**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(13) - 1406(18) - 1407(33) - 1408(50) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(17) - 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(16) - 1811(25) - 1812(29)

2019 - 1901(12) - 1902(11) - 1903(21) - 1904(25) - 1905(23) - 1906(43) - 1907(42) - 1908(21) - 1909(27)

Any replacements are listed farther down

[2120] **viXra:1909.0473 [pdf]**
*submitted on 2019-09-23 00:57:25*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

I published the odd value formula for ζ, but I realized that this was true even when it was even.
Therefore, it will be announced.

**Category:** Number Theory

[2119] **viXra:1909.0461 [pdf]**
*submitted on 2019-09-21 12:41:29*

**Authors:** Julia Beauchamp

**Comments:** 3 Pages.

In this paper, we ask whether a heuristic test for prime numbers can be derived from the Fibonacci numbers. The results below test for values up to $F_{75}$ show that we might have a heuristic test for prime numbers akin to Fermat's Little Theorem.

**Category:** Number Theory

[2118] **viXra:1909.0456 [pdf]**
*submitted on 2019-09-22 02:26:58*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 79 Pages.

In the present research thesis, we have obtained various and interesting new mathematical connections concerning the fundamental Ramanujan’s formula to obtain a highly precise golden ratio, some sectors of Particle Physics and Black Holes entropies.

**Category:** Number Theory

[2117] **viXra:1909.0385 [pdf]**
*submitted on 2019-09-18 20:47:21*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

I tried to find a new expression for zeta odd-numbers.
It may be a new expression and will be published here.
The correctness of this formula was confirmed by WolframAlpha to be numerically com-
pletely correct.

**Category:** Number Theory

[2116] **viXra:1909.0384 [pdf]**
*submitted on 2019-09-18 21:28:20*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

ζ(4), ζ(6).......ζ(108), ζ(110) considered.
From these equations, it can be said that ζ(4),ζ(6).......ζ(108),ζ(110) are irrational numbers.
ζ(112),ζ(114) etc. can also be expressed by these equations.
Because I use π2, these are to be irrational numbers.
The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of
ζ(2n) is multiplied by π2.

**Category:** Number Theory

[2115] **viXra:1909.0378 [pdf]**
*submitted on 2019-09-19 04:18:29*

**Authors:** Miguel Cerdá Bennassar

**Comments:** 34 Pages.

Propongo una tabla numérica en la que se demuestra visualmente que las secuencias formadas con el algoritmo de Collatz acaban siempre en el número 1.

**Category:** Number Theory

[2114] **viXra:1909.0370 [pdf]**
*submitted on 2019-09-17 13:19:02*

**Authors:** Miika Rankaviita

**Comments:** 20 Pages. Licencing: CC BY-SA

This thesis explains the solution to the problem of finding all of the integer pair solutions to the equation x!+x=x^n.
A detailed explanation is given so that anyone with high school mathematics background can follow the solution.
This paper is a translation of my diplom work in Vaasa Lyseo Upper Secondary School.

**Category:** Number Theory

[2113] **viXra:1909.0337 [pdf]**
*submitted on 2019-09-17 00:13:09*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages. We give another proof of the conjecture c

In this paper, we consider the $abc$ conjecture. Firstly, we give anelementaryproof the conjecture $c<rad^2(abc)$. Secondly, the proof of the $abc$ conjecture is given for $\epsilon \geq 1$, then for $\epsilon \in ]0,1[$. We choose the constant $K(\epsilon)$ as $K(\epsilon)=e^{\left(\frac{1}{\epsilon^2} \right)}$. Some numerical examples are presented.

**Category:** Number Theory

[2112] **viXra:1909.0334 [pdf]**
*submitted on 2019-09-17 02:04:19*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we propose the axiomatic regularity of prime numbers.

**Category:** Number Theory

[2111] **viXra:1909.0315 [pdf]**
*submitted on 2019-09-15 23:09:11*

**Authors:** Toshiro Takami

**Comments:** 24 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers.
ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement.
This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.

**Category:** Number Theory

[2110] **viXra:1909.0312 [pdf]**
*submitted on 2019-09-14 06:51:50*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

With help of theta functions, a generalization of the functional equation of the zeta Riemann function can be defined.

**Category:** Number Theory

[2109] **viXra:1909.0305 [pdf]**
*submitted on 2019-09-14 13:53:36*

**Authors:** Michele Nardelli, Antonio Nardelli

**Comments:** 196 Pages.

In the present research thesis, we have obtained various interesting new possible mathematical connections between the Ramanujan Modular Equations, Class Invariants, the Mock Theta Functions, some particle-like solutions, Black Holes entropies, ζ(2) and Golden Ratio

**Category:** Number Theory

[2108] **viXra:1909.0299 [pdf]**
*submitted on 2019-09-15 01:39:20*

**Authors:** Natalino Sapere

**Comments:** 9 Pages. None

This paper explains the Collatz Conjecture through the use of recursive functions.

**Category:** Number Theory

[2107] **viXra:1909.0297 [pdf]**
*submitted on 2019-09-15 02:21:15*

**Authors:** Wei Zhang

**Comments:** 5 Pages.

This paper gives the definition and nature of Φ(m) function, as well as the relationship between Φ(m) and Euler’s totient function φ(m). In number theory, Euler function φ(m) is widely used, Φ(m) function if there are other applications, also not clear.

**Category:** Number Theory

[2106] **viXra:1909.0295 [pdf]**
*submitted on 2019-09-15 05:25:01*

[2105] **viXra:1909.0285 [pdf]**
*submitted on 2019-09-13 19:27:39*

**Authors:** Alfredo Olmos, R. Romyna Olmos

**Comments:** 7 Pages.

In this article we study some characteristics of polygonal numbers, which are the positive integers that can be ordered, to form a regular polygon.
The article is closed, showing the relation of the polygonal numbers, with the Beta function when expressing any polygonal number, as a sum of terms of the Beta function.

**Category:** Number Theory

[2104] **viXra:1909.0232 [pdf]**
*submitted on 2019-09-10 23:51:37*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

I wondered why I would do this and I traced the course. However, I did not understand much.
Numerically, they match perfectly.
Also, this is considered to indicate that ζ(5), ζ(7), ζ(9), ζ(11), ζ(13)........ζ(197), ζ(199) are irrational numbers. Because, ζ(3) is irrational number.
It can also be said that it is expressed by an expression using π2.

**Category:** Number Theory

[2103] **viXra:1909.0178 [pdf]**
*submitted on 2019-09-08 12:33:13*

**Authors:** Shekhar Suman

**Comments:** 5 Pages.

Analytic continuation by hadamard product is strictly monotonic which implies RH

**Category:** Number Theory

[2102] **viXra:1909.0165 [pdf]**
*submitted on 2019-09-09 05:17:04*

**Authors:** Sitangsu Maitra

**Comments:** 3 page

Proof of Goldbach's strong conjecture in a different way

**Category:** Number Theory

[2101] **viXra:1909.0154 [pdf]**
*submitted on 2019-09-07 13:41:13*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is on primes3.

**Category:** Number Theory

[2100] **viXra:1909.0103 [pdf]**
*submitted on 2019-09-05 18:48:43*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This is on primes.

**Category:** Number Theory

[2099] **viXra:1909.0059 [pdf]**
*submitted on 2019-09-03 23:11:41*

**Authors:** Ayal Sharon

**Comments:** 32 Pages. Approx. 7500 words, and approx. 130 references in the bibliography

Riemann's "analytic continuation" produces a second definition of the Zeta function, that Riemann claimed is convergent throughout half-plane $s \in \mathbb{C}$, $\text{Re}(s)\le1$, (except at $s=1$). This contradicts the original definition of the Zeta function (the Dirichlet series), which is proven divergent there. Moreover, a function cannot be both convergent and divergent at any domain value. In other mathematics conjectures and assumed-proven theorems, and in physics, the Riemann Zeta function (or the class of $L$-functions that generalizes it) is assumed to be true. Here the author shows that the two contradictory definitions of Zeta violate Aristotle's Laws of Identity, Non-Contradiction, and Excluded Middle. The of Non-Contradiction is an axiom of classical and intuitionistic logics, and an inherent axiom of Zermelo-Fraenkel set theory (which was designed to avoid paradoxes). If Riemann's definition of Zeta is true, then the Zeta function is a contradiction that causes deductive "explosion", and the foundation logic of mathematics must be replaced with one that is paradox-tolerant. If Riemann's Zeta is false, it renders unsound all theorems and conjectures that falsely assume that it is true. Riemann's Zeta function appears to be false, because its derivation uses the Hankel contour, which violates the preconditions of Cauchy's integral theorem.

**Category:** Number Theory

[2098] **viXra:1909.0038 [pdf]**
*submitted on 2019-09-02 12:25:38*

**Authors:** Shekhar Suman

**Comments:** 5 Pages.

Modulus of Hadamard product is shown increasing which proves the Riemann Hypothesis

**Category:** Number Theory

[2097] **viXra:1909.0027 [pdf]**
*submitted on 2019-09-01 12:06:47*

**Authors:** Francis Maleval

**Comments:** 1 Page.

The sieve of the addition of two prime numbers and the sieve of the product of two natural numbers are linked by a paradox of symmetrical objects. Goldbach's conjecture, additive version of a property of primes, would then have no chance being demonstrated if its multiplicative alter ego remained impenetrable to the disorder of prime numbers.

**Category:** Number Theory

[2096] **viXra:1909.0019 [pdf]**
*submitted on 2019-09-01 21:24:11*

**Authors:** Zeolla Gabriel Martín

**Comments:** 4 Pages.

This document exposes the construction of infinite patterns for prime numbers smaller than P.
In this case, the pattern for prime numbers less than 11 is graphic.

**Category:** Number Theory

[2095] **viXra:1909.0013 [pdf]**
*submitted on 2019-09-02 04:05:12*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

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.
The real value is 0 only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero.

**Category:** Number Theory

[2094] **viXra:1909.0010 [pdf]**
*submitted on 2019-09-01 01:13:44*

**Authors:** Kurmet Sultan

**Comments:** 1 Page. This Russian version of the article.

The article reports on the new patterns of modular arithmetic.

**Category:** Number Theory

[2093] **viXra:1908.0617 [pdf]**
*submitted on 2019-08-30 17:11:23*

**Authors:** Francis Maleval

**Comments:** 1 Page.

Le crible de l’addition de deux nombres premiers et le crible du produit de deux nombres naturels sont liés par un paradoxe d’objets symétriques. La conjecture de Goldbach, version additive d’une propriété des premiers, n’aurait alors aucune chance d’être un jour démontrée si son alter ego multiplicatif demeurait également impénétrable au désordre, voire au chaos des nombres premiers.

**Category:** Number Theory

[2092] **viXra:1908.0614 [pdf]**
*submitted on 2019-08-31 04:36:55*

**Authors:** Galeotti Giuseppe

**Comments:** 2 Pages.

the C ensemble is considered close in all the operations but if you divide a number by 0 you will not get a complex number

**Category:** Number Theory

[2091] **viXra:1908.0586 [pdf]**
*submitted on 2019-08-28 08:36:03*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

We give some Fourier Series - Identities.

**Category:** Number Theory

[2090] **viXra:1908.0585 [pdf]**
*submitted on 2019-08-28 08:40:22*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

We recall a Ramanujan's integral: int(f(x),x=0..1)=(pi*pi)/15

**Category:** Number Theory

[2089] **viXra:1908.0568 [pdf]**
*submitted on 2019-08-29 06:21:02*

**Authors:** Yuji Masuda

**Comments:** 1 Page.

This study focuses on primes.

**Category:** Number Theory

[2088] **viXra:1908.0527 [pdf]**
*submitted on 2019-08-27 04:14:11*

**Authors:** Shekhar Suman

**Comments:** 4 Pages.

ANALYTIC CONTINUATION AND SIMPLE APPLICATION OF ROLLE'S THEOREM

**Category:** Number Theory

[2087] **viXra:1908.0474 [pdf]**
*submitted on 2019-08-24 02:25:01*

**Authors:** Shekhar suman

**Comments:** 11 Pages. Please send replies at shekharsuman068@gmail.com

Analytic continuation and monotonicity gives the zeroes

**Category:** Number Theory

[2086] **viXra:1908.0427 [pdf]**
*submitted on 2019-08-20 13:26:39*

**Authors:** Shekhar Suman

**Comments:** 9 Pages. Please read once

We take the integral representation of the Riemann Zeta Function over entire complex plane, except for a pole at 1.
Later we draw an equivalent to the Riemann Hypothesis by studying its monotonicity properties.

**Category:** Number Theory

[2085] **viXra:1908.0424 [pdf]**
*submitted on 2019-08-20 15:10:57*

**Authors:** Shekhar Suman

**Comments:** 7 Pages.

Analytical continuation gives a functional equation having nice properties. Further we give an equivalence of riemann hypotheis through its monotonicity in specific intervals

**Category:** Number Theory

[2084] **viXra:1908.0420 [pdf]**
*submitted on 2019-08-21 05:02:25*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages. Comments welcome. Submitted to the Ramanujan Journal.

In this paper, we consider the abc conjecture. As the conjecture c<rad^2(abc) is less open, we give firstly the proof of a modified conjecture that is c<2rad^2(abc). The factor 2 is important for the proof of the new conjecture that represents the key of the proof of the main conjecture. Secondly, the proof of the abc conjecture is given for \epsilon \geq 1, then for \epsilon \in ]0,1[. We choose the constant K(\epsion) as K(\epsilon)=2e^{\frac{1}{\epsilon^2} } for $\epsilon \geq 1 and K(\epsilon)=e^{\frac{1}{\epsilon^2}} for \epsilon \in ]0,1[. Some numerical examples are presented.

**Category:** Number Theory

[2083] **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

[2082] **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

[2081] **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

[2080] **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

[2079] **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

[2078] **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

[2077] **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

[2076] **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

[2075] **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

[2074] **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

[2073] **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°.

[2072] **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

[2071] **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

[2070] **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

[2069] **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

[2068] **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

[2067] **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

[2066] **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

[2065] **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

[2064] **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

[2063] **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

[2062] **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

[2061] **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

[2060] **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

[2059] **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

[2058] **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

[2057] **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

[2056] **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

[2055] **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

[2054] **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

[2053] **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

[2052] **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

[2051] **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

[2050] **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

[2049] **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

[2048] **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

[2047] **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

[2046] **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

[2045] **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

[2044] **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

[2043] **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

[2042] **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

[2041] **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

[2040] **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

[2039] **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

[2038] **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

[2037] **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

[2036] **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

[2035] **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

[2034] **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

[2033] **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

[2032] **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

[2031] **viXra:1906.0570 [pdf]**
*submitted on 2019-06-30 18:22:13*

**Authors:** Akalabu, Emmanuel Chukwuemeka

**Comments:** 8 Pages.

--

**Category:** Number Theory

[2030] **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

[2029] **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

[2028] **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

[2027] **viXra:1906.0498 [pdf]**
*submitted on 2019-06-27 08:44:40*

**Authors:** Nurlan Qasimli

**Comments:** 6 Pages.

History of conjecture

**Category:** Number Theory

[2026] **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

[2025] **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

[2024] **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

[2023] **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

[2022] **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

[2021] **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

[2020] **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

[2019] **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

[2018] **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

[2017] **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

[2016] **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

[2015] **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

[2014] **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

[2013] **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

[2012] **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

[2011] **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

[2010] **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

[2009] **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

[2008] **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

[1153] **viXra:1909.0385 [pdf]**
*replaced on 2019-09-22 00:51:56*

**Authors:** Toshiro Takami

**Comments:** 11 Pages.

I tried to find a new expression for zeta odd-numbers.
It may be a new expression and will be published here.
The correctness of this formula was confirmed by WolframAlpha to be numerically com-
pletely correct.

**Category:** Number Theory

[1152] **viXra:1909.0384 [pdf]**
*replaced on 2019-09-23 03:33:35*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

ζ(4), ζ(6).......ζ(108), ζ(110) considered.
From these equations, it can be said that ζ(4),ζ(6).......ζ(108),ζ(110) are irrational numbers.
ζ(112),ζ(114) etc. can also be expressed by these equations.
Because I use π2, these are to be irrational numbers.
The fact that the even value of ζ(2n) is irrational can also be explained by the fact that each even value of
ζ(2n) is multiplied by π2.

**Category:** Number Theory

[1151] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-20 03:32:04*

**Authors:** Toshiro Takami

**Comments:** 37 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers.
ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement.
This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.

**Category:** Number Theory

[1150] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-19 03:16:57*

**Authors:** Toshiro Takami

**Comments:** 35 Pages.

Using the fact that ζ(3) is an irrational number, I prove that ζ(5), ζ(7)...........ζ(331) and ζ(333) are irrational numbers.
ζ(5), ζ(7)...........ζ(331) and ζ(333) are confirmed that they were in perfect numerical agreement.
This is because I created an odd-number formula for ζ, and the formula was created by dividing the odd- number for ζ itself into odd and even numbers.

**Category:** Number Theory

[1149] **viXra:1909.0315 [pdf]**
*replaced on 2019-09-17 08:58:39*

**Authors:** Toshiro Takami

**Comments:** 38 Pages.

**Category:** Number Theory

[1148] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-15 21:52:30*

**Authors:** Toshiro Takami

**Comments:** 23 Pages.

I wondered why I would do this and I traced the course. However, I did not understand much.
Numerically, they match perfectly.
Also, this is considered to indicate that ζ(5), ζ(7), ζ(9), ζ(11), ζ(13)........ζ(197), ζ(199) are irrational numbers. Because, ζ(3) is irrational number.
It can also be said that it is expressed by an expression using π2.

**Category:** Number Theory

[1147] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-15 03:48:17*

**Authors:** Toshiro Takami

**Comments:** 25 Pages.

I wondered why I would do this and I traced the course. However, I did not understand much.
Numerically, they match perfectly.
Also, this is considered to indicate that ζ(5), ζ(7), ζ(9), ζ(11), ζ(13)........ζ(197), ζ(199) are irrational numbers. Because, ζ(3) is irrational number.
It can also be said that it is expressed by an expression using π2.

**Category:** Number Theory

[1146] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-14 01:48:36*

**Authors:** Toshiro Takami

**Comments:** 21 Pages.

**Category:** Number Theory

[1145] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-13 05:58:37*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

**Category:** Number Theory

[1144] **viXra:1909.0232 [pdf]**
*replaced on 2019-09-11 09:35:20*

**Authors:** Toshiro Takami

**Comments:** 12 Pages.

**Category:** Number Theory

[1143] **viXra:1909.0165 [pdf]**
*replaced on 2019-09-11 01:57:47*

**Authors:** Sitangsu Maitra

**Comments:** 4 pages

Proof of Goldbach's strong conjecture in a different way

**Category:** Number Theory

[1142] **viXra:1909.0013 [pdf]**
*replaced on 2019-09-07 05:20:27*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

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.
The real value is 0 only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero.

**Category:** Number Theory

[1141] **viXra:1909.0013 [pdf]**
*replaced on 2019-09-06 04:56:06*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

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.
The real value is 0 only when the real part of s is 1/2. Non-trivial zeros must always have a real value of zero.

**Category:** Number Theory

[1140] **viXra:1909.0013 [pdf]**
*replaced on 2019-09-05 04:47:51*

**Authors:** Toshiro Takami

**Comments:** 5 Pages.

**Category:** Number Theory

[1139] **viXra:1909.0013 [pdf]**
*replaced on 2019-09-03 00:03:42*

**Authors:** Toshiro Takami

**Comments:** 7 Pages.

**Category:** Number Theory

[1138] **viXra:1908.0302 [pdf]**
*replaced on 2019-08-29 03:23:02*

**Authors:** Kouider Mohammed Ridha

**Comments:** 3 Pages.

According to Josephuse history we present a new numbers called The josephuse numbers. Hence we give explicit formulas to compute the Josephus-numbers J(n)where n is positive integer . Furthermore we present a new fast algorithm to calculate J(n). 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

[1137] **viXra:1908.0191 [pdf]**
*replaced on 2019-08-31 16:21:49*

**Authors:** Toshiro Takami

**Comments:** 4 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

[1136] **viXra:1908.0191 [pdf]**
*replaced on 2019-08-29 03:52:28*

**Authors:** Toshiro Takami

**Comments:** 6 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

[1135] **viXra:1908.0191 [pdf]**
*replaced on 2019-08-27 21:20:01*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

**Category:** Number Theory

[1134] **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

[1133] **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

[1132] **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

[1131] **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

[1130] **viXra:1907.0378 [pdf]**
*replaced on 2019-09-22 12:08:14*

**Authors:** Horacio Useche Losada

**Comments:** 33 Pages.

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

[1129] **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

[1128] **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

[1127] **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

[1126] **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

[1125] **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

[1124] **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

[1123] **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

[1122] **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

[1121] **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

[1120] **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

[1119] **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

[1118] **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

[1117] **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

[1116] **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

[1115] **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

[1114] **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