**Previous months:**

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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(3)

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2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(10) - 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(27) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(52) - 1409(47) - 1410(17) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(35) - 1504(23) - 1505(19) - 1506(13) - 1507(16) - 1508(15) - 1509(15) - 1510(13) - 1511(9) - 1512(26)

2016 - 1601(14) - 1602(18) - 1603(78) - 1604(56) - 1605(29) - 1606(19) - 1607(24) - 1608(17)

Any replacements are listed further down

[1311] **viXra:1608.0390 [pdf]**
*submitted on 2016-08-28 19:28:48*

**Authors:** Lucas Allen

**Comments:** English, 4 pages, ideas and examples

This paper presents a method of calculating powers and sums of powers using binomial coefficients. The method involves finding analogues of Pascal's triangle for each power and then showing that powers and sums of powers are the sums of binomial coefficients multiplied by constants. The constants are unique for each power. This paper presents a general idea and not a formal proof.

**Category:** Number Theory

[1310] **viXra:1608.0376 [pdf]**
*submitted on 2016-08-28 05:41:42*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

First, we prove the relation of the sum of the mobius function and Riemann Hypothesis. This relationship is well known. I prove next section, without no tool we prove Riemann Hypothesis about mobius function. This is very chalenging attempt.And at ﬁnal,we reach skeptical answer.

**Category:** Number Theory

[1309] **viXra:1608.0375 [pdf]**
*submitted on 2016-08-28 00:35:34*

**Authors:** Nathan Sponder

**Comments:** 10 Pages.

We discuss the asymptotics of the sum $\sum_{k=1}^{m} e^{ \frac{{\ln(k)}^n}{k} }-1$ for $n \geq 0 $. Our main interest is to show the asymptotics of this sum and show expressions for the constants tied to the asymptotics of the sum as well as in particular show the properties of the constants associated with the sum.

**Category:** Number Theory

[1308] **viXra:1608.0367 [pdf]**
*submitted on 2016-08-26 21:10:58*

**Authors:** Raymond Aschheim, Carlos Castro Perelman, Klee Irwin

**Comments:** 28 Pages. Submitted to IJGMMP

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential,
and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the $asymptotic$
(very large $N$) region. The ordinates $\lambda_n$ are the positive imaginary parts of the nontrivial zeta zeros in the critical line : $ s_n = { 1\over 2} + i \lambda_n $.
The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches
not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula ($without$ any truncations) for the number $ N (E) $ of zeroes in the critical strip with imaginary part greater than $0$ and less than or equal to $E$.

**Category:** Number Theory

[1307] **viXra:1608.0356 [pdf]**
*submitted on 2016-08-25 20:25:15*

**Authors:** Zhang Tianshu

**Comments:** 15 Pages.

Positive integers which can operate to 1 by the set operational rule of the conjecture and positive integers got via contrary operations of the set operational rule are one-to-one correspondence unquestionably. In this article, we classify positive integers to prove the Collatz conjecture by the mathematical induction via operations of substep according to confirmed two theorems plus a lemma in advance.

**Category:** Number Theory

[1306] **viXra:1608.0144 [pdf]**
*submitted on 2016-08-12 21:14:51*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 06 Pages.

Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r t h order with some new interesting properties.

**Category:** Number Theory

[1305] **viXra:1608.0140 [pdf]**
*submitted on 2016-08-12 21:20:08*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, some properties of k Fibonacci and k Lucas numbers are derived and proved by using matrices S and M. The identities we proved are not encountered in the k Fibonacci
and k Lucasnumber literature.

**Category:** Number Theory

[1304] **viXra:1608.0139 [pdf]**
*submitted on 2016-08-12 21:22:02*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 04 Pages.

In this paper we defined general matrices Mk(n,m),
Tk,n and Sk(n,m) for k-Fibonacci number. Using these matrices we find some new summation properties for k-Fibonacci and k-Lucas
numbers.

**Category:** Number Theory

[1303] **viXra:1608.0138 [pdf]**
*submitted on 2016-08-12 21:23:18*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. In this paper, I present some properties of multiplicative coupled Fibonacci sequences of fourth order under two specific schemes.

**Category:** Number Theory

[1302] **viXra:1608.0137 [pdf]**
*submitted on 2016-08-12 21:24:35*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first introduced coupled Fibonacci sequences of second order in additive form. There are 32 different schemes of generalization for the Fibonacci sequences of fifth order in the case of two sequences [1]. I introduce their recurrent formulas below.

**Category:** Number Theory

[1301] **viXra:1608.0135 [pdf]**
*submitted on 2016-08-12 21:46:46*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 07 Pages.

In this paper, we de¯ned new relationship between k Lucas sequences and determinants of their associated matrices, this approach is di®erent and never tried in k Fibonacci sequence
literature.

**Category:** Number Theory

[1300] **viXra:1608.0134 [pdf]**
*submitted on 2016-08-12 21:48:49*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 14 Pages.

In this paper, we defined new relationship between k Fibonacci and k Lucas sequences using continued fractions and series of fractions, this approach is different and never tried in k Fibonacci sequence literature.

**Category:** Number Theory

[1299] **viXra:1608.0133 [pdf]**
*submitted on 2016-08-12 21:50:43*

**Authors:** A. D. Godase, M. B. Dhakne

**Comments:** 08 Pages.

Coupled Fibonacci sequences involve two sequences of integers in which the elements of one
sequence are part of the generalization of the other and vice versa. K. T. Atanassov was first
introduced coupled Fibonacci sequences of second order in additive form. There are 8 different schemes of generalization for the Tribonacci sequences in the case of two sequences & there are 16 different schemes of generalization for the Tetranacci sequences in the case of two sequences. I introduce their recurrent formulas below.

**Category:** Number Theory

[1298] **viXra:1608.0128 [pdf]**
*submitted on 2016-08-12 13:47:27*

**Authors:** Michael M. Ross

**Comments:** Pages.

By defining a function for a linear equation (of slope-intercept form) to be a composite generator, I am able to show that a subset of odd-value slopes must always have even solutions. I apply this function to generate a parity truth table for any perfect square interval that demonstrates the unequal cardinality of the subsets of odd and even composites. Using elementary set theory I demonstrate that this inequality is deterministic, eliminating the possibility of a prime-free interval. This method successfully attacks Legendre's conjecture, providing a logical-conceptual framework for a formal proof.

**Category:** Number Theory

[1297] **viXra:1608.0100 [pdf]**
*submitted on 2016-08-09 21:54:15*

**Authors:** Choe Ryong Gil

**Comments:** 20 pages, 4 tables

In this paper we consider the Riemann hypothesis (RH) by the Euler function and primorial numbers. The paper consists of two parts. In the first part, we find a new sufficient condition for the RH from well known Robin theorem and prove it under a certain condition, which would be called the condition (d). In the second one, we prove that the condition (d) holds.
Keywords; Riemann hypothesis, Euler function, Primorial number.

**Category:** Number Theory

[1296] **viXra:1608.0082 [pdf]**
*submitted on 2016-08-08 11:34:13*

**Authors:** Bengt Månsson

**Comments:** 10 Pages.

A formula giving the $n$:th number of a sequence defined by a recursion formula plus initial value is deduced using generating functions. Of particular interest is the possibility to get an exact expression for the n:th term by means a recursion formula of the same type as the original one. As for the sequence itself it is of some interest that the original recursion is non-linear and the fact that the sequence grows very fast, the number of digits increasing more or less exponentially. Other sequences with the same rekursion span can be treated similarly.

**Category:** Number Theory

[1295] **viXra:1608.0062 [pdf]**
*submitted on 2016-08-06 04:06:33*

**Authors:** Lucas Allen

**Comments:** English, 6 pages, equations and examples

This paper presents a “formula” (more or less) for prime numbers in a specific interval. This formula is then used to partially prove the Goldbach conjecture and the twin primes conjecture. The proofs are incomplete however and have not been reviewed by anyone.

**Category:** Number Theory

[1294] **viXra:1607.0569 [pdf]**
*submitted on 2016-07-31 17:30:32*

**Authors:** Hervé G.

**Comments:** 19 Pages.

It is presented an elementary proof that Beta(3)=Pi^3/32. Beta is the Dirichlet Beta function.

**Category:** Number Theory

[1293] **viXra:1607.0557 [pdf]**
*submitted on 2016-07-30 15:05:24*

**Authors:** Simon Plouffe

**Comments:** 3 Pages.

An extension of a known result of Ramanujan is used to produce sums with exponential terms that gives a representation of many prime numbers.

**Category:** Number Theory

[1292] **viXra:1607.0551 [pdf]**
*submitted on 2016-07-29 12:19:22*

**Authors:** Richard Broxley Omeston

**Comments:** 1 Page.

In this paper I show how the equivalence of the summation of the Móbius function with the Zeta function allows for proof of the Riemann Hypothesis.

**Category:** Number Theory

[1291] **viXra:1607.0536 [pdf]**
*submitted on 2016-07-28 15:35:29*

**Authors:** José de Jesús Camacho Medina

**Comments:** 1 Page.

The present article shows an unpublished formula to evaluate twin primes, the formula is based on the theorem of Wilson and contains mathematical functions such that greatest common divisor, factorial and floor function.

**Category:** Number Theory

[1290] **viXra:1607.0522 [pdf]**
*submitted on 2016-07-27 10:29:42*

**Authors:** Matilda Walter

**Comments:** 3 Pages.

Lemoine - Levy Conjecture, probably the least known of the 'Goldbach Conjectures', states that
every positive odd integer > 5 is a sum of a prime and double of a prime. We present a simple sieve
procedure for finding all existing solutions to the problem for any given odd number > 5.

**Category:** Number Theory

[1289] **viXra:1607.0468 [pdf]**
*submitted on 2016-07-25 01:02:22*

**Authors:** Dhananjay P. Mehendale

**Comments:** 3 Pages

This paper proposes a generalised ABC conjecture and assuming its validity settles a generalised version of Fermat’s last theorem.

**Category:** Number Theory

[1288] **viXra:1607.0437 [pdf]**
*submitted on 2016-07-23 12:06:52*

**Authors:** Kunle Adegoke

**Comments:** 15 Pages.

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here.

**Category:** Number Theory

[1287] **viXra:1607.0434 [pdf]**
*submitted on 2016-07-23 12:24:01*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

Chapter12. Challenge "proof of the Lehman expectation".
A mathematics difficult problem biggest in history.
[1] With the mathematics difficult problem "proof of the Lehman expectation" biggest in history.
[2] I challenge the difficult problem Lehman expectation that rejected the geniuses challenge for 150 years.
[3] It is challenged the mystery of the prime number, a mathematics difficult problem biggest in history, proof of the Lehman expectation.
[4] Neology of the Lehman expectation. A point of intersection that all 0 points are straight.
[5] An elementary particle pulsation principle founds a door of the Lehman expected proof.

**Category:** Number Theory

[1286] **viXra:1607.0400 [pdf]**
*submitted on 2016-07-21 22:44:27*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We give an illogical point in Dirichlet's proof, therefore the used infininite descent is not
powered in his proof

**Category:** Number Theory

[1285] **viXra:1607.0381 [pdf]**
*submitted on 2016-07-20 11:59:45*

**Authors:** Peter Bissonnet

**Comments:** 10 Pages.

Prime products are analyzed from various points of view, with an emphasis on graphical representation and analysis. A prime product N is determined to have two integer coordinates D and m. These coordinates are related to the solutions of a parabola, as well as to right triangles, in what the author calls a ‘backbone - rib’ representation. A prime number or a prime product fall on three dimensional helices, which can be represented in two dimensions as sets of parallel lines. If a prime or a prime product can be represented by 6s - 1, then helix 1 or H1 is designated; if a prime or a prime product can be represented by 6s + 1, then helix 2 or H2 is designated. The integer s is really a composite number, which can be represented as s = r + n, where r is the row number and n is the grouping number called the complex number, both determined from the two dimensional representation of the double helices.
It is also discovered that, due to the mathematical form relating N to D and m, that there must be Lorentz - like transformations between N, D, and m and a new set Nʹ, Dʹ and mʹ; however, the concept of velocity and the speed of light seem out of place in this instance. Nevertheless, the question is asked as to whether or not prime products can be considered to be away to unite relativity and quantum mechanics, which also depends upon integers in a large measure.

**Category:** Number Theory

[1284] **viXra:1607.0360 [pdf]**
*submitted on 2016-07-18 13:58:50*

**Authors:** Reuven Tint

**Comments:** 13 Pages. Original written in Russian

A variant of the solution with the help of Bill hypothesis direct evidence "Great" Fermat's theorem elementary methods rows. New are "invariant identity" (keyword) and obtained by us in the text, the identity of the work, which allowed directly to solve the FLT, and several others.

**Category:** Number Theory

[1283] **viXra:1607.0359 [pdf]**
*submitted on 2016-07-18 15:28:12*

**Authors:** Matilda Walter

**Comments:** 2 Pages.

We present a simple sieve algorithm for finding all existing solutions to the binary Goldbach
problem for a given even number 2N > 4.

**Category:** Number Theory

[1282] **viXra:1607.0178 [pdf]**
*submitted on 2016-07-15 06:08:08*

**Authors:** Zhang Tianshu

**Comments:** 25 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds from AX+BY=CZ. Then, affirmed AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, proved AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric relations of positive odd numbers concerned after divide the inequality in four. Finally, reached a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.

**Category:** Number Theory

[1281] **viXra:1607.0161 [pdf]**
*submitted on 2016-07-12 19:35:56*

**Authors:** Michael Pogorsky

**Comments:** 7 Pages.

The theorem is proved by means of general algebra. The proof is based on set of polynomials a=uwv+v^n; b=uwv+w^n; c=uwv+v^n+w^n and their modifications deduced as required for a,b,c to meet condition of being integers and to satisfy equation a^n+b^n=c^n. The equation also requires existence of positive integers u_p and c_p such that a+b is divisible by u_p^n and c is divisible by u_p c_p. Then the contradiction revealed between expressed by deduced polynomials a^n+b^n and c^n.

**Category:** Number Theory

[1280] **viXra:1607.0159 [pdf]**
*submitted on 2016-07-13 11:11:18*

**Authors:** Maaninou Youssef

**Comments:** 1 Page.

by this theory you can chek any number if it is a prime or not
also you can generat a new prime number

**Category:** Number Theory

[1279] **viXra:1607.0095 [pdf]**
*submitted on 2016-07-07 16:22:11*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, Finite Difference, starting from line ”An n-th power has a constant
nth finite difference. For example, take n=3 and make a difference table”.
The entry appears to to wide the properties of high order difference of
n-th power.

**Category:** Number Theory

[1278] **viXra:1607.0094 [pdf]**
*submitted on 2016-07-07 16:24:21*

**Authors:** Kolosov Petro

**Comments:** 2 Pages.

The basic and short description of proposed entry to Wolfram MathWorld, ”Power”, properties section. The entry appears to connect the
high order finite difference and derivative of Power function.

**Category:** Number Theory

[1277] **viXra:1607.0093 [pdf]**
*submitted on 2016-07-07 16:26:41*

**Authors:** Kolosov Petro

**Comments:** 12 Pages.

This paper presents the way to make expansion for the next form function: y =
x n , ∀(x,n) ∈ N to the numerical series. The most widely used methods to solve this
problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that
is, derivative and integral are inverse operators). The paper provides the other kind of
solution, except above described theorems.

**Category:** Number Theory

[1276] **viXra:1607.0087 [pdf]**
*submitted on 2016-07-07 12:02:36*

**Authors:** Ryan Zielinski

**Comments:** 81 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.

**Category:** Number Theory

[1275] **viXra:1607.0072 [pdf]**
*submitted on 2016-07-07 03:56:39*

**Authors:** Korn Rakpradit

**Comments:** 47 Pages.

The opinions of this work are stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals and the undefined or divergent functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for mathematicians and physicists because it is widespread in use though it is unproved officially for a long time.

**Category:** Number Theory

[1274] **viXra:1607.0058 [pdf]**
*submitted on 2016-07-06 02:42:43*

**Authors:** Tomer Shushi

**Comments:** 4 Pages.

In this paper we give a general proof for the irrationality property of numbers which take a certain form of infinite sums. As special cases we prove the irrationality of infinite sums involving pi and e constants. This proof is obtained in the spirit of the well known Fourier's proof for the irrationality of e.

**Category:** Number Theory

[1273] **viXra:1607.0047 [pdf]**
*submitted on 2016-07-04 21:15:33*

**Authors:** T.Nakashima

**Comments:** 1 Page.

The half proof of the Gilbreath’s onjeture

**Category:** Number Theory

[1272] **viXra:1607.0029 [pdf]**
*submitted on 2016-07-03 05:30:52*

**Authors:** Andrea OSSICINI

**Comments:** 16 Pages.

It is shown that an appropriate use of so-called << double equations >> by Diophantus provides the origin of the Frey elliptic curve and from it we can deduce an elementary proof of Fermat’s Last Theorem

**Category:** Number Theory

[1271] **viXra:1607.0003 [pdf]**
*submitted on 2016-07-01 03:27:48*

**Authors:** Pingyuan Zhou

**Comments:** 13 Pages. If so-called strong Goldbach number sequence introduced in this paper is acceptable then the existence of strong Goldbach number sequence will imply both Goldbach and twin prime conjectures.

Abstract: In this note, we present a new and direct appraoch to prove the Goldbach conjecture that if the existence of the limit of

**Category:** Number Theory

[1270] **viXra:1606.0345 [pdf]**
*submitted on 2016-06-30 11:00:58*

**Authors:** Scott B. Guthery

**Comments:** 8 Pages.

An algorithm for recursively generating the sequence of solutions of a prime constellation is described. The algorithm is based on an polynomial equation formed from the first n elements of the constellation. A root of this equation is the next element of the sequence.

**Category:** Number Theory

[1269] **viXra:1606.0327 [pdf]**
*submitted on 2016-06-29 07:46:25*

**Authors:** Mohamed Ouiter

**Comments:** 7 Pages.

From the definition that a number is Prime if he admits that two dividers and any number not prime can it write as a product of prime factors, I was able to identify a simple law that governs the distribution of prime numbers.

**Category:** Number Theory

[1268] **viXra:1606.0310 [pdf]**
*submitted on 2016-06-28 09:09:00*

**Authors:** Peter Bissonnet

**Comments:** 14 Pages.

When a subset of sequential integers is arranged in a specific way, there appears a paired set of slanted straight lines along which prime numbers seem to naturally arrange themselves in a repeated fashion. This arrangement can further be observed to be a two dimensional surface applicable to the cylinder. If this arrangement is on a piece of paper, then one can fold the paper in the form of a cylinder, and the paired set of slanted straight lines unite at the page edges to form a double helix winding down the cylinder. Paired primes are thus seen to be an association between primes residing on these two paired helixes, and further analysis shows that there appears to be two types of paired primes. Is the prime number double helix nature’s secret for jump starting life (DNA) and thereby defeating entropy (at least in the initial stage) by creating order from order instead of order from disorder? The prime number trends examined exhibit right handed chirality; left handed chirality is obtained by symmetry considerations.

**Category:** Number Theory

[1267] **viXra:1606.0306 [pdf]**
*submitted on 2016-06-28 09:30:16*

**Authors:** Peter Bissonnet

**Comments:** 2 Pages. This is a Russian journal named "Almanac of Modern Science and Education" and can be found at the following link: http://www.gramota.net/eng/editions/5.html

This very short paper illustrates a mathematical method which illustrates the possibility of reducing a prime product (N = PQ) from two independent variables, namely the two original prime numbers P and Q, to only one independent variable which determines both prime numbers.

**Category:** Number Theory

[1266] **viXra:1606.0302 [pdf]**
*submitted on 2016-06-27 21:24:29*

**Authors:** T.Nakashima

**Comments:** 1 Page.

reration of recurring decimal and primitive root

**Category:** Number Theory

[1265] **viXra:1606.0271 [pdf]**
*submitted on 2016-06-25 20:21:56*

**Authors:** T.Nakashima

**Comments:** 1 Page.

Conway’s problem is correct.

**Category:** Number Theory

[1264] **viXra:1606.0264 [pdf]**
*submitted on 2016-06-24 20:45:44*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

Near m,the destance of primes is lower order than logm.This is the Legendre’s conjecture.

**Category:** Number Theory

[580] **viXra:1607.0359 [pdf]**
*replaced on 2016-07-20 07:41:06*

**Authors:** Matilda Walter

**Comments:** 2 Pages.

We present a simple sieve algorithm for finding all existing solutions to the binary Goldbach
problem for a given even number 2N > 4.

**Category:** Number Theory

[579] **viXra:1607.0087 [pdf]**
*replaced on 2016-07-25 10:05:25*

**Authors:** Ryan Zielinski

**Comments:** 84 Pages. This work is licensed under the CC BY 4.0, a Creative Commons Attribution License.

What is a general expression for the sum of the first n integers, each raised to the mth power, where m is a positive integer? Answering this question will be the aim of the paper....We will take the unorthodox approach of presenting the material from the point of view of someone who is trying to solve the problem himself. Keywords: analogy, Johann Faulhaber, finite sums, heuristics, inductive reasoning, number theory, George Polya, problem solving, teaching of mathematics.

**Category:** Number Theory

[578] **viXra:1607.0072 [pdf]**
*replaced on 2016-07-15 07:54:41*

**Authors:** Korn Rakpradit

**Comments:** 47 Pages.

The opinions of this work are revising, stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals that was found and those undefined (and/or multiplied by zero) functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for many mathematicians and physicists nowadays because the Hypothesis is widely used in many subjects and works, unaware of risks, thought it is not officially proved right or wrong.

**Category:** Number Theory

[577] **viXra:1607.0072 [pdf]**
*replaced on 2016-07-08 11:10:23*

**Authors:** Korn Rakpradit

**Comments:** 48 Pages.

The opinions of this work are stalking and proving in details the derivation of Riemann Zeta Function and Riemann Hypothesis, which Riemann did roughly for more than 150 years ago without proof, and correcting all mistakes about the boundaries of the integrals and the undefined or divergent functional equations which caused very big problems to this Riemann Hypothesis.
Proof or disproof of Riemann Hypothesis’s derivation will be very useful for mathematicians and physicists because it is widespread in use though it is unproved officially for a long time.

**Category:** Number Theory

[576] **viXra:1607.0047 [pdf]**
*replaced on 2016-07-27 04:35:14*

**Authors:** T.Nakashima

**Comments:** 1 Page.

The half proof of the Gilbreath’s onjeture

**Category:** Number Theory

[575] **viXra:1606.0302 [pdf]**
*replaced on 2016-07-21 20:07:29*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

reration of recurring decimal and primitive root

**Category:** Number Theory