**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(1)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(2) - 1110(5) - 1111(4) - 1112(4)

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

2013 - 1301(5) - 1302(9) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(26) - 1404(10) - 1405(17) - 1406(20) - 1407(33) - 1408(51) - 1409(47) - 1410(16) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(15) - 1508(14) - 1509(13) - 1510(11) - 1511(9) - 1512(25)

2016 - 1601(14) - 1602(17) - 1603(77) - 1604(54) - 1605(28) - 1606(17) - 1607(17) - 1608(15) - 1609(22) - 1610(22) - 1611(12) - 1612(19)

2017 - 1701(19) - 1702(24) - 1703(25) - 1704(32) - 1705(25) - 1706(25) - 1707(21) - 1708(26) - 1709(17) - 1710(26) - 1711(23) - 1712(34)

2018 - 1801(32) - 1802(22) - 1803(23) - 1804(28) - 1805(33) - 1806(16) - 1807(18) - 1808(15) - 1809(23) - 1810(20) - 1811(18)

Any replacements are listed farther down

[1890] **viXra:1811.0211 [pdf]**
*submitted on 2018-11-13 08:01:46*

**Authors:** Zeolla Gabriel Martín

**Comments:** 9 Pages. Idioma: Español

Este documento desarrolla y demuestra el descubrimiento de un nuevo algoritmo de multiplicación que funciona absolutamente con todos los números.

**Category:** Number Theory

[1889] **viXra:1811.0207 [pdf]**
*submitted on 2018-11-13 11:20:00*

**Authors:** Thomas Pierre Nicolas Jean Brouard

**Comments:** 2 Pages.

We may confirm the Riemann hypothesis by proving that two nontrivial zeros have the same real part.

**Category:** Number Theory

[1888] **viXra:1811.0179 [pdf]**
*submitted on 2018-11-11 19:03:33*

[1887] **viXra:1811.0159 [pdf]**
*submitted on 2018-11-11 04:13:06*

**Authors:** Zach Don

**Comments:** 1 Page.

In this paper, I will be presenting an alternative way of writing the Riemann zeta function in terms of Euler's constant, e.

**Category:** Number Theory

[1886] **viXra:1811.0158 [pdf]**
*submitted on 2018-11-11 04:19:06*

**Authors:** Zach Don

**Comments:** 1 Page.

In this paper, I will propose a legitimate way of re-writing the Riemann zeta function in terms of Euler's constant, e.

**Category:** Number Theory

[1885] **viXra:1811.0145 [pdf]**
*submitted on 2018-11-10 00:53:50*

**Authors:** Toshiro Takami

**Comments:** 14 Pages.

I built a prime number production formula (interim report)
22 + (a ^ 2 + 8) / 24 (a is integer)
This can be used to generate huge prime numbers.
This is an interim report to the last, it seems far to completion.
The previous
√ 24 a + 1
I got an e-mail saying that it was the one that was already mentioned,
However, √ 24 a + 1
Neither this equation nor the equations were found in the wiki or the paper.

**Category:** Number Theory

[1884] **viXra:1811.0119 [pdf]**
*submitted on 2018-11-07 10:57:26*

**Authors:** Viktor Strohm

**Comments:** 5 Pages.

In accordance with the General Theory Systems of Urmantsev (GTSU) [1, 2, 3], the set of primes is considered as a system of objects. For the relationship between objects taken the difference of prime numbers. Revealed periodicity of pairs of intervals.

**Category:** Number Theory

[1883] **viXra:1811.0116 [pdf]**
*submitted on 2018-11-07 11:25:55*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Properties of the zeta function of the Riemann hypothesis are not confirmed as tautologous and hence refute it.

**Category:** Number Theory

[1882] **viXra:1811.0112 [pdf]**
*submitted on 2018-11-07 18:02:13*

**Authors:** Es-said En-naoui

**Comments:** 5 Pages.

The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture.\\
The only best known result so far is that of Chen [3], proving that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to $4.10^{18}$ in 2014 , J\"erg [4] and Tom\'as [5]. In this paper, we prove that the conjecture is true for all even integers greater than 8.

**Category:** Number Theory

[1881] **viXra:1811.0099 [pdf]**
*submitted on 2018-11-06 12:55:12*

**Authors:** Nicolò Rigamonti

**Comments:** 3 Pages.

This paper shows the importance of two properties, which are at the base of the Riemann hypothesis. The key point of all the reasoning about the validity of the Riemann hypothesis is in the fact that only if the Riemann hypothesis is true, these two properties, which are satisfied by the non-trivial zeros, are both true.

**Category:** Number Theory

[1880] **viXra:1811.0080 [pdf]**
*submitted on 2018-11-05 12:05:54*

**Authors:** Ralf Wüsthofen

**Comments:** 1 Page.

This note proves the inconsistency of the Peano arithmetic (PA) by deriving both the strong Goldbach conjecture and its negation.

**Category:** Number Theory

[1879] **viXra:1811.0072 [pdf]**
*submitted on 2018-11-06 02:39:26*

**Authors:** Quang Nguyen Van

**Comments:** 4 Pages.

We give expression of w^n and the possible to apply for solving Fermat's Last theorem.

**Category:** Number Theory

[1878] **viXra:1811.0046 [pdf]**
*submitted on 2018-11-03 21:04:42*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Karush-Kuhn-Tucker constraints for linear programming are not tautologous.

**Category:** Number Theory

[1877] **viXra:1811.0032 [pdf]**
*submitted on 2018-11-02 17:55:05*

**Authors:** Hassine Saidane

**Comments:** 4 Pages.

Optimization of relevant concepts such as action or utility functions enabled the derivation of theories and laws in some scientific fields such as physics and economics. This fact suggested that the problem of the location of the Riemann Zeta Function’s (RZF) nontrivial zeros can be addressed in a mathematical programming framework. Using a constrained nonlinear optimization formulation of the problem, we prove that RZF’s nontrivial zeros are located on the critical line, thereby confirming the Riemann Hypothesis. This result is a direct implication of the Kuhn-Tucker necessary optimality conditions for the formulated nonlinear program.
Keywords: Riemann Zeta function, Riemann Hypothesis, Optimization, Kuhn-Tucker conditions

**Category:** Number Theory

[1876] **viXra:1811.0031 [pdf]**
*submitted on 2018-11-02 18:27:16*

**Authors:** César Aguilera

**Comments:** 9 pages, 4 figures, 4 tables.

A set of relations between perfect numbers is presented. Then some properties of this relations and how they behave, next, a geometric interpretation, a function and finally, the way this function works.

**Category:** Number Theory

[1875] **viXra:1811.0026 [pdf]**
*submitted on 2018-11-03 05:56:56*

**Authors:** Toshiro Takami

**Comments:** 25 Pages.

All prime number expressed
18a + p (a is integer include 0, p is prime number less than 18 include 1, and a continues to infinity).
Prime numbers cycle at 18.
24a + p (a is integer include 0, p is prime number less than 24 include 1, and a continues to infinity).
Prime numbers cycle at 24.
and 30a + p (a is integer include 0, p is prime number less than 30 include 1, and a continues to infinity).
Prime numbers cycle at 30.
There is no exception.
I noticed while building prime number production formula. Prime numbers arise from several lines. Therefore, I believe it is extremely difficult to build a prime number production formula.

**Category:** Number Theory

[1874] **viXra:1811.0017 [pdf]**
*submitted on 2018-11-01 12:03:50*

**Authors:** Salvatore Gerard Micheal

**Comments:** 2 Pages.

a brief exposition on quantifying irrational density within the reals - and - attempt to categorize groups of irrationals

**Category:** Number Theory

[1873] **viXra:1811.0016 [pdf]**
*submitted on 2018-11-01 12:17:02*

**Authors:** Clemens Kroll

**Comments:** 6 Pages.

It is shown that Riemann’s hypothesis is true by showing that an equivalent statement is true. Even more, it is shown that Stieltjes’ conjecture is true.

**Category:** Number Theory

[1872] **viXra:1810.0497 [pdf]**
*submitted on 2018-10-31 06:49:56*

**Authors:** M. Sghiar

**Comments:** 5 Pages.

If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles , the puspose of this article is to give a simple demonstration and deduce a proof of the Beal conjecture.

**Category:** Number Theory

[1871] **viXra:1810.0487 [pdf]**
*submitted on 2018-10-29 13:14:19*

**Authors:** Ankur Shukla, Anurag Singh

**Comments:** 20 Pages.

This article provide unique methods of creating perfect magic squares of order 4 and using certain conditions the magic square can be made more precise and varied. The article contains rules which connects all the perfect squares of order 4 and bring them under one roof. The rules contained in this article deals with various methods of arranging a square maintaining the perfectness and bringing new appearance every time using symmetry. The main objective is to create a method for generation of perfect formula of magic square.

**Category:** Number Theory

[1870] **viXra:1810.0479 [pdf]**
*submitted on 2018-10-28 10:11:24*

**Authors:** Victor Sorokine

**Comments:** 1 Page.

The number D=[(A+B)^n-(C-B)^n-(C-A)^n+(A^n+B^n-C^n)] ends with k+2 zeroes, where k is the number of zeroes at the end of the number A (and in the number A+B-C). However, after the opening of the binomials in the number D its (k+2)-th digit is not equal to zero.

**Category:** Number Theory

[1869] **viXra:1810.0478 [pdf]**
*submitted on 2018-10-28 10:12:26*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Число D=[(A+B)^n-(C-B)^n-(C-A)^n+(A^n+B^n-C^n)] оканчивается на k+2 нулей, где k – число нулей на конце числа А (и в числе A+B-C). Однако после раскрытия биномов в числе D его (k+2)-я цифра нулю не равна.

**Category:** Number Theory

[1868] **viXra:1810.0459 [pdf]**
*submitted on 2018-10-27 12:00:17*

**Authors:** Di Pietro Gabriele

**Comments:** 9 Pages.

This paper gives us an application of Eratosthenes sieve to distribution mean
distance between primes using first and upper orders of Gauss integral log-
arithm Li(x).We define function Υ in section 5. Sections 1 − 4 give us an
introduction to the terminology and a clarification on Υ terms. Section 6
reassumes foregoing explanations and gives us two theorems using first and
upper integral logarithm orders.

**Category:** Number Theory

[1867] **viXra:1810.0457 [pdf]**
*submitted on 2018-10-27 14:12:29*

**Authors:** Toshiro Takami

**Comments:** 19 Pages.

All prime number expressed
24a + p (a is integer include 0, p is prime number less than 24 include 1, and a continues to infinity).
Prime numbers cycle at 24.
and 30a + p (a is integer include 0, p is prime number less than 30 include 1, and a continues to infinity).
Prime numbers cycle at 30, too.
There is no exception.

**Category:** Number Theory

[1866] **viXra:1810.0442 [pdf]**
*submitted on 2018-10-26 15:59:26*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The conjectured proof of the Riemann hypothesis using the excluded middle is refuted by the Meth8/VŁ4 modal logic model checker.

**Category:** Number Theory

[1865] **viXra:1810.0418 [pdf]**
*submitted on 2018-10-24 06:14:11*

**Authors:** Matan Cohen

**Comments:** 9 Pages.

In this paper, I will describe and demonstrate a new method to prove Goldbach’s Conjecture. The idea behind my method is to organize all natural numbers in a binary tree, and to find the connections between the even numbers and the prime numbers by using the characteristics of the tree structure.

**Category:** Number Theory

[1864] **viXra:1810.0390 [pdf]**
*submitted on 2018-10-23 07:31:38*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

This note presents a definite integral for pi.

**Category:** Number Theory

[1863] **viXra:1810.0384 [pdf]**
*submitted on 2018-10-23 19:27:56*

**Authors:** Toshiro Takami

**Comments:** 2 Pages.

If the sum of the digits of a natural number is 6, 9, 12, 15, 18, 21, 24, 27 ... ... (multiples of 3 excluding 3), it is definitely not a prime number

**Category:** Number Theory

[1862] **viXra:1810.0368 [pdf]**
*submitted on 2018-10-22 14:22:06*

**Authors:** A.I.Somsikov

**Comments:** 2 Pages. -

Examples of the seeming contradictions of mathematical transformations

**Category:** Number Theory

[1861] **viXra:1810.0335 [pdf]**
*submitted on 2018-10-20 11:17:11*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages.

e look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We then give an application of Cantor's Diagonal Method that shows zeta(2) is irrational.

**Category:** Number Theory

[1860] **viXra:1810.0288 [pdf]**
*submitted on 2018-10-19 02:09:09*

**Authors:** Toshiro Takami

**Comments:** 4 Pages.

If prime, we found that the remainder is a prime number including 1 when divided by 30, 60 and 90.
This is called toshichan-man's small theorem.
It seems to be useful for prime number determination (especially huge prime number determination).

**Category:** Number Theory

[1859] **viXra:1810.0282 [pdf]**
*submitted on 2018-10-17 06:31:37*

**Authors:** Vyacheslav Telnin

**Comments:** 116 Pages. format A5

In this paper, we analyze the construction of numbers from natural to complex. A close relationship between these numbers and three direct operations ([1] – addition, [2] – multiplication, [3] – exponentiation) is revealed. A method of constructing new direct operations is found. Two new direct operations ([4] and [0]) are constructed. Two inverse operations are constructed for each of them. If equations that are unsolvable in complex numbers are found for them, new numbers can be constructed on the basis of these unsolvabilities.
So far in this way in this work the question numbers are constructed (?- numbers) for [3] direct operation.
Along the way, numbers-strings and N-numbers are described. The topology of the numerical line is traced on the basis of N-numbers.
In this paper we find the relation of N-numbers and ?-numbers'.

**Category:** Number Theory

[1858] **viXra:1810.0255 [pdf]**
*submitted on 2018-10-16 17:40:35*

**Authors:** Zeolla Gabriel Martín

**Comments:** 3 Pages.

This paper develops the construction of the Golden Pattern for different prime divisors, the discovery of patterns towards infinity. The discovery of infinite harmony represented in fractal numbers and patterns. These patterns form the sequence of prime numbers.

**Category:** Number Theory

[1857] **viXra:1810.0240 [pdf]**
*submitted on 2018-10-15 14:59:35*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I found out my original make prime number built method.
30n+7 (n is positive integer, including 0).
30n+17 (n is positive integer, including 0).
Prime has a period of 30. We focused on that point and developed it.
However, those that are not prime are also quite included.
And,
30n+11 (n is positive integer, including 0).
30n+13 (n is positive integer, including 0).
Also considered.

**Category:** Number Theory

[1856] **viXra:1810.0223 [pdf]**
*submitted on 2018-10-14 17:11:04*

**Authors:** Marco Ripà

**Comments:** 4 Pages.

In 2011, in his book “La strana coda della serie n^n^...^n", M. Ripà analyzed some properties involving the rightmost figures of integer tetration, the iterated exponentiation a^^b, characterized by an increasing number of stable digits for any base a > 1.
A few conjectures arose about how many new stable digits are generated by any unitary increment of the hyperexponent b, and Ripà indicated this value as V(a) or “convergence speed” of a. In fact, when b is large enough, V(a) seems to not depend from b, taking on a (strictly positive) unique value, and many observations supported this claim. Moreover, we claim that V(a) = 1 for any a(mod 25) congruent to {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a)>=2 otherwise.

**Category:** Number Theory

[1855] **viXra:1810.0175 [pdf]**
*submitted on 2018-10-12 02:45:55*

**Authors:** Anna Povazanova, Ivo Povazan

**Comments:** 13 Pages.

p is prime.The article describes the new Abelian groups of type
p=4k+1 and p = 4k-1, for which a theorem similar to the Fermat's
little theorem applies. The multiplicative group (Z/pZ)* in some sense
similar to the Abelian group of type p = 4k+1. Abelian group of type
p = 4k-1 is a different structure compared to group (Z/pZ)*. This
fact is used for the primality test of integer N = 4k-1. The primality
test was veried up to N = 2^(64).

**Category:** Number Theory

[1854] **viXra:1810.0153 [pdf]**
*submitted on 2018-10-09 07:38:30*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents an integral for Catalan's constant: G=0.915965...

**Category:** Number Theory

[1853] **viXra:1810.0141 [pdf]**
*submitted on 2018-10-09 16:57:44*

**Authors:** Jonathan Trousdale

**Comments:** 6 Pages.

This paper sets forth a representation of the hyperbolic substratum that defines order on $\mathbb{N}$. Degeneracy of $\mathbb{N}$ at points of intersection with the substratum is observed as violations of the fundamental theorem of arithmetic in the form of mutable prime factorization. At a point of maximum symmetry on the representation manifold, an exact expression of $\pi$ is available as a combination of three integers.

**Category:** Number Theory

[1852] **viXra:1810.0046 [pdf]**
*submitted on 2018-10-05 01:33:59*

**Authors:** Idriss Olivier Bado

**Comments:** 5 Pages.

The main contribution of this paper is to achieve the proof of Riemann hypothesis. The key idea is based on new formulation of the problem
$$\zeta(s)=\zeta(1-s) \Leftrightarrow re(s)=\frac{1}{2}$$. This proof is considered as a great discovery in mathematic.

**Category:** Number Theory

[1851] **viXra:1809.0600 [pdf]**
*submitted on 2018-09-30 07:35:40*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

The totherian set definition is not tautologous.

**Category:** Number Theory

[1850] **viXra:1809.0576 [pdf]**
*submitted on 2018-09-30 02:18:20*

**Authors:** Bado Idriss Olivier

**Comments:** 4 Pages.

The purpose of this article is to introduce the theory of totherian analysis in order to provide proof of the Riemann hypothesis: the concepts introduced have been so effective and we can use it to build a coherent and tangible analysis. Totherian analysis can be considered as an effective remedy in solving a lot of problems in mathematics

**Category:** Number Theory

[1849] **viXra:1809.0571 [pdf]**
*submitted on 2018-09-28 08:11:33*

**Authors:** Victor Sorokine

**Comments:** 2 Pages.

If we leave only the last digits A', B', C' in the numbers A, B, C, then the third digit in the left
part of one of the equivalent Fermat’s equations will be greater than 1. And the inclusion of
the following digits in the numbers A, B, C can reduce the value of this digit only by 1.

**Category:** Number Theory

[1848] **viXra:1809.0570 [pdf]**
*submitted on 2018-09-28 08:12:33*

**Authors:** Victor Sorokine

**Comments:** 1 Page. Russian version

Если в числах A, B, C оставить лишь последние цифры A', B', C', то третья цифра в
левой части одного из эквивалентных равенств Ферма будет больше 1. А включение в
числа A, B, C последующих цифр могут уменьшить значение этой цифры лишь на 1.

**Category:** Number Theory

[1847] **viXra:1809.0554 [pdf]**
*submitted on 2018-09-29 05:26:24*

**Authors:** Bado Idriss Olivier

**Comments:** 6 Pages.

The purpose of this article is to introduce the theory of totherian analysis in order to provide proof of the Riemann hypothesis: the concepts introduced have been so effective and we can use it to build a coherent and tangible analysis. Totherian analysis can be considered as an effective remedy in solving a lot of problems in mathematics

**Category:** Number Theory

[1846] **viXra:1809.0522 [pdf]**
*submitted on 2018-09-25 13:29:51*

**Authors:** Colin James III

**Comments:** 1 Page. © Copyright 2016, 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Conjecture sentence "If p or q is equivalent to s and p,q,s are relatively prime, then p or q is tautologous" is not tautologous, and it deviates by one value: FTTT TTTT TTTT TTTT.

**Category:** Number Theory

[1845] **viXra:1809.0514 [pdf]**
*submitted on 2018-09-24 07:49:36*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents an integral for Catalan's constant.

**Category:** Number Theory

[1844] **viXra:1809.0505 [pdf]**
*submitted on 2018-09-24 13:03:16*

**Authors:** Joseph Dise

**Comments:** 2 Pages.

A short demonstration of the title claim, which proves the Legendre conjecture and potentially other conjectures.

**Category:** Number Theory

[1843] **viXra:1809.0503 [pdf]**
*submitted on 2018-09-24 16:12:05*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 4 Pages. Comments welcome.

We give an elementary proof of the ABC conjecture.

**Category:** Number Theory

[1842] **viXra:1809.0488 [pdf]**
*submitted on 2018-09-23 10:45:03*

**Authors:** Emmanuil Manousos

**Comments:** 34 Pages.

It holds that every product of natural numbers can also be written as a sum. The inverse does not hold when 1 is excluded from the product. For this reason, the investigation of natural numbers should be done through their sum and not through their product. Such an investigation is presented in the present article. We prove that primes play the same role for odd numbers as the powers of 2 for even numbers, and vice versa. The following theorem is proven: “Every natural number, except for 0 and 1, can be uniquely written as a linear combination of consecutive powers of 2 with the coefficients of the linear combination being -1 or +1.” This theorem reveals a set of symmetries in the internal order of natural numbers which cannot be derived when studying natural numbers on the basis of the product. From such a symmetry a method for identifying large prime numbers is derived. We prove a factorization test for the natural numbers.

**Category:** Number Theory

[1841] **viXra:1809.0479 [pdf]**
*submitted on 2018-09-22 05:35:56*

**Authors:** Nicolò Rigamonti

**Comments:** 4 Pages.

In this paper we will resume and conclude the last work.

**Category:** Number Theory

[1840] **viXra:1809.0449 [pdf]**
*submitted on 2018-09-20 16:53:30*

**Authors:** Juan Moreno Borrallo

**Comments:** 5 Pages. (Spanish version)

In this paper it is demonstrated Fermat's last theorem only using elementary methods.

**Category:** Number Theory

[1839] **viXra:1809.0351 [pdf]**
*submitted on 2018-09-17 17:16:10*

**Authors:** S.I. Nada, A. Elrokh, R. Hamza

**Comments:** 16 pages

Abstract
A graph is called cordial if it has a 0 - 1 labeling such that the number
of vertices (edges) labeled with ones and zeros dier by at most one. The
conjunction of two graphs (V1;E1) and (V2;E2) is the graph G = (V;E),
where V = V1 x V2 and u = (a1; a2), v = (b1; b2) are two vertices, then
uv belongs to E if aibi belongs to Ei for i = 1 or 2. In this paper, we present necessary and
sucient condition for cordial labeling for the conjunction of two paths,
denoted by Pn ^ Pm. Also, we drive an algorithm to generate cordial labeling for the conjunction Pn ^ Pm.

**Category:** Number Theory

[1838] **viXra:1809.0341 [pdf]**
*submitted on 2018-09-16 11:10:33*

**Authors:** Pablo Hernan Pereyra

**Comments:** 8 Pages. Copyright 2007

In the present work is demonstrate that the so called Goldbach conjecture from 1742 -- “All positive even numbers greater than two can be expressed as a sum of two primes” – due to Leonhard Euler, is a true statement. This result is partially based on the Wilson theorem, and complementary on our reasoning to cast the problem into a diophantine equation. The latter is the master equation for the conjectures proof.

**Category:** Number Theory

[1837] **viXra:1809.0338 [pdf]**
*submitted on 2018-09-16 13:40:39*

**Authors:** Paco Derek Tucker

**Comments:** 1 Page.

A simple and fast factoring algorithm that requires only copy paste, the modular operation, and paired additive partitions of the number.

**Category:** Number Theory

[1836] **viXra:1809.0322 [pdf]**
*submitted on 2018-09-15 10:35:14*

**Authors:** Hajime Mashima

**Comments:** 9 Pages.

Legendre's conjecture argues that there is always a prime number between n2 and (n + 1)2 when natural number n.

**Category:** Number Theory

[1835] **viXra:1809.0207 [pdf]**
*submitted on 2018-09-10 14:02:28*

**Authors:** Samia Lakehal

**Comments:** 13 Pages ; 2 Tables ; 766 Ko

The Sieve of Eratosthenes finds all the prime numbers up to any given limit by eliminating all non-primes from the list of all natural numbers. A list of natural numbers containing no multiples of 2, 3 or 5 is created, it is obtained by 8 formulas. The elimination of non-primes is effectuated from this new list. It then appears that it is possible to express all the non-primes belonging to this list by 36 other formulas, the prime numbers being the numbers satisfying the 8 formulas but not the 36.

**Category:** Number Theory

[1834] **viXra:1809.0158 [pdf]**
*submitted on 2018-09-07 14:10:16*

**Authors:** Robert C. Hall

**Comments:** 27 Pages.

The Summation test consists of adding all numbers that begin with a particular first digit or first two digits and determining its distribution with respect to these first or first two digits numbers. Most people familiar with this test believe that the distribution is a uniform distribution for any distribution that conforms to Benford's law i.e. the distribution of the mantissas of the logarithm of the data set is uniform U[0,12). The summation test that results in a uniform distribution is true for an exponential function (geometric progression) but not true for a data set that conforms to a Log Normal distribution even when the Log Normal distribution itself closely approximates Benford's Law.

**Category:** Number Theory

[1833] **viXra:1809.0139 [pdf]**
*submitted on 2018-09-08 05:07:23*

**Authors:** Ryujin Choe

**Comments:** 1 Page.

Solution of Goldbach's Conjecture

**Category:** Number Theory

[1832] **viXra:1809.0086 [pdf]**
*submitted on 2018-09-04 16:35:05*

**Authors:** Kunle Adegoke

**Comments:** 7 pages, no figures

We derive an identity connecting any two Horadam sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed.

**Category:** Number Theory

[1831] **viXra:1809.0060 [pdf]**
*submitted on 2018-09-03 15:11:34*

**Authors:** Pablo Hernan Pereyra

**Comments:** 5 Pages. 2009 Copyright

This paper aims to present an elementary demonstration of the Fermat-Wiles theorem, also known as Fermat's Last Theorem, using a different equivalent statement and a parameterization method.

**Category:** Number Theory

[1830] **viXra:1809.0059 [pdf]**
*submitted on 2018-09-03 20:40:44*

**Authors:** Jeozadaque Marcos

**Comments:** 4 pages

We show in this paper another proof of Benford’s Law. The idea starts with the
problem of to find the first digit of a power. Then we deduced a function to calculate
the first digit of any power a j called L f function. The theorem 1.2 its a consequence
of the periodicity of the $L_f$ function.

**Category:** Number Theory

[1829] **viXra:1809.0008 [pdf]**
*submitted on 2018-09-01 03:59:30*

**Authors:** Radomir Majkic

**Comments:** 3 Pages.

The minimal homothetic integer solution of the Fermat equation is zero and the Fermat last theorem is true.

**Category:** Number Theory

[1828] **viXra:1808.0635 [pdf]**
*submitted on 2018-08-30 00:59:38*

**Authors:** A.I.Somsikov

**Comments:** 3 Pages. -

The solution of the problem of irrational numbers is proposed

**Category:** Number Theory

[1827] **viXra:1808.0634 [pdf]**
*submitted on 2018-08-30 01:07:48*

**Authors:** A.I.Somsikov

**Comments:** 4 Pages. -

"the physical sense" (the logical contents) of complex numbers is revealed.

**Category:** Number Theory

[1826] **viXra:1808.0633 [pdf]**
*submitted on 2018-08-30 01:14:33*

**Authors:** A.I.Somsikov

**Comments:** 10 Pages. -

The sense (the logical contents) of concept of numbers is revealed. Definition of arithmetic actions is given.

**Category:** Number Theory

[1825] **viXra:1808.0628 [pdf]**
*submitted on 2018-08-28 07:37:10*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

In this note we recall a formula for pi.The distinctive feature of these formula is that pi is expressed in terms of the Lerch Transcendent Function.

**Category:** Number Theory

[1824] **viXra:1808.0567 [pdf]**
*submitted on 2018-08-26 17:11:59*

**Authors:** Julian Beauchamp

**Comments:** 5 Pages.

In the first part of this paper, we show how a^x - b^y can be expressed as a binomial expansion (to an indeterminate power, $z$). In the second part we will show how this leads to a proof for the Beal Conjecture.

**Category:** Number Theory

[1823] **viXra:1808.0509 [pdf]**
*submitted on 2018-08-21 08:06:10*

**Authors:** Timothy W. Jones

**Comments:** 4 Pages.

We introduce an unaccustomed number system, H±, and show how it can be used to prove gamma
is irrational. This number system consists of
plus and minus multiplies of the terms of the harmonic series. Using some properties of ln, this system can depict the harmonic series and
lim as n goes to infinity of ln n at the same time, giving gamma as an infinite decimal. The
harmonic series converges to infinity so negative terms are forced. As all rationals can be given in H± without negative terms, it follows that must be irrational.

**Category:** Number Theory

[1822] **viXra:1808.0284 [pdf]**
*submitted on 2018-08-20 01:46:43*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

【Abstract】
I found a prime number equation. All prime numbers except 2 and 3 are expressed by the following formula.
(a = positive integer, t = prime number) For other positive integers, t is an irrational number. As an exception, This generates all prime numbers except 2 and 3, but also generates a composite number of prime numbers. The composite number of the prime has regularity.

**Category:** Number Theory

[1821] **viXra:1808.0254 [pdf]**
*submitted on 2018-08-18 07:57:07*

**Authors:** Bado Idriss Olivier

**Comments:** 7 Pages.

Goldbach's famous conjecture has always fascinated eminent mathematicians. In this paper we give a rigorous proof based on a new formulation, namely, that every even integer has a primo-raduis. Our proof is mainly based on the application of Chebotarev-Artin's theorem, Mertens' formula and the Principle exclusion-inclusion of Moivre

**Category:** Number Theory

[1820] **viXra:1808.0201 [pdf]**
*submitted on 2018-08-15 22:03:17*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[1819] **viXra:1808.0193 [pdf]**
*submitted on 2018-08-14 06:41:55*

**Authors:** Nicolò Rigamonti

**Comments:** 7 Pages.

In these papers we will try to face the Riemann hypothesis, basing on the study of the functional equation of the Riemann zeta function.

**Category:** Number Theory

[1818] **viXra:1808.0190 [pdf]**
*submitted on 2018-08-14 07:42:35*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

Some remarks on the integral 4.371.1 in G&R table of integrals.

**Category:** Number Theory

[1817] **viXra:1808.0180 [pdf]**
*submitted on 2018-08-15 04:02:30*

**Authors:** Hajime Mashima

**Comments:** 2 Pages.

Brocard's problem was presented by Henri Brocard in 1876 and 1885.
n! + 1 = m2. The number that satisfies this is called "Brown numbers"
and three are known: (n;m) = (4; 5); (5; 11); (7:71).

**Category:** Number Theory

[1816] **viXra:1808.0158 [pdf]**
*submitted on 2018-08-12 11:45:22*

**Authors:** Radomir Majkic

**Comments:** 4 Pages.

There is no cuboid with all integer edges and face diagonals.

**Category:** Number Theory

[1815] **viXra:1808.0074 [pdf]**
*submitted on 2018-08-07 02:00:58*

**Authors:** Angel Isaac Cruz Escalante

**Comments:** 3 Pages.

Fermat's last theorem states there are not solutions for a^x+b^x=c^x if (a,b,c,x) are positive integers and x>2, we can consider two possible cases for Fermat's last theorem, when x=4, and x=2n+1, n is natural numbers. case x=4 was proved by Fermat, here is a proof for case x=2n+1.

**Category:** Number Theory

[1814] **viXra:1807.0533 [pdf]**
*submitted on 2018-07-31 08:23:22*

**Authors:** Andrey B. Skrypnik

**Comments:** 2 Pages.

This is the third result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1813] **viXra:1807.0512 [pdf]**
*submitted on 2018-07-30 22:07:40*

**Authors:** Andrey B. Skrypnik

**Comments:** 2 Pages.

This is the second result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1812] **viXra:1807.0510 [pdf]**
*submitted on 2018-07-31 02:52:22*

**Authors:** Elhadj Zeraoulia

**Comments:** 11 Pages.

The main objective of this short note is prove that some statements concerning the represenation of positive integers by the sum of prime numbers are equivalent to some true trivial cases. This implies that these statements are also true. The analysis is based on a new prime formula and some trigonometric expressions.

**Category:** Number Theory

[1811] **viXra:1807.0494 [pdf]**
*submitted on 2018-07-30 04:59:52*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

This is the first result of applying Formula of Disjoint Sets of Odd Numbers

**Category:** Number Theory

[1810] **viXra:1807.0484 [pdf]**
*submitted on 2018-07-28 07:29:36*

**Authors:** Andrey B. Skrypnik

**Comments:** 4 Pages.

Destroyed another fortress of unproven tasks

**Category:** Number Theory

[916] **viXra:1811.0145 [pdf]**
*replaced on 2018-11-12 06:05:25*

**Authors:** Toshiro Takami

**Comments:** 9 Pages.

I tried variously.
(30a+bi)(30a-bi)+k
(24a+bi)(24a-bi)+k
(1007a+bi)(1007a-bi)+k
(60a+bi)(60a-bi)+k
(a, b and k are positive integer.)
However, in the above formula it did not work well.
and, It settled down.
(√24a+4i)(√24a-4i)+1
and
(√6a+4i)(√6a-4i)+1
I half successful.
√ 8, √ 12, √ 14, √ 18 did not succeed.
Last,
(√10a+4i)(√10a-4i)-3
I half successful.
However, a relatively large number of things that are not prime numbers are still included.
The challenge to my prime production ceremony will continue.

**Category:** Number Theory

[915] **viXra:1811.0145 [pdf]**
*replaced on 2018-11-11 22:07:38*

**Authors:** Toshiro Takami

**Comments:** 6 Pages.

I tried variously.
(30a+bi)(30a-bi)+k
(24a+bi)(24a-bi)+k
(1007a+bi)(1007a-bi)+k
(60a+bi)(60a-bi)+k
(a, b and k are positive integer. )But, I did not succeed.
and, It settled down.
(√24a+4i)(√24a-4i)+1
and
(√6a+4i)(√6a-4i)+1
I almost succeed.
However, a relatively large number of things that are not prime numbers are still included.
The challenge to my prime production ceremony will continue.

**Category:** Number Theory

[914] **viXra:1811.0145 [pdf]**
*replaced on 2018-11-10 01:54:22*

**Authors:** Toshiro Takami

**Comments:** 15 Pages.

I built a prime number production formula (interim report)
22 + (a ^ 2 + 8) / 24 (a is integer)
This can be used to generate huge prime numbers.
This is an interim report to the last, it seems far to completion.
The previous
√ 24 a + 1
I got an e-mail saying that it was the one that was already mentioned,
However, √ 24 a + 1
Neither this equation nor the equations were found in the wiki or the paper.

**Category:** Number Theory

[913] **viXra:1811.0112 [pdf]**
*replaced on 2018-11-09 19:22:45*

**Authors:** Es-said En-naoui

**Comments:** 5 Pages.

The Goldbach conjecture dates back to 1742 ; we refer the reader to [1]-[2] for a history of the conjecture. Christian Goldbach stated that every odd integer greater than seven can be written as the sum of at most three prime numbers. Leonhard Euler then made a stronger conjecture that every even integer greater than four can be written as the sum of two primes. Since then, no one has been able to prove the Strong Goldbach Conjecture.\\
The only best known result so far is that of Chen [3], proving that every sufficiently large even integer N can be written as the sum of a prime number and the product of at most two prime numbers. Additionally, the conjecture has been verified to be true for all even integers up to $4.10^{18}$ in 2014 , J\"erg [4] and Tom\'as [5]. In this paper, we prove that the conjecture is true for all even integers greater than 8.

**Category:** Number Theory

[912] **viXra:1811.0032 [pdf]**
*replaced on 2018-11-04 14:37:02*

**Authors:** Hassine Saidane

**Comments:** 4 Pages.

Abstract. Optimization of relevant concepts such as action or utility functions enabled the derivation of theories and laws in some scientific fields such as physics and economics. This fact suggested that the problem of the location of the Riemann Zeta Function’s (RZF) nontrivial zeros can be addressed in a mathematical programming framework. Using a constrained nonlinear optimization formulation of the problem, we prove that RZF’s nontrivial zeros are located on the critical line, thereby confirming the Riemann Hypothesis. This result is a direct implication of the Kuhn-Tucker necessary optimality conditions for the formulated nonlinear program.
Keywords: Riemann Zeta function, Riemann Hypothesis, Optimization, Kuhn-Tucker conditions.

**Category:** Number Theory

[911] **viXra:1810.0497 [pdf]**
*replaced on 2018-11-01 07:30:51*

**Authors:** M. Sghiar

**Comments:** 5 Pages.

If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles [2], the puspose of this article is to give a simple demonstration and deduce a proof of the
Beal conjecture.

**Category:** Number Theory

[910] **viXra:1810.0457 [pdf]**
*replaced on 2018-10-27 14:23:02*

**Authors:** Toshiro Takami

**Comments:** 19 Pages.

All prime number expressed
24a + p (a is integer include 0, p is prime number less than 24 include 1, and a continues to infinity).
Prime numbers cycle at 24.
and 30a + p (a is integer include 0, p is prime number less than 30 include 1, and a continues to infinity).
Prime numbers cycle at 30, too.
There is no exception.

**Category:** Number Theory

[909] **viXra:1810.0335 [pdf]**
*replaced on 2018-11-11 12:07:54*

**Authors:** Timothy W. Jones

**Comments:** 10 Pages. Yet more clarifications.

We look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We also puzzle out why the convergence of the constructed number, its value, is of no concern. We next review general properties of decimals and prove the existence of an irrational number with a modified version of Cantor's diagonal method. Finally, we show, with yet another modification of the method, that zeta(2) is irrational.

**Category:** Number Theory

[908] **viXra:1810.0335 [pdf]**
*replaced on 2018-11-09 08:01:59*

**Authors:** Timothy W. Jones

**Comments:** 11 Pages. Exposition is clearer that previous drafts.

We look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We next review general properties of decimals and prove the existence of an irrational number with a modified version of Cantor's diagonal method. Finally, we show, with yet another modification of the argument, that zeta(2) is irrational.

**Category:** Number Theory

[907] **viXra:1810.0335 [pdf]**
*replaced on 2018-11-01 08:29:23*

**Authors:** Timothy W. Jones

**Comments:** 8 Pages. More details and history.

We look at some of the details of Cantor's Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We next review general properties of decimals and prove the existence of an irrational number with a modified version of Cantor's diagonal method. Finally, we show, with yet another modification of the argument, that zeta(2) is irrational.

**Category:** Number Theory

[906] **viXra:1810.0335 [pdf]**
*replaced on 2018-10-28 12:02:04*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages. Further arguments and examples given.

We look at some of the details of Cantor’s Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We then give a application of Cantor’s Diagonal Method that shows zeta(2) is irrational.

**Category:** Number Theory

[905] **viXra:1810.0335 [pdf]**
*replaced on 2018-10-23 12:29:15*

**Authors:** Timothy W. Jones

**Comments:** 5 Pages. Some tightening of the argument.

We look at some of the details of Cantor’s Diagonal Method and argue that the swap function given does not have to exclude 9 and 0, base 10. We then give a application of Cantor’s Diagonal Method that shows zeta(2) is irrational.

**Category:** Number Theory

[904] **viXra:1810.0282 [pdf]**
*replaced on 2018-11-06 12:17:49*

**Authors:** Vyacheslav Telnin

**Comments:** 122 Pages. Format A5. Content is at the end of the paper.

In this paper, we analyze the construction of numbers from natural to complex. A close relationship between these numbers and three direct operations ([1] – addition, [2] – multiplication, [3] – exponentiation) is revealed. A method of constructing new direct operations is found. Two new direct operations ([4] and [0]) are constructed. Two inverse operations are constructed for each of them. If equations that are unsolvable in complex numbers are found for them, new numbers can be constructed on the basis of these unsolvabilities.
So far in this way in this work the question numbers are constructed (?- numbers) for [3] direct operation.
Along the way, numbers-strings and N-numbers are described. The topology of the numerical line is traced on the basis of N-numbers.
In this paper we find the relation of N-numbers and ?-numbers'.

**Category:** Number Theory

[903] **viXra:1810.0282 [pdf]**
*replaced on 2018-10-26 02:06:30*

**Authors:** Vyacheslav Telnin

**Comments:** 120 Pages. Content is at the end of the paper. Format A5

In this paper, we analyze the construction of numbers from natural to complex. A close relationship between these numbers and three direct operations ([1] – addition, [2] – multiplication, [3] – exponentiation) is revealed. A method of constructing new direct operations is found. Two new direct operations ([4] and [0]) are constructed. Two inverse operations are constructed for each of them. If equations that are unsolvable in complex numbers are found for them, new numbers can be constructed on the basis of these unsolvabilities.
So far in this way in this work the question numbers are constructed (?- numbers) for [3] direct operation.
Along the way, numbers-strings and N-numbers are described. The topology of the numerical line is traced on the basis of N-numbers.
In this paper we find the relation of N-numbers and ?-numbers'.

**Category:** Number Theory

[902] **viXra:1810.0223 [pdf]**
*replaced on 2018-10-23 16:14:29*

**Authors:** Marco Ripà

**Comments:** 4 Pages.

In 2011, in his book “La strana coda della serie n^n^...^n", M. Ripà analyzed some properties involving the rightmost figures of integer tetration, the iterated exponentiation a^^b, characterized by an increasing number of stable digits for any base a > 1.
A few conjectures arose about how many new stable digits are generated by any unitary increment of the hyperexponent b, and Ripà indicated this value as V(a) or “convergence speed” of a. In fact, when b is large enough, V(a) seems to not depend from b, taking on a (strictly positive) unique value, and many observations supported this claim. Moreover, we claim that V(a) = 1 for any a(mod 25) congruent to {2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23} and V(a)>=2 otherwise.

**Category:** Number Theory

[901] **viXra:1809.0488 [pdf]**
*replaced on 2018-10-15 13:52:47*

**Authors:** Emmanuil Manousos

**Comments:** 35 Pages.

It holds that every product of natural numbers can also be written as a sum. The inverse does not hold when 1 is excluded from the product. For this reason, the investigation of natural numbers should be done through their sum and not through their product. Such an investigation is presented in the present article. We prove that primes play the same role for odd numbers as the powers of 2 for even numbers, and vice versa. The following theorem is proven: “Every natural number, except for 0 and 1, can be uniquely written as a linear combination of consecutive powers of 2 with the coefficients of the linear combination being -1 or +1.” This theorem reveals a set of symmetries in the internal order of natural numbers which cannot be derived when studying natural numbers on the basis of the product. From such a symmetry a method for identifying large prime numbers is derived. We prove a factorization test for the natural numbers.

**Category:** Number Theory

[900] **viXra:1808.0567 [pdf]**
*replaced on 2018-09-05 10:25:08*

**Authors:** Julian Beauchamp

**Comments:** 6 Pages.

In the first part of this paper, we show how a^x - b^y can be expressed as a new non-standard binomial formula (to an indeterminate power, n). In the second part, by fixing n to the value of z we compare this binomial formula to the standard binomial formula for c^z to prove the Beal Conjecture.

**Category:** Number Theory

[899] **viXra:1808.0531 [pdf]**
*replaced on 2018-09-20 19:53:51*

**Authors:** Toshiro Takami

**Comments:** 15 Pages.

I proved the Goldbach's conjecture.
Even numbers are prime numbers and prime numbers added, but it has not been proven yet whether it can be true even for a huge number (forever huge number).
All prime numbers are included in (6n - 1) or (6n + 1) except 2 and 3 (n is a positive integer).
All numbers are executed in hexadecimal notation. This does not change even in a huge number (forever huge number).
2 (6n + 2), 4 (6n - 2), 6 (6n) in the figure are even numbers. 1 (6n + 1), 3 (6n + 3), 5 (6n - 1) are odd numbers.

**Category:** Number Theory

[898] **viXra:1808.0201 [pdf]**
*replaced on 2018-10-27 01:43:28*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[897] **viXra:1808.0201 [pdf]**
*replaced on 2018-08-18 00:13:24*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we find the axiomatic pattern of prime numbers.

**Category:** Number Theory

[896] **viXra:1808.0193 [pdf]**
*replaced on 2018-08-27 07:09:00*

**Authors:** Nicolò Rigamonti

**Comments:** 7 Pages.

The last version includes an improved and better explanation.

**Category:** Number Theory

[895] **viXra:1808.0193 [pdf]**
*replaced on 2018-08-17 04:45:35*

**Authors:** Nicolò Rigamonti

**Comments:** Pages.

In these papers we will try to face the Riemann hypothesis, basing on the study of the functional equation of the Riemann zeta function.

**Category:** Number Theory

[894] **viXra:1807.0510 [pdf]**
*replaced on 2018-08-25 05:18:13*

**Authors:** Elhadj Zeraoulia

**Comments:** 13 Pages.

The main objective of this short note is prove that some statements
concerning the representation of even integers by the sum of prime
numbers are equivalent to some true trivial case. This implies that
these statements are also true. The analysis is based on a new prime
formula and some trigonometric expressions.

**Category:** Number Theory