square Root Spiral.

**Category:** Number Theory

[2059] **viXra:1907.0345 [pdf]**
*submitted on 2019-07-17 08:31:03*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

We give a formula for Pi.

**Category:** Number Theory

[2058] **viXra:1907.0303 [pdf]**
*submitted on 2019-07-17 05:02:05*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This Relative formula shows the relationship between e and π without i.

**Category:** Number Theory

[2057] **viXra:1907.0288 [pdf]**
*submitted on 2019-07-15 08:52:01*

**Authors:** Igor Hrnčić

**Comments:** 29 Pages.

In this manuscript we use the Perron formula to connect zeta evaluated on the root free halfplane to zeta evaluated on the critical strip. This is possible since the Perron formula is of the form f(s)=O f(s+w) with O being an integral operator. The variable s+w is on the root free halfplane, and yet s can be on the critical strip. Hence, the Perron formula serves as a form of a functional equation that connects the critical strip with the root free halfplane. Then, one simply notices that in the Perron formula, the left hand side converges only conditionally, whilst the right hand side converges absolutely. This, of course, cannot be, since the left side of an equation is always equal to the right side. This contradiction when examined in detail disproves the Riemann hypothesis. This method is employed on an arbitrary distribution of zeta roots as well, concluding that zeta has a root arbitrarily close to the vertical line passing through unity.

**Category:** Number Theory

[2056] **viXra:1907.0221 [pdf]**
*submitted on 2019-07-13 10:26:58*

**Authors:** Kamal Barghout

**Comments:** 5 Pages. The manuscript is not to be copied or used in whole or part. The manuscript is copyrighted.

In this note I will show how Beal’s conjecture can be used to study abc conjecture. I will first show how Beal’s conjecture was proved and derive the necessary steps that will lead to further understand the abc conjecture hoping this will aid in proving it. In short, Beal’s conjecture was identified as a univariate Diophantine polynomial identity derived from the binomial identity by expansion of powers of binomials, e.g. the binomial〖 (λx^l+γy^l )〗^n; λ,γ,l,n are positive integers. The idea is that upon expansion and reduction to two terms we can cancel the gcd from the identity equation which leaves the coefficient terms coprime and effectively describes the abc conjecture. To further study the abc terms we need to specifically look for criterion upon which the general property of abc conjecture that states that if the two numbers a and b of the conjecture are divisible by large powers of small primes, a+b tends to be divisible by small powers of large primes which leads to a+b be divisible by large powers of small primes. In this note I only open the door to investigate related possible criterions that may lead to further understand the abc conjecture by expressing it in terms of binomial expansions as Beal’s conjecture was handled.

**Category:** Number Theory

[2055] **viXra:1907.0206 [pdf]**
*submitted on 2019-07-12 23:13:57*

**Authors:** Toshiro Takami

**Comments:** 10 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive.
We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5.
This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value.
We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.

**Category:** Number Theory

[2054] **viXra:1907.0191 [pdf]**
*submitted on 2019-07-12 02:40:19*

**Authors:** Labib Zakaria

**Comments:** 12 Pages. Hopefully this is obvious from the abstract & a quick overview of the paper, but this is not meant to be an immensely technical paper. It is simply meant to be so that people can nurture an appreciation for math. Constructive criticism appreciated.

There exist many algorithms to test the primality of positive natural numbers both proved and unproved, as well as in base 10 and outside base 10. Once the primality of a number has been determined, natural questions are $(1)$ what the unique prime factors of it are and $(2)$ their degree, according to the fundamental theorem of arithmetic.
These questions can prove to be useful in beginning to analyze the properties of the number by allowing us to determine the number of (proper) divisors of a number as well as their sum and product. In regards to $(1)$, there are many algorithms that could be applied to determine these prime factors through modular arithmetic algorithms. We will be tackling this question in base 10 specifically by constructing functions as curious mathematicians.

**Category:** Number Theory

[2053] **viXra:1907.0171 [pdf]**
*submitted on 2019-07-11 00:49:20*

**Authors:** Surajit Ghosh

**Comments:** 19 Pages.

Riemann hypothesis stands proved in three diﬀerent ways.To prove Riemann hypothesis from the functional equation concept of Delta function is introduced similar to Gamma and Pi function. Zeta values are renormalised to remove the poles of zeta function. Extending sum to product rule fundamental formula of numbers are deﬁned which then helps proving other prime conjectures namely goldbach conjecture, twin prime conjecture etc.

**Category:** Number Theory

[2052] **viXra:1907.0154 [pdf]**
*submitted on 2019-07-09 18:42:44*

**Authors:** Viktor Kalaj

**Comments:** 10 Pages. This paper is rather succinct; it deals with a contradiction while testing the Riemann Zeta function valid on 0 < Re(s) < 1

In this paper, we summarize results of a contradiction while testing the Riemann Hypothesis

**Category:** Number Theory

[2051] **viXra:1907.0126 [pdf]**
*submitted on 2019-07-09 01:25:02*

**Authors:** Darrin Taylor

**Comments:** Pages.

In base 3, the presence of leading 1s during division has a one to one correlation with the 3n+1 operation.
This is because dividing a leading 1 in base 3 is the only way to lose a digit and 3n+1 shifts are the only way to gain a digit. Total digit length doesn't change around a loop so they must equal each other.
Because the leading 1 pattern among a series of divides only has 2 segments either 1->2 or 1->2->1 there are a limited number of patterns that can make up a loop. Naming 1->2->(next segments leading 1) as segment A Naming 1->2->1->(next segments leading 1) as segment B We can see that A is 2 divides and 1 non localized shift while B is 3 divides and 2 non localized shifts. The pattern ABB...ABB descends because 8 divides and 5 shifts descends for numbers larger than 1000 and lower then 1000 have been numerically disqualified previously.
So the sequence BBB must exist at least once in every loop. BBB implies ABBB or BBBB if BBBB then "expel" a B which ascends and keep searching for the segment before the sequence. Once ABBB is found this implies AABBB or BABBB and AABBB is disproven as not possible. Once BABBB is known this implies ABABBB or BBABBB and ABABBB is disproven. Once BBABBB is known this implies ABBABBB or BBBABBB and ABBABBB is disproven. Once BBBABBB is found we can "expel" ABBB which ascends and BBB(ABBB) becomes BBB and we are back where we started. Once the entire loop has been traversed this way the sequence has expelled only (B) or (ABBB) and the remaining sequence is BBB and all of these ascend. Loops must have ascending and descending segments for a total non ascending and non descending but this loop always ascends.
Thus it cannot be a loop and no loops of As and Bs can exist as those with fewer Bs than ABBABBABB…..always descend and adding a single B makes it always ascend. And As and Bs are the only possible segments to add.

**Category:** Number Theory

[2050] **viXra:1907.0109 [pdf]**
*submitted on 2019-07-06 06:57:31*

**Authors:** Victor Sorokine

**Comments:** 4 Pages.

В ПЕРВОМ СЛУЧАЕ каждое число (А) заменяется на сумму (A'+A°n) последней цифры и остатка. После раскрытия биномов в равенстве Ферма все члены объединятся в два слагаемых: E=A'^n+B'^n-C'^n с третьей цифрой E''', которая в одном из n-1 эквивалентных равенств Ферма равна 2, и остаток D с третьей цифрой D''', равной либо 0, либо n-1, и, следовательно, третья цифра в числе A^n+B^n-C^n не равна 0.
ВО ВТОРОМ СЛУЧАЕ (например A=A°n^k, но (BС)'≠0, ) после преобразования 3kn-значного окончания числа B в 1 и оставления в числах А, В, С лишь последних значащих цифр простейшие расчёты показывают, что (3kn-2)-я цифра числа A^n+B^n-C^n нулю не равна и не меняется после восстановления всех остальных цифр в числах A, B, C, т.к. является функцией только последней цифры числа A°.

**Category:** Number Theory

[2049] **viXra:1907.0108 [pdf]**
*submitted on 2019-07-06 11:07:22*

**Authors:** Simon Plouffe

**Comments:** 53 Pages.

Conference in Montreal, Canada to be held on July 17 2019. The subject is Pi , the prime numbers and the Lambert W function

**Category:** Number Theory

[2048] **viXra:1907.0091 [pdf]**
*submitted on 2019-07-05 13:23:11*

**Authors:** Viktor Kalaj

**Comments:** 11 Pages. Notify me, the author, Viktor Kalaj, if this paper is in anyway difficult to read by the print (font, size, etc.)

This paper deals with a proposed contradiction to the Riemann Hypothesis. We see by a deductive approach the necessity of no zeroes for the entire critical strip, including for the critical line.

**Category:** Number Theory

[2047] **viXra:1907.0089 [pdf]**
*submitted on 2019-07-05 17:23:01*

**Authors:** Viktor Kalaj

**Comments:** 1 Page. Minor typo correction in my paper "A technical procedure for the Riemann Hypothesis".

There was a minor typographical error in my paper entitled "A technical procedure for the Riemann Hypothesis". It does not affect the technical procedure of the paper.

**Category:** Number Theory

[2046] **viXra:1907.0088 [pdf]**
*submitted on 2019-07-05 17:28:12*

**Authors:** Viktor Kalaj

**Comments:** A minor typographical correction to my 11-page paper

I made a typographical error that is now corrected. There is no change in the flow of the paper entitled "A technical procedure for the Riemann Hypothesis".

**Category:** Number Theory

[2045] **viXra:1907.0087 [pdf]**
*submitted on 2019-07-05 20:53:17*

**Authors:** Toshiro Takami

**Comments:** 5 Pages. Riemann's hypothesis really proved.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
In this paper, we give a proof of mathematical expression.
“the non-trivial zero values of all positive infinity and negative infinity lie on the real value 0.5” I am here mathematically proved.

**Category:** Number Theory

[2044] **viXra:1907.0063 [pdf]**
*submitted on 2019-07-04 01:20:32*

**Authors:** Predrag Terzic

**Comments:** 4 Pages.

General,deterministic,unconditional,polynomial time primality test is introduced.

**Category:** Number Theory

[2043] **viXra:1907.0055 [pdf]**
*submitted on 2019-07-03 10:09:12*

**Authors:** Http://vixra.org/author/andrew_w_ivashenko

**Comments:** 1 Page.

Decomposition of integer powers of a mersenne number into binomial coefficients

**Category:** Number Theory

[2042] **viXra:1907.0046 [pdf]**
*submitted on 2019-07-02 08:37:34*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some integrals for Pi.

**Category:** Number Theory

[2041] **viXra:1907.0045 [pdf]**
*submitted on 2019-07-02 08:40:14*

**Authors:** Edgar Valdebenito

**Comments:** 1 Page.

This note presents two identities for Pi.

**Category:** Number Theory

[2040] **viXra:1907.0037 [pdf]**
*submitted on 2019-07-02 16:29:32*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
It is impossible to make c = 0.5 exactly like this. c can only be 0.5 and its edge.
It is considered that “when the imaginary value increases to infinity, the denominator of the number becomes infinity and shifts from 0.5 to 0”.

**Category:** Number Theory

[2039] **viXra:1907.0018 [pdf]**
*submitted on 2019-07-01 23:59:43*

**Authors:** Simon Plouffe

**Comments:** 58 Pages.

Une revue historique du nombre Pi faite à l'IUT de Nantes.
A presentation of Pi made at Université de Nantes (IUT) on April 25 2019.

**Category:** Number Theory

[2038] **viXra:1906.0570 [pdf]**
*submitted on 2019-06-30 18:22:13*

**Authors:** Akalabu, Emmanuel Chukwuemeka

**Comments:** 8 Pages.

--

**Category:** Number Theory

[2037] **viXra:1906.0544 [pdf]**
*submitted on 2019-06-28 11:10:36*

**Authors:** Simon Plouffe

**Comments:** 8 Pages.

Un nouveau modèle est proposé pour représenter ces quantités. En premier lieu, 4 formules sont données qui sont déduites des résultats classiques, ensuite un principe est appliqué, appelé matriochkas ou des poupées russes qui permet de trouver des développements asymptotiques remarquablement simples et élégants. De plus, les développements obtenus sont tous très similaires.
A new model is proposed to represent these quantities. In the first place, 4 formulas are given which are deduced from the classical results, then a principle is applied, called matriochkas or Russian dolls which allows to find remarkably simple and elegant asymptotic expansions. Moreover, the developments obtained are all very similar.

**Category:** Number Theory

[2036] **viXra:1906.0531 [pdf]**
*submitted on 2019-06-27 18:15:41*

**Authors:** Xuan Zhong Ni

**Comments:** 1 Page.

In this article, we use the sieve of Eratosthenes to prove the Oppermann Conjecture.

**Category:** Number Theory

[2035] **viXra:1906.0508 [pdf]**
*submitted on 2019-06-27 04:33:42*

**Authors:** Oksana Vozniuk, Bogdana Oliynyk, Roman Yavorskyi

**Comments:** 5 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152597

iotope spaces were introduced by Marchevsky-Steinhaus in for the needs of mathematical biology, namely the study of ecosystems. Biotope distance is defined on the set of all subsets of some finite set X. The distance between any subsets A1 and A2 of X is calculated by the rule: d(A1, A2) = (0, if A1 = A2 = ∅; |A1⊕A2| |A1∪A2| , if A1, A2 ∈ B(X)).We introduce a new generalization of a biotope metric to the infinite case using supernatural or Steinitz numbers. A supernatural number (or Steinitz number) is an infinite formal product of the form Y p∈P p kp where P is the set of all primes and kp ∈ N ∪ {0, ∞}. On the set of all periodic {0, 1}-sequences with the period that is a divisor of some supernatural u; we define the metric dB for any infinite periodic sequences x¯ and y¯ by the rule: dB(¯x, y¯) = dBn (¯xn, y¯n) where n is a common period of periodic sequences x¯ and y¯, and the formula dB(¯xn, y¯n) denotes the biotope distance between the first n coordinates of sequences x¯ and y¯ in the finite biotope metric space Bn. We denote the periodic biotope space that is defined by some Steinitz number u as B(u). If u is a finite Steinitz number, i.e. u is a positive integer, then B(u) is isometric finite biotope space Bu. We also prove that the introduced metric between such two periodic sequences does not depend on a choice of a common period.
A family of such introduced periodic biotope spaces is naturally parametrized by supernatural numbers. More precisely, the family of these spaces forms a lattice that is isomorphic to the lattice of supernatural numbers. Moreover, each of these spaces B(u) is invariant with respect to the shift.
We prove that the diametr of any periodic biotope space equals 1. We also show that any finite subset of a countable biotope space introduced in is isometric embedding in the periodic biotope space B(u) for any u.

**Category:** Number Theory

[2034] **viXra:1906.0498 [pdf]**
*submitted on 2019-06-27 08:44:40*

**Authors:** Nurlan Qasimli

**Comments:** 6 Pages.

History of conjecture

**Category:** Number Theory

[2033] **viXra:1906.0488 [pdf]**
*submitted on 2019-06-25 08:29:56*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a simple formula for Pi.

**Category:** Number Theory

[2032] **viXra:1906.0463 [pdf]**
*submitted on 2019-06-24 20:24:24*

**Authors:** H. Tran

**Comments:** 12 Pages. Proof of the Riemann hypothesis

We first find a Hamiltonian H that has the Hurwitz zeta functions ζ(s,x) as eigenfunctions. Then we continue constructing an operator G that is self-adjoint, with appropriate boundary conditions. We will find that the ζ(s,x)-functions do not meet these boundary conditions, except for the ones where s is a nontrivial zero of the Riemann zeta, with the real part of s being greater than 1/2. Finally, we find that these exceptional functions cannot exist, proving the Riemann hypothesis, that all nontrivial zeros have real part equal to 1/2.

**Category:** Number Theory

[2031] **viXra:1906.0426 [pdf]**
*submitted on 2019-06-22 12:54:14*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of a modiﬁed sieve of Eratosthenes to prove that any large even numbers can always be expressed as sums of two prime numbers.

**Category:** Number Theory

[2030] **viXra:1906.0424 [pdf]**
*submitted on 2019-06-22 15:55:33*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of a modiﬁed sieve of Eratosthenes to prove the cousin prime conjecture.

**Category:** Number Theory

[2029] **viXra:1906.0423 [pdf]**
*submitted on 2019-06-22 16:50:25*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages.

Mostre que o seno de um arco na forma 1/p, com p inteiro, resulta em um
irracional.
Observe que

**Category:** Number Theory

[2028] **viXra:1906.0422 [pdf]**
*submitted on 2019-06-22 20:43:47*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages.

Title, authors and abstract should also be included in the PdF file. These should be in English. If the submission is not in English please translate the title and abstract here.

**Category:** Number Theory

[2027] **viXra:1906.0421 [pdf]**
*submitted on 2019-06-22 20:58:02*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use method of sieve of Eratosthenes to prove that there is a larger prime gap near any primorial number.

**Category:** Number Theory

[2026] **viXra:1906.0420 [pdf]**
*submitted on 2019-06-22 21:11:07*

**Authors:** Israel Meireles Chrisostomo

**Comments:** 2 Pages. irrationality and pi other transformation

irrationality and pi other transformationirrationality and pi other transformationirrationality and pi other transformationirrationality and pi other transformation

**Category:** Number Theory

[2025] **viXra:1906.0418 [pdf]**
*submitted on 2019-06-22 22:15:12*

**Authors:** Pedro Hugo García Peláez

**Comments:** 3 Pages.

What I try to prove is that there are infinite number of Lucas primes

**Category:** Number Theory

[2024] **viXra:1906.0408 [pdf]**
*submitted on 2019-06-20 13:40:49*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

We show that attempting to map the set of real numbers to the natural numbers by listing them as infinite decimal fractions is futile. The real numbers are represented as the limit of partial decimal sums. This allows them to be explicitly referenced and makes them into a countable set. We conjecture that the Pi, i, and e generate the Real Numbers.

**Category:** Number Theory

[2023] **viXra:1906.0391 [pdf]**
*submitted on 2019-06-21 08:18:38*

**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, that is, a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[2022] **viXra:1906.0378 [pdf]**
*submitted on 2019-06-21 21:47:44*

**Authors:** Xuan Zhong Ni

**Comments:** 2 Pages.

In this article, we use a modified sieve of Eratosthenes to prove twin prime conjecture.

**Category:** Number Theory

[2021] **viXra:1906.0377 [pdf]**
*submitted on 2019-06-21 22:02:41*

**Authors:** Xuan Zhong Ni

**Comments:** 4 Pages.

In this article, we assume that the Riemann Zeta Function equals to the Euler product at the non zero points of the Riemann Zeta function. From this assumption we can prove that there are no zero points of Riemann Zeta function, ς(s) in Re(s) > 1/2. We applied proof by contradiction.

**Category:** Number Theory

[2020] **viXra:1906.0374 [pdf]**
*submitted on 2019-06-22 06:25:55*

**Authors:** Julian TP Beauchamp

**Comments:** 6 Pages.

Catalan's Conjecture was first made by Belgian mathematician Eugène Charles Catalan in 1844, and states that 8 and 9 (2^3 and 3^2) are the only consecutive powers, excluding 0 and 1. That is to say, that the only solution in the natural numbers of a^x - b^y=1 for a,b,x,y > 1 is a=3, x=2, b=2, y=3. In other words, Catalan conjectured that 3^2-2^3=1 is the only nontrivial solution. It was finally proved in 2002 by number theorist Preda Mihailescu making extensive use of the theory of cyclotomic fields and Galois modules.

**Category:** Number Theory

[2019] **viXra:1906.0373 [pdf]**
*submitted on 2019-06-19 07:35:33*

**Authors:** Méhdi Pascal

**Comments:** 20 Pages.

The bute of this algebra is to give a tool which makes it possible to find new formulas for the sequences of the numbers, for example, I take the numbers of Bernoulli (Bn), and the numbers of Fibonacci (Fn), and this algebra allows us the following formula:
n*F(n)=sum(binomial(n,j)*(F(2n-2j+1)-F(n-j+1))*B(j)), From j=0 to j=n.

**Category:** Number Theory

[2018] **viXra:1906.0322 [pdf]**
*submitted on 2019-06-17 08:54:10*

**Authors:** James Edwin Rock

**Comments:** 1 Page.

We exploit some rudimentary facts about the number one: (-1)(-1) = 1, 1 = sqrt(1 squared), and 1 squared = 1 to show an anomaly in the set of Complex Numbers.

**Category:** Number Theory

[2017] **viXra:1906.0315 [pdf]**
*submitted on 2019-06-17 22:43:25*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

All prime numbers are represented as factors of Fibonacci numbers, following a relationship with the corresponding Fibonacci number index.

**Category:** Number Theory

[2016] **viXra:1906.0282 [pdf]**
*submitted on 2019-06-15 15:16:11*

**Authors:** Sally Myers Moite

**Comments:** 6 Pages.

For a fixed last prime, sieve the positive integers as follows. For every prime up to and including that last prime, choose one arbitrary remainder and its negative. Sieve the positive integers by eliminating all numbers congruent to the chosen remainders modulo their prime. Consider the maximum of the first open numbers left by all such sieves for a particular last prime. Computations for small last primes support a conjecture that the maximum first open number is less than (last prime)^1.75. If this conjecture could be proved, it would imply Goldbach’s Theorem is true.

**Category:** Number Theory

[2015] **viXra:1906.0273 [pdf]**
*submitted on 2019-06-16 04:31:04*

**Authors:** Silvio Gabbianelli

**Comments:** 14 Pages.

By arranging the prime numbers on four columns ten-to-ten (columns of one, three, seven, nine) and establishing a suitable correspondence between the quadruples obtained and the numbers between zero and fifteen, we obtain a synthetic representation of them which allows to establish that the order in the distribution of prime numbers among positive natural numbers is not random.

**Category:** Number Theory

[2014] **viXra:1906.0243 [pdf]**
*submitted on 2019-06-13 11:24:19*

**Authors:** David Rudisill

**Comments:** 14 Pages.

We prove an important new result on this problem: Given any epsilon > 0 and k >= 5, and given any set of speeds s_1 < s_2 < ... < s_k, there is a set of speeds v_1 < v_2 < ... < v_k for which the lonely runner conjecture is true and for which |s_i - v_i| < epsilon. We also prove some measure theorems.

**Category:** Number Theory

[2013] **viXra:1906.0242 [pdf]**
*submitted on 2019-06-13 11:35:10*

**Authors:** David Rudisill

**Comments:** 10 Pages.

We prove that the lonely runner conjecture is equivalent to a set of Diophantine approximation problems.

**Category:** Number Theory

[2012] **viXra:1906.0241 [pdf]**
*submitted on 2019-06-13 11:51:58*

**Authors:** David v. Rudisill

**Comments:** 8 Pages.

We prove some measure and covering problems related to the lonely runner conjecture.

**Category:** Number Theory

[2011] **viXra:1906.0199 [pdf]**
*submitted on 2019-06-13 05:44:33*

**Authors:** Julian TP Beauchamp

**Comments:** 4 Pages.

In this paper, we show how a^x - b^y can be expressed as a binomial expansion (to an indeterminate power, z, and use it as the basis for a proof for the Beal Conjecture.

**Category:** Number Theory

[2010] **viXra:1906.0195 [pdf]**
*submitted on 2019-06-11 07:12:25*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages.

The rational root test gives a means for determining if a root of a polynomial is rational. If none of the tests possible rational roots are roots, then if the roots are real, they must be irrational. Combining this observation with Taylor polynomials and the Taylor series for sin(x) gives an intimation that pi, and e, are likely irrational.

**Category:** Number Theory

[2009] **viXra:1906.0131 [pdf]**
*submitted on 2019-06-08 13:38:48*

**Authors:** Robert C. Hall

**Comments:** 46 Pages.

The concept and application of Benford's Law have been examined a lot in the last 10 years or so, especially with regard to accounting forensics. There have been many papers written as to why Benford's Law is so prevalent and the concomitant reasons why(proofs). There are, unfortunately, many misconceptions such as the newly coined phrase "the Summation theorem", which states that if a data set conforms to Benford's Law then the sum of all numbers that begin with a particular digit (1,2,3,4,5,6,7,8,9) should be equal. Such is usually not the case. For exponential functions (y=aexp(x) it is but not for most other functions. I will show as to why this is the case. The distribution tends to be a Benford instead of a Uniform distribution.
Also, I will show that if the probability density function (pdf) of the logarithm of a data set begins and ends on the x axis and if the the values of the pdf between all integral powers of ten can be approximated with a straight line then the data set will tend to conform to Benford's Law.

**Category:** Number Theory

[2008] **viXra:1906.0121 [pdf]**
*submitted on 2019-06-07 08:28:35*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We recall a simple representation for Pi.

**Category:** Number Theory

[2007] **viXra:1906.0114 [pdf]**
*submitted on 2019-06-07 09:50:33*

**Authors:** Igor Hrnčić

**Comments:** 4 Pages.

This paper disproves the Riemann hypothesis by generalizing the results from Titchmarsh’s book The Theory of the Riemann Zeta-Function to rearrangements of conditionally convergent series that represent the reciprocal function of zeta. When one replaces the conditionally convergent series in Titchmarsh’s theorems and consequent proofs by its rearrangements, the left hand sides of equations change, but the right hand sides remain invariant. This contradiction disproves the Riemann hypothesis.

**Category:** Number Theory

[2006] **viXra:1906.0111 [pdf]**
*submitted on 2019-06-07 11:38:42*

**Authors:** Arthur Shevenyonov

**Comments:** 8 Pages. bridging

Some testing criteria or decision-procedures, notably when deployed as part of automated proving vehicles, might pose more of an AI threat than they do in terms of an opportunity leverage. In particular, tautology, unless rethought, will likely prove just that--irrelevant and inefficient. Mochizuki's IUT, referred to for benchmarking and illustration purposes, may well bear fruit beyond ABC if shown to be Teichmueller legacy-invariant.

**Category:** Number Theory

[2005] **viXra:1906.0103 [pdf]**
*submitted on 2019-06-07 23:49:45*

**Authors:** Franco Sabino Stoianoff Lindstron

**Comments:** 4 Pages.

The method used in this article is based on analytical geometry, abstract algebra and number theory.

**Category:** Number Theory

[2004] **viXra:1906.0069 [pdf]**
*submitted on 2019-06-05 23:59:34*

**Authors:** Sally Myers Moite

**Comments:** 8 Pages.

For the n-th prime P, P# or P primorial is the product of all the primes up to and including P. Let (c, d) be a pair of integers that represents a point in the primorial square, 1 < c, d < P#. For each prime p, 2 < p < P, the remainders of c and d mod p may be the same, opposite (sum to a multiple of p) or neither. Count the number of remainders of (c, d) which have same, opposite or either agreement for any such P. This gives three partitions of the primorial square, by counts for same, opposite and either agreement. Polynomial multiplication is used to find the number of points in each part of these partitions.

**Category:** Number Theory

[2003] **viXra:1906.0066 [pdf]**
*submitted on 2019-06-06 03:14:54*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 9 Pages.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other
words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin
prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid disproof for twin prime conjecture.

**Category:** Number Theory

[2002] **viXra:1906.0044 [pdf]**
*submitted on 2019-06-05 00:21:36*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

Con este algoritmo podrás encontrar el máximo común divisor de dos polinomios o de números complejos y por supuesto también de números naturales de una manera fácil.

**Category:** Number Theory

[2001] **viXra:1906.0042 [pdf]**
*submitted on 2019-06-05 01:16:23*

**Authors:** Pedro Hugo García Peláez

**Comments:** 6 Pages.

With this algorithm you can find the greatest common divisor of two polynomials or complex numbers and of course also natural numbers in an easy way.

**Category:** Number Theory

[2000] **viXra:1906.0028 [pdf]**
*submitted on 2019-06-03 18:13:17*

**Authors:** Bertrand Wong

**Comments:** 20 Pages.

This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in a journal of number theory.]

**Category:** Number Theory

[1999] **viXra:1906.0025 [pdf]**
*submitted on 2019-06-04 03:57:36*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 7 Pages. Submitted to the Ramanujan Journal. Comments welcome.

In this paper, using the recent result that $c<rad(abc)^2$, we will give the proof of the $abc$ conjecture for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\frac{1}{\epsilon^2} $. Some numerical examples are presented.

**Category:** Number Theory

[1998] **viXra:1906.0018 [pdf]**
*submitted on 2019-06-02 15:45:53*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The six seminal equations evaluated are not tautologous, refuting the subsequent claimed proof of the ABC conjecture, and forming a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[1997] **viXra:1906.0010 [pdf]**
*submitted on 2019-06-01 14:44:08*

**Authors:** Arthur Shevenyonov

**Comments:** 5 Pages. pre-ordual

While seeking to bypass the complex matching/ordering/comparability issue, the paper appears to have straddled areas seemingly as diverse as RH, Mikusinski operators, Euler equation for variations, and Veblen ordinals.

**Category:** Number Theory

[1996] **viXra:1905.0614 [pdf]**
*submitted on 2019-05-31 08:28:49*

**Authors:** Surajit Ghosh

**Comments:** 32 Pages.

Based on Eulers formula a concept of dually unit or d-unit circle is discovered. Continuing with, Riemann hypothesis is proved from diﬀerent angles, Zeta values are renormalised to remove the poles of Zeta function and relationships between numbers and primes is discovered. 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 without needing branch cuts. Pi can also be a base to natural logarithm and complement complex logarithm.Grand integrated scale is discovered which can reconcile the scale diﬀerence between very big and very small. Complex constants derived from complex logarithm following Goldbach partition theorem and Eulers Sum to product and product to unity can explain lot of mysteries in the universe.

**Category:** Number Theory

[1995] **viXra:1905.0584 [pdf]**
*submitted on 2019-05-29 09:05:21*

**Authors:** Henry Wong

**Comments:** 2 Pages.

An addendum to elementary number theory.

**Category:** Number Theory

[1994] **viXra:1905.0574 [pdf]**
*submitted on 2019-05-29 17:57:53*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

The root in partition Jensen polynomials for hyperbolicity is not tautologous. Hence its use to prove the Riemann hypothesis is denied. These conjectures form a non tautologous fragment of the universal logic VŁ4.

**Category:** Number Theory

[1993] **viXra:1905.0571 [pdf]**
*submitted on 2019-05-29 20:34:27*

**Authors:** Pedro Hugo García Peláez

**Comments:** Pages.

With this algorithm you can easily find the greatest common divisor of two numbers even with large numbers of figures and the same can be done if you want to find the greatest common divisor of polynomials easily and also complex numbers.

**Category:** Number Theory

[1992] **viXra:1905.0570 [pdf]**
*submitted on 2019-05-29 20:36:13*

**Authors:** Pedro Hugo García Peláez

**Comments:** 5 Pages.

Con este algoritmo podrás hallar fácilmente el máximo comun divisor de dos números incluso con gran cantidad de cifras y lo mismo podrás hacer si quieres hallar el máximo común divisor de polinomios fácilmente.

**Category:** Number Theory

[1991] **viXra:1905.0565 [pdf]**
*submitted on 2019-05-30 02:35:37*

**Authors:** Aryan Phadke

**Comments:** 12 Pages.

Set of Pythagorean triple consists of three values such that they comprise the three sides of a right angled triangle. Euclid gave a formula to find Pythagorean Triples for any given number. Motive of this paper is to find number of possible Pythagorean Triples for a given number. I have been able to provide a different proof for Euclid’s formula, as well as find the number of triples for any given number. Euclid’s formula is altered a little and is expanded with a variable ‘x’. When ‘x’ follows the conditions mentioned the result is always a Pythagorean Triple.

**Category:** Number Theory

[1990] **viXra:1905.0560 [pdf]**
*submitted on 2019-05-28 08:35:26*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give some infinite series for Pi.

**Category:** Number Theory

[1989] **viXra:1905.0559 [pdf]**
*submitted on 2019-05-28 08:38:47*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

This note presents some remarks on the equation: x^x-x-1=0,x>0

**Category:** Number Theory

[1988] **viXra:1905.0546 [pdf]**
*submitted on 2019-05-28 18:08:43*

**Authors:** Toshiro Takami

**Comments:** 50 Pages.

I considered Riemann’s hypothesis. At first, the purpose was to prove, but can not to prove.
It is written in the middle of the proof, but it can not been proved at all.
(The calculation formula is also written, but the real value 0.5 was not shown at all) The non-trivial zero values match perfectly in the formula of this paper.
However, the formula did not reach the real value 0.5.
In this case, it only reaches the pole near the real value 0.5.

**Category:** Number Theory

[1987] **viXra:1905.0502 [pdf]**
*submitted on 2019-05-25 17:36:55*

**Authors:** Gang tae geuk

**Comments:** 1 Page.

리만가설이란 ζ(s)=0를 만족하는 모든 자명하지 않은 근의 실수부는 0.5이라는 가설이다
이것을 존 비더셔와 데니스 헤이셜, 두분의 아이디어로 일반인에게 설명하기 위해 변형한 (이하 일반인을 위한 설명)으로 바꾼다면
"임의의 자연수를 골라 소인수분해했을때(1과 소수의 거듭제곱인 약수가 포함된 수는 제외한다) 약수의 개수가 짝수 또는 홀수일 확률은
0.5이다"라고 할수있다
여기서 나는 리만가설을 푸는것이 아니라 일반인을 위한 설명을 풀어낼것이다
그것은 곧 리만가설의 해결로 이어질것이다
이미 자명한 사실인 이항계수의 성질에 의하면 C(n,1)+C(n,3)+C(n,5)+...+(홀수번째 항의 계수의 합)=2^(n-1),C(n,0)+C(n,2)+C(n,4)+...+(짝수번째 항의 계수의 합)=2^(n-1)이다.
이를 언급하였던 일반인을 위한 설명에 사용할것이다
소수의 개수 = n일때 모든 수는 소수들의 중복을 허용한 조합으로 표현가능하다
만약 중복을 허용하지 않는다면 소수의 거듭제곱인 약수가 포함되지 않은 수들을 얻을수 있다.
우리는 이 숫자들에 전부 문자를 붙일것이다
이를 위 언급한 이항계수의 성질에 대입하고 n을 무한대로 보낸다면 홀수번째 항의 계수의 합은 계수가 홀수인 문자조합의 개수가 될것이고 짝수번째 항의 계수의 합은 계수가 짝수인 문자조합의 개수가 될것이다
이는
계수가 홀수인 문자조합의 계수 = 계수가 짝수인 문자조합의 계수+1이다
(오른쪽 항에 1을 더한 이유는 계수가 짝수인 문자조합의 계수 계산에 C(n,0)을 포함하지 않았기 때문이다)
라는 식을 얻을수 있다
결국 '약수의 개수가 홀수인경우가 짝수보다 1경우 많다'라는 사실을 알수있다
이로써 리만가설은 증명되었다
가족분들 감사드리고 선생님들 모두 감사드리고 내 친구들에게도 감사를 표한다

**Category:** Number Theory

[1986] **viXra:1905.0501 [pdf]**
*submitted on 2019-05-25 22:28:46*

**Authors:** Toshiro Takami

**Comments:** 2 Pages. for I am first.

I proved the Twin Prime Conjecture.\\
All Twin Prime are executed in hexadecimal notation. For example, it does not change in a huge number (forever huge number).\\
In a hexagonal diagram, (6n -1) and (6n+1), many are prime numbers.\\
Since the positive integers keep spinning around this hexagon forever, Twin Primes exist forever.
All Twin Prime numbers are consist in (6n -1) or (6n +1) (n is a positive integer).\\
All numbers are executed in hexadecimal notation. This does not change even in a huge number (forever huge number).\\

**Category:** Number Theory

[1985] **viXra:1905.0498 [pdf]**
*submitted on 2019-05-26 04:53:08*

**Authors:** Esteve J., Martinez J.E.

**Comments:** 6 Pages.

By using results obtained by Srinivāsa A. Rāmānujan (specifically in his paper A Proof of Bertrand's Postulate), we made a proof of Goldbach's Conjecture. A generalization of the conjecture is also proven for every natural not coprime with a natural m > 1 and greater or equal than 2m.

**Category:** Number Theory

[1984] **viXra:1905.0485 [pdf]**
*submitted on 2019-05-25 02:34:55*

**Authors:** Aryan Phadke

**Comments:** 10 Pages.

Sum of Harmonic Progression is an old problem. While a few complex approximations have surfaced, a simple and efficient formula hasn’t. Motive of the paper is to find a general formula for sum of harmonic progression without using ‘summation’ as a tool. This is an approximation for sum of Harmonic Progression for numerical terms. The formula was obtained by equating the areas of graphs of Harmonic Progression and curve of equation (y=1/x). Formula also has a variability that makes it more suitable for different users with different priorities in terms of accuracy and complexity.

**Category:** Number Theory

[1983] **viXra:1905.0468 [pdf]**
*submitted on 2019-05-23 19:26:05*

**Authors:** Bambore Dawit Geinamo

**Comments:** 9 Pages. If there is any correction and comment welcom

This paper magically shows very interesting and simple proof of Fermat’s Last Theorem. The proof identifies sufficient derivations of equations that holds the statement true and describes contradictions on them to satisfy the theorem. If Fermat had proof, his proof is most probably
similar to this one. The proof does not require any higher field of mathematics and it can be understood in high school level of mathematics. It
uses only modular arithmetic, factorization and some logical statements.

**Category:** Number Theory

[1982] **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

[1981] **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

[1980] **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

[1979] **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

[1127] **viXra:1908.0186 [pdf]**
*replaced on 2019-08-13 16:53:26*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist between successive primes, so that we can be sure that there is atleast one prime number between their squares.

**Category:** Number Theory

[1126] **viXra:1907.0558 [pdf]**
*replaced on 2019-07-29 10:29:00*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.

**Category:** Number Theory

[1125] **viXra:1907.0558 [pdf]**
*replaced on 2019-07-28 15:32:20*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is a proof of ∑(n=1,∞)(-1)^n=-1/2.

**Category:** Number Theory

[1124] **viXra:1907.0539 [pdf]**
*replaced on 2019-08-07 23:22:18*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

I discovered “mirror effect" of the Riemann zeta function on the critical line.
I investigated with many non-trivial zeros, the Riemann zeta function has a mirror image of about axis 1/2.
I named it, “mirror effect".
On non-trivial zero imaginary values, it is possible that one zero on the real value 1/2 or two with the real value 1/2 symmetrical may exist, but there is no possibility of the latter. Because “mirror effect" consist.
That is, on the non-trivial zero imaginary values, the real value 1/2 becomes the lowest absolute value which equal to zero, and the absolute real and imaginary value larger as it gets farther from the real value 1/2.
So, on non-trivial zero imaginary values, two with the real value 1/2 symmetrical can not exist, I conclude.

**Category:** Number Theory

[1123] **viXra:1907.0521 [pdf]**
*replaced on 2019-07-26 22:32:15*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.

**Category:** Number Theory

[1122] **viXra:1907.0303 [pdf]**
*replaced on 2019-07-19 23:41:18*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This relative formula shows The relationship between napier number e and π without imaginary unit i.

**Category:** Number Theory

[1121] **viXra:1907.0206 [pdf]**
*replaced on 2019-07-25 07:13:55*

**Authors:** Toshiro Takami

**Comments:** 39 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive.
We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5.
This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value.
We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.

**Category:** Number Theory

[1120] **viXra:1907.0206 [pdf]**
*replaced on 2019-07-24 01:12:26*

**Authors:** Toshiro Takami

**Comments:** 24 Pages.

In the Riemann zeta function, when the value of the nontrivial zero is zero, the value of the real part of the function is negative from 0 to 0.5, but the value of the real part of the function is 0.5 to 1 I found it to be positive.
We also found that the positive and negative of the imaginary part also interchanged with the real part 0.5.
This tendency is seen as a tendency near the non-trivial zero value, but becomes less and less as it deviates from the non-trivial zero value.
We present and discuss the case of four non-trivial zero values. This seems to be an important finding and will be announced here.

**Category:** Number Theory

[1119] **viXra:1907.0126 [pdf]**
*replaced on 2019-08-01 03:39:38*

**Authors:** Darrin Taylor

**Comments:** 68 Pages. Last step of proof broke so this is merely a new mathematical framework to attack Collatz type problems

A new form of mathematics is explored where a sequence of values are acted on by a set of rules (in this case the 3n+1 rules) and each digit within the values is acted on by a subordinate set of rules which produce the same values.
But the digit rules allow patterns to be identified and calculations to be performed on mostly unknown values.
Proves that loop length must be 13x + 18y
Proves that loop is made up of segments of 8 and 11 values and names the leading digits of each value in the segments.
Shows that base 4 descent is favored on average by a factor of 5.
Shows that if the base 4 upper digits were always even sequence would always eventually descend.
Possible future work may link the leading 0s which are infinitely even can be reflected to the least significant digits over time so that over infinity the effect approaches the always even which drives descent.
Predicts loop values based on most significant base 3 digit and show quantized loop leading digits and possible pattern of increasing smallest segments.
Predict general sequence based on least significant digit base 3.
Predict general sequence based on least significant digit base 4.

**Category:** Number Theory

[1118] **viXra:1907.0109 [pdf]**
*replaced on 2019-07-18 07:34:38*

**Authors:** Victor Sorokine

**Comments:** 4 Pages. Russian version

В ПЕРВОМ СЛУЧАЕ каждое число (А) заменяется на сумму (A'+A°n) последней цифры и остатка. После раскрытия биномов в равенстве Ферма все члены объединятся в два слагаемых: E=A'^n+B'^n-C'^n с третьей цифрой E''', которая в одном из n-1 эквивалентных равенств Ферма равна 2, и остаток D с третьей цифрой D''', равной либо 0, либо n-1, и, следовательно, третья цифра в числе A^n+B^n-C^n не равна 0. ВО ВТОРОМ СЛУЧАЕ (например A=A°n^k, но (BС)'≠0, ) после преобразования 3kn-значного окончания числа B в 1 и оставления в числах А, В, С лишь последних значащих цифр простейшие расчёты показывают, что (3kn-2)-я цифра числа A^n+B^n-C^n нулю не равна и не меняется после восстановления всех остальных цифр в числах A, B, C, т.к. является функцией только последней цифры числа A°.

**Category:** Number Theory

[1117] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-17 12:48:59*

**Authors:** Simon Plouffe

**Comments:** 77 Pages.

Conference held in Montréal at the ACA 2019, ETS.

**Category:** Number Theory

[1116] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-17 03:11:40*

**Authors:** Simon Plouffe

**Comments:** 77 Pages.

Conference in Montreal at the ACA 2019 (ETS) on July 17 2019.

**Category:** Number Theory

[1115] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-15 06:03:35*

**Authors:** Simon Plouffe

**Comments:** 75 Pages.

Pi, the primes and the Lambert W function, conference in Montréal at the ACA 2019. July 17.

**Category:** Number Theory

[1114] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-13 14:04:05*

**Authors:** Simon Plouffe

**Comments:** 76 Pages.

Pi, primes and the Lambert W function, conference in Montréal, July 17 2019 (update)

**Category:** Number Theory

[1113] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-10 02:23:10*

**Authors:** Simon Plouffe

**Comments:** 75 Pages.

This is a conference to be hold in Montréal on July 17, 2019.

**Category:** Number Theory

[1112] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-08 10:24:42*

**Authors:** Simon Plouffe

**Comments:** 71 Pages.

Pi the primes and the Lambert W function,
a conference to be hold in Montréal on July 17 2019.

**Category:** Number Theory

[1111] **viXra:1907.0108 [pdf]**
*replaced on 2019-07-07 09:28:39*

**Authors:** Simon Plouffe

**Comments:** 67 Pages.

Conference to be given in Montréal , july 17 2019.
The talk is in english.
Subject : Pi, the primes and the Lambert W function.

**Category:** Number Theory

[1110] **viXra:1907.0087 [pdf]**
*replaced on 2019-08-08 01:51:35*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
In this paper, we give a proof of mathematical expression.
“the non-trivial zero values of all positive infinity and negative infinity lie on the real value 0.5” I am here mathematically proved.

**Category:** Number Theory

[1109] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-25 07:11:39*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

In my previous paper “Consideration of the Riemann hypothesis” c=0.5 and x is non- trivial zero value, and it was described that it converges to almost 0, but a serious proof in mathematical expression could not be obtained.
In this paper, we give a proof of mathematical expression.
“the non-trivial zero values of all positive infinity and negative infinity lie on the real value 0.5” I am here mathematically proved.

**Category:** Number Theory

[1108] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-24 11:32:21*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1107] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-22 04:09:10*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1106] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-21 03:56:58*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1105] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-17 02:03:38*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1104] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-15 03:34:44*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

**Category:** Number Theory

[1103] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-10 02:18:50*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

**Category:** Number Theory

[1102] **viXra:1907.0087 [pdf]**
*replaced on 2019-07-06 16:10:13*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1101] **viXra:1906.0463 [pdf]**
*replaced on 2019-06-25 11:55:41*

**Authors:** Hung Tran

**Comments:** 5 Pages. Proof of the Riemann hypothesis using a Hamiltonian and a self-adjoint operator

We first find a Hamiltonian H that has the Hurwitz zeta functions ζ(s,x) as eigenfunctions. Then we continue constructing an operator G that is self-adjoint, with appropriate boundary conditions. We will find that the ζ(s,x)-functions do not meet these boundary conditions, except for the ones where s is a nontrivial zero of the Riemann zeta, with the real part of s being greater than 1/2. Finally, we find that these exceptional functions cannot exist, proving the Riemann hypothesis, that all nontrivial zeros have real part equal to 1/2.

**Category:** Number Theory

[1100] **viXra:1906.0418 [pdf]**
*replaced on 2019-06-23 17:48:55*

**Authors:** Pedro Hugo García Peláez

**Comments:** 4 Pages.

What I try to prove is if there are infinite number of Fibonacci and Lucas primes

**Category:** Number Theory

[1099] **viXra:1906.0391 [pdf]**
*replaced on 2019-08-05 10:36:01*

**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, that is, a structuring of the natural numbers starting from 3.

**Category:** Number Theory

[1098] **viXra:1906.0318 [pdf]**
*replaced on 2019-06-18 12:20:34*

**Authors:** Alan M. Gómez

**Comments:** 2 Pages.

Assuming the Riemann Hypothesis to be true, we propose an asymptotic and closed-form formula to find the imaginary part for non-trivial zeros of the Riemann Zeta Function.

**Category:** Number Theory

[1097] **viXra:1906.0195 [pdf]**
*replaced on 2019-06-13 07:12:16*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages. Additional comments and examples added.

The rational root test gives a means for determining if a root of a polynomial is rational. If none of the possible rational roots are roots, then if the roots are real, they must be irrational. Combining this observation with Taylor polynomials and the Taylor series for $\sin (x)$ gives intimations that $\pi$, and $e$, are likely irrational.

**Category:** Number Theory

[1096] **viXra:1906.0114 [pdf]**
*replaced on 2019-07-16 14:40:47*

**Authors:** Igor Hrnčić

**Comments:** 4 Pages. Rectified an obvious small error, sigma>1 instead of sigma>1/2, in the section Disproof of RH.

This paper disproves the Riemann hypothesis by generalizing the results from Titchmarsh's book The Theory of the Riemann Zeta-Function to rearrangements of conditionally convergent series that represent the reciprocal function of zeta. When one replaces the conditionally convergent series in Titchmarsh's theorems and consequent proofs by its rearrangements, the left hand sides of equations change, but the right hand sides remain invariant. This contradiction disproves the Riemann hypothesis.

**Category:** Number Theory

[1095] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-20 03:03:17*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 11 Pages.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other
words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin
prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid proof for twin prime conjecture.

**Category:** Number Theory

[1094] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-19 04:10:03*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 11 Pages.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other
words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin
prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid proof for twin prime conjecture.

**Category:** Number Theory

[1093] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-15 23:19:02*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 10 Pages.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid proof for twin prime conjecture.

**Category:** Number Theory

[1092] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-13 03:52:18*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 10 Pages.

**Category:** Number Theory

[1091] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-09 06:26:54*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 9 Pages.

**Category:** Number Theory

[1090] **viXra:1906.0066 [pdf]**
*replaced on 2019-08-07 09:08:00*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 9 Pages.

**Category:** Number Theory

[1089] **viXra:1906.0066 [pdf]**
*replaced on 2019-07-18 06:18:30*

**Authors:** K.H.K. Geerasee Wijesuriya

**Comments:** 09 Pages. Instead of previous paper, please upload this new version

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other
words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin
prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research
paper, my attempt is to provide a valid disproof for twin prime conjecture.

**Category:** Number Theory

[1088] **viXra:1905.0614 [pdf]**
*replaced on 2019-06-04 08:31:22*

**Authors:** Surajit Ghosh

**Comments:** 36 Pages.

Based on Eulers formula a concept of dually unit or d-unit circle is discovered. Continuing with, Riemann hypothesis is proved from diﬀerent angles, Zeta values are renormalised to remove the poles of Zeta function and relationships between numbers and primes is discovered. 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 without needing branch cuts. Pi can also be a base to natural logarithm and complement complex logarithm.Grand integrated scale is discovered which can reconcile the scale diﬀerence between very big and very small. Complex constants derived from complex logarithm following Goldbach partition theorem and Eulers Sum to product and product to unity can explain lot of mysteries in the universe.

**Category:** Number Theory

[1087] **viXra:1905.0546 [pdf]**
*replaced on 2019-07-30 20:11:59*

**Authors:** Toshiro Takami

**Comments:** 68 Pages.

I considered Riemann’s hypothesis. At first, the purpose was to prove, but can not to prove.
It is written in the middle of the proof, but it can not been proved at all.
(The calculation formula is also written, but the real value 0.5 was not shown at all) The non-trivial zero values match perfectly in the formula of this paper.
However, the formula did not reach the real value 0.5.
In this case, it only reaches the pole near the real value 0.5.

**Category:** Number Theory

[1086] **viXra:1905.0546 [pdf]**
*replaced on 2019-07-29 03:41:33*

**Authors:** Toshiro Takami

**Comments:** 68 Pages.

I considered Riemann’s hypothesis. At first, the purpose was to prove, but can not to prove.
It is written in the middle of the proof, but it can not been proved at all.
(The calculation formula is also written, but the real value 0.5 was not shown at all) The non-trivial zero values match perfectly in the formula of this paper.
However, the formula did not reach the real value 0.5.
In this case, it only reaches the pole near the real value 0.5.

**Category:** Number Theory

[1085] **viXra:1905.0546 [pdf]**
*replaced on 2019-07-10 02:46:08*

**Authors:** Toshiro Takami

**Comments:** 75 Pages.

**Category:** Number Theory

[1084] **viXra:1905.0546 [pdf]**
*replaced on 2019-07-07 02:21:12*

**Authors:** Toshiro Takami

**Comments:** 78 Pages. refined

**Category:** Number Theory

[1083] **viXra:1905.0546 [pdf]**
*replaced on 2019-06-26 17:52:06*

**Authors:** Toshiro Takami

**Comments:** 50 Pages. refined

**Category:** Number Theory

[1082] **viXra:1905.0546 [pdf]**
*replaced on 2019-06-24 23:32:14*

**Authors:** Toshiro Takami

**Comments:** 35 Pages. I polished it.

**Category:** Number Theory

[1081] **viXra:1905.0546 [pdf]**
*replaced on 2019-06-23 02:44:18*

**Authors:** Toshiro Takami

**Comments:** 55 Pages. It has been refined

**Category:** Number Theory

[1080] **viXra:1905.0546 [pdf]**
*replaced on 2019-06-06 22:39:33*

**Authors:** Toshiro Takami

**Comments:** 45 Pages.

**Category:** Number Theory

[1079] **viXra:1905.0502 [pdf]**
*replaced on 2019-05-30 09:52:05*

**Authors:** Gang tae geuk

**Comments:** 1 Page.

Riemann hypothesis means that satisfying ζ(s)=0(ζ(s) means Riemann Zeta function) unselfevidenceable root's part of true numbers are 1/2.
Dennis Hejhal, and John Dubisher explained this hypothesis to :
"Choosed Any natural numbers(exclude 1 and constructed with two or higher powered prime numbers) then the probability of numbers that choosed number's forming prime factor become an even number is 1/2."
I'll prove this explain to prove Riemann hypothesis indirectly.
In binomial coefficient, C(n,0)+C(n+1)+...+C(n,n)=2^n. And C(n,1)+C(n,3)+C(n,5)+...+C(n,n) and C(n,0)+C(n,2)+C(n,4)+...+C(n,n) is 2^(n-1).
If you pick up 8 prime numbers, then you can make numbers that exclude 1 and constructed with two or higher powered prime numbers, and the total amount of numbers that you made is 2^8.
Same principle, if you pick the numbers in k times(k is a variable), the total amount of numbers you made is C(8,k).
If k is an even number, the total amount of numbers you can make is C(8,0)+C(8,2)+...+C(8,8)-1(because we must exclude 1,same for C(8,0)), and as what i said, it equals to 2^(8-1)-1.
So, the probability of the numbers that forming prime factor's numbers is an even number is 2^(8-1)-1/2^8
If there are amount of prime numbers exist, and we say that amount to n(n is a variable, as the k so), and sequence of upper works sameas we did, so the probability is 2^(n-1)/2^n.
If you limits n to inf, then probability convergents to 1/2.
This answer coincident with the explain above, so explain is established, same as the Riemann hypothesis is.

**Category:** Number Theory

[1078] **viXra:1905.0501 [pdf]**
*replaced on 2019-08-02 02:53:06*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

I proved the Twin Prime Conjecture.\\
All Twin Prime are executed in hexadecimal notation. For example, it does not change in a huge number (forever huge number).\\
In a hexagonal diagram, (6n -1) and (6n+1), many are prime numbers.\\
Since the positive integers keep spinning around this hexagon forever, Twin Primes exist forever.
All Twin Prime numbers are consist in (6n -1) or (6n +1) (n is a positive integer).\\
All numbers are executed in hexadecimal notation. This does not change even in a huge number (forever huge number).\\

**Category:** Number Theory

[1077] **viXra:1905.0501 [pdf]**
*replaced on 2019-07-22 01:24:56*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

I proved the Twin Prime Conjecture.\\
All Twin Prime are executed in hexadecimal notation. For example, it does not change in a huge number (forever huge number).\\
In a hexagonal diagram, (6n -1) and (6n+1), many are prime numbers.\\
Since the positive integers keep spinning around this hexagon forever, Twin Primes exist forever.
All Twin Prime numbers are consist in (6n -1) or (6n +1) (n is a positive integer).\\
All numbers are executed in hexadecimal notation. This does not change even in a huge number (forever huge number).\\

**Category:** Number Theory

[1076] **viXra:1905.0501 [pdf]**
*replaced on 2019-07-15 15:11:14*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

**Category:** Number Theory

[1075] **viXra:1905.0501 [pdf]**
*replaced on 2019-06-29 17:53:42*

**Authors:** Toshiro Takami

**Comments:** 8 Pages.

**Category:** Number Theory

[1074] **viXra:1905.0501 [pdf]**
*replaced on 2019-06-19 04:55:57*

**Authors:** Toshiro Takami

**Comments:** 4 Pages. It has been refined

**Category:** Number Theory

[1073] **viXra:1905.0501 [pdf]**
*replaced on 2019-06-16 02:36:29*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

**Category:** Number Theory

[1072] **viXra:1905.0501 [pdf]**
*replaced on 2019-06-14 02:39:50*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

**Category:** Number Theory

[1071] **viXra:1905.0498 [pdf]**
*replaced on 2019-07-24 13:26:15*

**Authors:** Esteve J., Martinez J.E.

**Comments:** 7 Pages. Minor corrections were made on the latest version.

We proof Goldbach's Conjecture. We use results obtained by Srinivasa A. Ramanujan (specifically in his paper A Proof of Bertrand's Postulate). A generalization of the conjecture is also proven for every natural not coprime with a natural m > 1 and greater or equal than 2m.

**Category:** Number Theory

[1070] **viXra:1905.0498 [pdf]**
*replaced on 2019-07-20 20:38:20*

**Authors:** Esteve J., Martinez J. E.

**Comments:** 6 Pages.

We proof Goldbach's Conjecture. We use results obtained by Srinivāsa A. Rāmānujan (specifically in his paper A Proof of Bertrand's Postulate). A generalization of the conjecture is also proven for every natural not coprime with a natural m > 1 and greater or equal than 2m.

**Category:** Number Theory

[1069] **viXra:1905.0498 [pdf]**
*replaced on 2019-05-30 06:22:41*

**Authors:** Esteve J., Martinez J. E.

**Comments:** 6 Pages. A correction in the logical argumentation of the main theorem was made.

We proof Goldbach's Conjecture. We use results obtained by Srinivāsa A. Rāmānujan (specifically in his paper A Proof of Bertrand's Postulate). A generalization of the conjeture is also proven for every natural not coprime with a natural m > 1 and greater or equal than 2m.

**Category:** Number Theory

[1068] **viXra:1905.0468 [pdf]**
*replaced on 2019-05-28 18:53:14*

**Authors:** Bambore Dawit Geinamo

**Comments:** 9 Pages.

This paper magically shows very interesting and simple proof of Fermat’s Last Theorem. The proof identifies sufficient derivations of equations that holds the statement true and describes contradictions on them
to satisfy the theorem. If Fermat had proof, his proof is most probably
similar to this one. The proof does not require any higher field of mathematics and it can be understood in high school level of mathematics. It uses only modular arithmetic, factorization and some logical statements.

**Category:** Number Theory

[1067] **viXra:1905.0365 [pdf]**
*replaced on 2019-08-02 13:21:54*

**Authors:** Emmanuil Manousos

**Comments:** 22 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, an algorithm for the factorization of Fermat numbers is derived. Also, we determine a sequence of prime numbers, and we prove an essential corollary for the composite Mersenn numbers.

**Category:** Number Theory

[1066] **viXra:1905.0365 [pdf]**
*replaced on 2019-05-26 05:54:55*

**Authors:** Emmanuil Manousos

**Comments:** 21 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