# Number Theory

## 1907 Submissions

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

### Simplified Research for the Constellation of All Roots of Dirichlet Eta Function in Critical Strip

Authors: Leonid Vakhov

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

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

### Prime Numbers and Twin Prime Numbers Algorithm.

Authors: Zeolla Gabriel Martin

: 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

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

### An Exact Formula for the Prime Counting Function

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

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

### On the Limits of a Generalized Harmonic Progression

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

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

### Complex Harmonic Progression

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

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

### Generalized Harmonic Progression

Authors: Jose R. Sousa

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

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

### Proof of ∑(n=1,∞)(-1)^n=-1/2

Authors: Yuji Masuda

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

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

### Hypergeometric Function Identities pi

Authors: Edgar Valdebenito

We give some identities for Pi.
Category: Number Theory

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

### Infinity and Pythagorean Theorem

Authors: Yuji Masuda

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.
Category: Number Theory

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

### Infinity and Pythagorean Theory

Authors: Yuji Masuda

Introducing infinity into the Pythagorean theorem provides the Pythagorean theorem even for triangles that are not right triangles.
Category: Number Theory

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

### Analytic Continuation of the Zeta Function Violates the Law of Non-Contradiction (LNC)

Authors: Ayal Sharon

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

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

### Values of the Riemann Zeta Function by Means of Division by Zero Calculus

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

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

### 多与少直接推翻黎曼猜想

Authors: Aaron chau

Category: Number Theory

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

### Suggestion to Study Number Theory of Non-Decimal Number Systems

Authors: Jian-ping Gu

This paper suggests extending the studies of number theory to non-decimal number systems.
Category: Number Theory

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

### Equation of Zero

Authors: Yuji Masuda

First, ±∞ is constant at any observation point (position).
Category: Number Theory

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

### Un Millón de Precisiones en Torno a pi

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

Category: Number Theory

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

### The Goldbach Theorem

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

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

### Natural Numbers and Their Square Roots Expressed by Constant Phi and 1

Authors: Harry K. Hahn

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

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

### The Ordered Distribution of Natural Numbers on the Square Root Spiral

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

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

### The Distribution of Prime Numbers on the Square Root Spiral

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

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

### About the Logic of the Prime Number Distribution

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

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

### The Distribution of Natural Numbers Divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral

Authors: Harry K. Hahn

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

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

### On: Z*z*z-7*z*z+3*z-1=0

Authors: Edgar Valdebenito

We give a formula for Pi.
Category: Number Theory

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

### Relative Formula (Relationship Between e and π Without i)

Authors: Yuji Masuda

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

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

### Disproof of the Riemann Hypothesis, Long Version

Authors: Igor Hrnčić

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

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

### The Abc Conjecture as Expansion of Powers of Binomials

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

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

### On Non-Trivial Zero Point

Authors: Toshiro Takami

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

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

### Algorithms for Testing Prime Factors Against Positive Composite Numbers (Finding the Unique Factorization Domain for Said Composite Numbers) in Base 10: a First Course Into Formal Mathematics

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

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

### Proof of Riemann Hypothesis and Other Prime Conjectures

Authors: Surajit Ghosh

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

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

### A Concise Paper Summarizing a Contradiction with the Riemann Hypothesis

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

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

### Collatz Conjecture a New Mathematical Approach.

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

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

### Fermat's Last Theorem. Unified Method (Russian)

Authors: Victor Sorokine

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

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

### Pi, the Primes and the Lambert W Function

Authors: Simon Plouffe

Conference held in Montréal at the ACA 2019, ETS.
Category: Number Theory

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

### A Technical Procedure for the Riemann Hypothesis

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

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

### A Minor Typographical Error in Paper "A Technical Procedure for the Riemann Hypothesis"

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

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

### An Minor Typographical Correction to Paper Riemann Hypothesis

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

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

### Polynomials of the Form P_n^{(a)}(x)=(1/2)*((x-Sqrt{x^2+a})^n+(x+sqrt{x^2+a})^n) and Primality Testing

Authors: Predrag Terzic

General,deterministic,unconditional,polynomial time primality test is introduced.
Category: Number Theory

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

### Decomposition of Integer Powers of a Mersenne Number Into Binomial Coefficients

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

Decomposition of integer powers of a mersenne number into binomial coefficients
Category: Number Theory

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

### Integrals - Identities - Pi

Authors: Edgar Valdebenito

We give some integrals for Pi.
Category: Number Theory

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

### Two Identities

Authors: Edgar Valdebenito

This note presents two identities for Pi.
Category: Number Theory

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

### Disproof of the Riemann Hypothesis

Authors: Toshiro Takami