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2016 - 1601(14) - 1602(18) - 1603(77) - 1604(55) - 1605(28) - 1606(18) - 1607(22) - 1608(18) - 1609(25) - 1610(24) - 1611(13)

Any replacements are listed further down

[1363] **viXra:1611.0410 [pdf]**
*submitted on 2016-11-30 07:48:39*

**Authors:** Zhang Tianshu

**Comments:** 18 Pages.

The ABC conjecture seemingly is difficult to carry conviction in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we select and adopt a specific equality 1+2N(2N-2)=(2N-1)2 with N≥2 satisfying 2N-1>(Rad(2N-2))1+ ε. Then, proceed from the analysis of the limits of values of ε to find its certain particular values, thereby finally negate the ABC conjecture once and for all.

**Category:** Number Theory

[1362] **viXra:1611.0390 [pdf]**
*submitted on 2016-11-29 03:29:40*

**Authors:** Robert Deloin

**Comments:** 13 Pages.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate infinitely many primes.
The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new
approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences.
As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and
the binary Goldbach conjecture.
The method is also used to prove that there are infinitely many primorial and factorial primes.

**Category:** Number Theory

[1361] **viXra:1611.0373 [pdf]**
*submitted on 2016-11-27 08:39:53*

**Authors:** Victor Christianto

**Comments:** 4 Pages. This paper will be submitted to Annals of Mathematics

In this paper we will give an outline of proof of Fermat’s Last Theorem using a graphical method. Although an exact proof can be given using differential calculus, we choose to use a more intuitive graphical method.

**Category:** Number Theory

[1360] **viXra:1611.0337 [pdf]**
*submitted on 2016-11-24 15:17:34*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

The Odd Goldbach conjecture has been proved to be correct and it states that every odd positive integer greater than seven may be written as a sum of three odd primes. However, the even Goldbach conjecture which states that every even positive integer greater than four may be written as the sum of two odd primes. However, we do not know if the Even Goldbach conjecture is true. In this paper, we use the result that the Odd Goldbach conjecture is true and analyze the consequences of having atleast a single counterexample of the Even Goldbach conjecture.

**Category:** Number Theory

[1359] **viXra:1611.0224 [pdf]**
*submitted on 2016-11-14 18:05:57*

**Authors:** Jonas Kaiser

**Comments:** 11 Pages.

The sieve of Collatz is a new algorithm to trace back the non-linear Collatz problem to a linear cross out algorithm. Until now it is unproved.

**Category:** Number Theory

[1358] **viXra:1611.0178 [pdf]**
*submitted on 2016-11-12 09:51:56*

**Authors:** Aaron Chau

**Comments:** 3 Pages.

十分幸运，本文应用的是永不改变的定律（多与少），而不再是重复那类受局限的定理。
感谢数学的美妙，因为多与少的个数区别永远会造成二个质数的距离= 2。简述，= 2。

**Category:** Number Theory

[1357] **viXra:1611.0176 [pdf]**
*submitted on 2016-11-12 04:58:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of even numbers n for which n^2 is a Harshad-Coman number and I also make a classification in four classes of all the even numbers.

**Category:** Number Theory

[1356] **viXra:1611.0175 [pdf]**
*submitted on 2016-11-12 05:01:08*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of odd numbers n for which n^2 is a Harshad-Coman number and I also make a classification in three classes of all the odd numbers greater than 1.

**Category:** Number Theory

[1355] **viXra:1611.0172 [pdf]**
*submitted on 2016-11-11 15:58:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) If P is both a Poulet number and a Harshad number, than the number P – 1 is also a Harshad number; (II) If P is a Poulet number divisible by 5 under the condition that the sum of the digits of P – 1 is not divisible by 5 than P – 1 is a Harshad number; (III) There exist an infinity of Harshad numbers of the form P – 1, where P is a Poulet number.

**Category:** Number Theory

[1354] **viXra:1611.0171 [pdf]**
*submitted on 2016-11-11 16:00:16*

**Authors:** Marius Coman

**Comments:** 2 Pages.

OEIS defines the notion of Harshad numbers as the numbers n with the property that n/s(n), where s(n) is the sum of the digits of n, is integer (see the sequence A005349). In this paper I define the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer and I make the conjecture that there exist an infinity of Poulet numbers which are also Harshad-Coman numbers.

**Category:** Number Theory

[1353] **viXra:1611.0120 [pdf]**
*submitted on 2016-11-09 07:22:21*

**Authors:** Jian Ye

**Comments:** 3 Pages.

Goldbach’s conjecture: symmetrical primes exists in natural numbers. the generalized Goldbach’s conjecture: symmetry of prime number in the former and tolerance coprime to arithmetic progression still exists.

**Category:** Number Theory

[1352] **viXra:1611.0089 [pdf]**
*submitted on 2016-11-07 11:29:42*

**Authors:** W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache

**Comments:** 10 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo
Kakutani) and so on. Several various generalization of the Collatz conjecture
has been carried. In this paper a new generalization of the Collatz conjecture
called as the 3n ± p conjecture; where p is a prime is proposed. It functions on
3n + p and 3n - p, and for any starting number n, its sequence eventually enters
a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is
a special case of the 3n ± p conjecture when p is 1.

**Category:** Number Theory

[1351] **viXra:1611.0085 [pdf]**
*submitted on 2016-11-07 06:46:24*

**Authors:** Predrag Terzic

**Comments:** 32 Pages.

Some theorems and conjectures concerning prime numbers .

**Category:** Number Theory

[1350] **viXra:1610.0356 [pdf]**
*submitted on 2016-10-29 14:52:21*

**Authors:** Caitherine Gormaund

**Comments:** 2 Pages.

In which the Collatz Conjecture is proven using fairly simple mathematics.

**Category:** Number Theory

[1349] **viXra:1610.0349 [pdf]**
*submitted on 2016-10-28 13:23:48*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper we offer the some details and particulars about some famous conjectures in relative to consecutive primes.

**Category:** Number Theory

[1348] **viXra:1610.0313 [pdf]**
*submitted on 2016-10-26 05:42:56*

**Authors:** Jared Beal

**Comments:** 14 Pages.

This paper describes an algorithm for finding all the prime numbers. It also describes how this pattern among primes can be used to show the ratio of primes to not primes in an infinite set of X integers. It can also be used to show that the ratio of twin primes to not twin primes in an infinite set of X integers is always going to be greater than zero.

**Category:** Number Theory

[1347] **viXra:1610.0284 [pdf]**
*submitted on 2016-10-24 03:05:49*

**Authors:** Reuven Tint

**Comments:** Updates: 4.3.2 - 4.3.5.. page 7

Аннотация. Предложен вариант решения гипотезы Била с помощью прямого доказательства» Великой» теоремы Ферма элементарными методами. Новыми являются «инвариантное тождество « (ключевое слово) и полученные нами приведенные в тексте работы тождества, позволившие напрямую решить ВТФ и гипотезу Била,и ряд других. Предложены также новая формулировка теорем ( п.2.1.4.), ,доказательства для n= 1,2,3,..n>2 и x,y,z>2.

**Category:** Number Theory

[1346] **viXra:1610.0276 [pdf]**
*submitted on 2016-10-24 00:02:00*

**Authors:** John Smith

**Comments:** 19 Pages.

Riemann's prime-counting function R(x) looks good for every value of x we can compute, but in the light of Littlewood's result its superiority over li(x) is illusory: Ingram (1938) pointed out that 'for special values of x (as large as we please), the one approximation will deviate as widely as the other from the true value'. This note introduces a type of prime-counting function that is always better than li(x)...

**Category:** Number Theory

[1345] **viXra:1610.0275 [pdf]**
*submitted on 2016-10-23 13:15:42*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Аннотация. Интерес к названной в заглавии проблеме вызван следующими соображениями:
1) Возьмем, к примеру, «пифагорово» уравнение, все взаимно простые решения которого опре-
деляются формулами A= a^2- b^2 и B=2ab. Но если мы выберем A≠a^2- b^2 и B≠2ab как гипо-
тетически «верные» решения этого уравнения, то, наверное, можно будет доказать, что, в этом
случае, «пифагорово» уравнение не существует. Но оно действительно не существует для гипотетически выбранных «верных» решений.
2) Уравнение A^N+B^N = C^N и уравнение эллиптической кривой Фрея (как будет показано ниже для предложенного варианта их решения) не совместны.
3) Поэтому, как представляется, выглядит не совсем убедительной связь между уравнением
эллиптической кривой Фрея и соответствующим уравнением Ферма.
4) Приведено приложение.

**Category:** Number Theory

[1344] **viXra:1610.0274 [pdf]**
*submitted on 2016-10-23 13:19:39*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Annotation. Interest in the title problem is caused by the following considerations:
1) Take, for example, "Pythagoras' equation, all of which are relatively prime solutions determined
Delyan formulas A= a^2- b^2 and B=2ab. But if we choose A≠a^2- b^2 and B≠2ab both hypo-
Tethyan "correct" solutions of this equation, then perhaps it will be possible to prove that, in this
case, "Pythagoras" equation exists. But it really does not exist for the selected hypothetically "true" solutions.
2) The equation A^N+B^N = C^N and the equation of the elliptic curve Frey (as will be shown below for the proposed options to solve them) are not compatible.
3) Therefore, it seems, it does not look quite convincing relationship between the equation
elliptic curve Frey Farm and the corresponding equation.
4) Supplement.

**Category:** Number Theory

[1343] **viXra:1610.0272 [pdf]**
*submitted on 2016-10-23 13:58:45*

**Authors:** Luca Nascimbene

**Comments:** 13 Pages.

In this paper the author continue the works [6] [11] [12] and present a proposal for a demonstration on the Riemann Hypothesis and the conjecture on the multiplicity of non-trivial zeros of the Riemann s zeta.

**Category:** Number Theory

[1342] **viXra:1610.0253 [pdf]**
*submitted on 2016-10-21 18:17:51*

**Authors:** Filippos Nikolaidis

**Comments:** 10 Pages. fil_nikolaidis@yahoo.com

The present study is an effort for giving some evidence that the goldbach conjecture is not true, by showing that not all even natural numbers greater than two can be expressed as a sum of two primes. This conclusion can be drawn by showing that prime numbers are not enough –in population- so that, when added in couples, to give all the even numbers.

**Category:** Number Theory

[1341] **viXra:1610.0183 [pdf]**
*submitted on 2016-10-17 05:37:47*

**Authors:** Edward Szaraniec

**Comments:** 5 Pages.

Equation constituting the Beal conjecture is rearranged and squared, then rearranged
again and raised to power 4. The result, standing as an equivalent having the same
property, is emerging as a singular primitive Pythagorean equation with no solution.
So, the conjecture is proved. General line of proving the Pythagorean equation is
observed as a moving spirit.

**Category:** Number Theory

[1340] **viXra:1610.0172 [pdf]**
*submitted on 2016-10-16 05:13:25*

**Authors:** Mugur B. Răuţ

**Comments:** 5 Pages.

In this paper we propose another proof for Fermat’s Last Theorem (FLT). We found a simpler approach through Pythagorean Theorem, so our demonstration would be close to the times FLT was formulated. On the other hand it seems the Pythagoras’ Theorem was the inspiration for FLT. It resulted one of the most difficult mathematical problem of all times, as it was considered. Pythagorean triples existence seems to support the claims of the previous phrase.

**Category:** Number Theory

[1339] **viXra:1610.0106 [pdf]**
*submitted on 2016-10-10 03:35:21*

**Authors:** W.B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache

**Comments:** 9 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani's problem (after Shizuo Kakutani) and so on.
In this paper a new conjecture called as the 3n-1 conjecture which is akin to the Collatz conjecture is proposed. It functions on 3n -1, for any starting number n, its sequence eventually reaches either 1, 5 or 17. The 3n-1 conjecture is compared with the Collatz conjecture.

**Category:** Number Theory

[1338] **viXra:1610.0099 [pdf]**
*submitted on 2016-10-08 17:28:15*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve de la conjecture de Sophie Germain en utilisant le theoreme de densité de Chebotarev ,le principe d' inclusion d'exclusion de Moivre ,la formule de Mertens . en 13 pages nous donnons une preuve convaincante

In this paper We give Sophie Germain 's conjecture proof by using Chebotarev density theorem, principle inclusion -exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1337] **viXra:1610.0083 [pdf]**
*submitted on 2016-10-07 06:34:33*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

ζ(s)=1/(((1/(2))/log(2)))+ 1/(((1/(3))/log(3)))+ 1/(((1/(4))/log(4)))+1/(((1/(5))/log(5))) is a form of Riemann Zeta Function and it shows an approximate relationship between the Riemann Zeta Function and Prime Numbers.

**Category:** Number Theory

[1336] **viXra:1610.0082 [pdf]**
*submitted on 2016-10-07 06:37:51*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The classical Distribution of Primes Equation can be modified to make an Nth Prime Equation which generates the Nth Prime.

**Category:** Number Theory

[1335] **viXra:1610.0065 [pdf]**
*submitted on 2016-10-05 09:48:06*

**Authors:** Bing He

**Comments:** 14 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. These generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[1334] **viXra:1610.0042 [pdf]**
*submitted on 2016-10-04 12:01:31*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve du théorème de Mertens en utilisant le théorème de densité de Chebotarev ,principle d'inclusion - exclusion de Moivre,formule de Mertens en 15 pages nous donnons une élégante preuve

In this paper we give the proof of Sophie Germain's conjecture by using Chebotarev density theorem, the principle inclusion-exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1333] **viXra:1610.0034 [pdf]**
*submitted on 2016-10-03 19:56:15*

**Authors:** Chunxuan Jiang

**Comments:** 6 Pages.

using complex hyperbolic function we prove Fermat last theorem

**Category:** Number Theory

[1332] **viXra:1610.0033 [pdf]**
*submitted on 2016-10-03 20:01:14*

**Authors:** Chunxuan Jiang

**Comments:** 5 Pages.

using trogonometric function we prove Fermat last theorem

**Category:** Number Theory

[1331] **viXra:1610.0024 [pdf]**
*submitted on 2016-10-03 09:06:13*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

(1/2 Part)>1.002 (1.002, 2.16, 4.008 & 6.012) Generate Riemann Non Trivial Zero’s Off Of Critical Line. A Riemann Non Trivial Zero off the Critical Line occurs between 1 /2 or .50 and Gamma 0.577215664901532860606512090 08240243104 215 93 359399.When (1/2 Part) = (1.002 , 2.16, 4.008 & 6.012) Riemann Non Trivial Zero’s Are Off .001 To The Rt. Of The Critical Line & When (1/2 Part)= (1 / 2) A Riemann Non Trivial Zero’s Will Be On Critical Line.

**Category:** Number Theory

[1330] **viXra:1610.0016 [pdf]**
*submitted on 2016-10-02 14:25:25*

**Authors:** Philip E Gibbs

**Comments:** 14 Pages.

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search is used to find over 300 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[1329] **viXra:1610.0009 [pdf]**
*submitted on 2016-10-01 19:37:45*

**Authors:** Liujingru

**Comments:** 4 Pages.

This work reveals the intrinsic relationship of numbers with the conception of “prime multiple” to prove the “hypothesis of twin primes”. Based on this proof, “Goldbach conjecture” is proved with the “Odd-Gaussian Corresponding”. The nature of “prime number” can be thus obtained.Paper is using the axiom Ⅶ twice. For the first time: high high more than nonsingular group, according to the axiom Ⅶ get there will be a (high + high group). Second: high + high group) will be (prime number + prime)

**Category:** Number Theory

[1328] **viXra:1610.0008 [pdf]**
*submitted on 2016-10-01 20:19:40*

**Authors:** 刘静儒

**Comments:** 4 Pages.

通过“素数的倍数”这一概念，揭示了数的内在关系，论证了“孪生素数猜想”，并在此基础上给出了“奇高组”的定义，并结合“高斯对应”，论文只是两次运用公理Ⅶ。第一次：奇高组多于非奇高组，根据公理Ⅶ得到必有这样的结果：（奇高组+奇高组）。第二次：（奇高组+奇高组）必有这样的结果：（素数+素数），这就证明了“哥德巴赫猜想”。

**Category:** Number Theory

[1327] **viXra:1610.0001 [pdf]**
*submitted on 2016-10-01 01:46:45*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Let us consider positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments of same permutations of c kinds of integers’ points, where c≥1. In this article we proved Grimm’s conjecture by stepwise change symbols of each kind of composite numbers’ points at the number axis, so as to form consecutive composite numbers’ points under the qualification of proven Legendre-Zhang conjecture as the true.

**Category:** Number Theory

[1326] **viXra:1609.0425 [pdf]**
*submitted on 2016-09-29 11:39:24*

**Authors:** Philip E Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided

**Category:** Number Theory

[1325] **viXra:1609.0398 [pdf]**
*submitted on 2016-09-27 14:41:12*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand ...

**Category:** Number Theory

[1324] **viXra:1609.0384 [pdf]**
*submitted on 2016-09-26 21:46:39*

**Authors:** Bing He, Long Li

**Comments:** 16 Pages.

In this paper we give a finite field analogue of one of the Appell series and obtain some transformation and reduction formulae and the generating functions for the Appell series over finite fields.

**Category:** Number Theory

[1323] **viXra:1609.0383 [pdf]**
*submitted on 2016-09-26 23:16:52*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[1322] **viXra:1609.0377 [pdf]**
*submitted on 2016-09-26 11:05:09*

**Authors:** Brekouk

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[1321] **viXra:1609.0374 [pdf]**
*submitted on 2016-09-26 10:09:55*

**Authors:** Wei Ren

**Comments:** 17 Pages.

Collatz conjecture (or 3x+1 problem) is out for about 80 years. The
verification of Collatz conjecture has reached to the number about
60bits until now. In this paper, we propose new algorithms that can
verify whether the number that is about 100000bits (30000 digits)
can return 1 after 3*x+1 and x/2 computations. This is the largest
number that has been verified currently. The proposed algorithm
changes numerical computation to bit computation, so that extremely
large numbers (without upper bound) becomes possible to be verified.
We discovered that $2^{100000}-1$ can return to 1 after 481603 times
of 3*x+1 computation, and 863323 times of x/2 computation.

**Category:** Number Theory

[1320] **viXra:1609.0373 [pdf]**
*submitted on 2016-09-26 10:14:45*

**Authors:** Wei Ren

**Comments:** 22 Pages.

Collatz conjecture (or 3x+1 problem) has not been proved to be true
or false for about 80 years. The exploration on this problem seems
to ask for introducing a totally new method. In this paper, a
mathematical induction method is proposed, whose proof can lead to
the proof of the conjecture. According to the induction, a new
representation (for dynamics) called ``code'' is introduced, to
represent the occurred $3*x+1$ and $x/2$ computations during the
process from starting number to the first transformed number that is
less than the starting number. In a code $3*x+1$ is represented by 1
and $x/2$ is represented by 0. We find that code is a building block
of the original dynamics from starting number to 1, and thus is more
primitive for modeling quantitative properties. Some properties only
exist in dynamics represented by code, but not in original dynamics.
We discover and prove some inherent laws of code formally. Code as a
whole is prefix-free, and has a unified form. Every code can be
divided into code segments and each segment has a form $\{10\}^{p
\geq 0}0^{q \geq 1}$. Besides, $p$ can be computed by judging
whether $x \in[0]_2$, $x\in[1]_4$, or computed by $t=(x-3)/4$,
without any concrete computation of $3*x+1$ or $x/2$. Especially,
starting numbers in certain residue class have the same code, and
their code has a short length. That is, $CODE(x \in [1]_4)=100,$
$CODE((x-3)/4 \in [0]_4)=101000,$ $CODE((x-3)/4 \in
[2]_8)=10100100,$ $CODE((x-3)/4 \in [5]_8)=10101000,$ $CODE((x-3)/4
\in [1]_{32})=10101001000,$ $CODE((x-3)/4\in [3]_{32})=10101010000,$
$CODE((x-3)/4\in [14]_{32})=10100101000.$ The experiment results
again confirm above discoveries. We also give a conjecture on $x \in
[3]_4$ and an approach to the proof of Collatz conjecture. Those
discoveries support the proposed induction and are helpful to the
final proof of Collatz conjecture.

**Category:** Number Theory

[1319] **viXra:1609.0358 [pdf]**
*submitted on 2016-09-25 11:28:38*

**Authors:** N.Prosh

**Comments:** 6 Pages.

About prime numbers and new way of find prime numbers

**Category:** Number Theory

[1318] **viXra:1609.0353 [pdf]**
*submitted on 2016-09-25 09:09:01*

**Authors:** Brekouk

**Comments:** 12 Pages.

ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[1317] **viXra:1609.0263 [pdf]**
*submitted on 2016-09-18 00:13:23*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Honorable Pierre de Fermat could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Three similar versions of the proof are presented, using a single page for each version. The approach used in each proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary conditions in the solution for n = 2. The necessary conditions in the solutions for n = 2. will guide one to determine if there are solutions when n > 2.. For the first two versions, the proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1; and for the third version, on (a^2 + b^2)/c^2 = 1, with n = 2, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary conditions in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. For the first version of the proof, the proof began with reference to a right triangle. The second version of the proof began with ratio terms without any reference to a geometric figure. The third version occupies about half of a page. The third version of the proof began without any reference to a geometric figure or ratio terms. The second and third versions confirmed the proof in the first version. Each proof version is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, would be $107,250,000.

**Category:** Number Theory

[1316] **viXra:1609.0258 [pdf]**
*submitted on 2016-09-17 09:37:47*

**Authors:** Junnichi Fujii

**Comments:** 2 Pages.

The definition in time in the present-day physics is insufficient. Several problems which are to reconsider a definition in time and concern in time can be settled.

**Category:** Number Theory

[1315] **viXra:1609.0157 [pdf]**
*submitted on 2016-09-13 00:19:51*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[1314] **viXra:1609.0123 [pdf]**
*submitted on 2016-09-09 13:09:01*

**Authors:** T.Nakashima

**Comments:** 6 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.

**Category:** Number Theory

[1313] **viXra:1609.0121 [pdf]**
*submitted on 2016-09-09 13:54:25*

**Authors:** Bijoy Rahman Arif

**Comments:** 5 Pages.

In this paper, we are going to prove Oppermann’s conjecture which states there are at least one prime presents between first and second halves of two consecutive pronic numbers greater than one. Subsequently, we are going to prove the logarithmic sum of primes between two pronic numbers increase highest magnitude of log(4).

**Category:** Number Theory

[1312] **viXra:1609.0115 [pdf]**
*submitted on 2016-09-09 08:08:37*

**Authors:** Bijoy Rahman Arif

**Comments:** 4 Pages.

In this paper, we are going to find the number of primes between consecutive squares. We are going to prove a special case: Brocard’s conjecture which states between the square of two consecutive primes greater than 2 at least four primes will present. Subsequently, we will approximate the number of primes between consecutive square

**Category:** Number Theory

[1311] **viXra:1609.0112 [pdf]**
*submitted on 2016-09-09 06:28:05*

**Authors:** Bijoy Rahman Arif

**Comments:** 3 Pages.

In this paper, we are going to prove a famous problem concerning prime numbers. Legendre’s conjecture states that there is always a prime p with n^2 < p < (n+1)^2, if n > 0. In 1912, Landau called this problem along with other three problems “unattackable at the presesnt state of mathematics.” Our approach to solve this problem is very simple. We will find a lower bound of the difference of second Chebyshev functions using a better Moiver-Stirling approximation and finally, we transfer it to the difference of first Chebyshev functions. The final difference is always greater than zero will prove Legendre’s conjecture.

**Category:** Number Theory

[1310] **viXra:1609.0080 [pdf]**
*submitted on 2016-09-06 23:20:51*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[1309] **viXra:1609.0059 [pdf]**
*submitted on 2016-09-05 11:45:39*

**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

[1308] **viXra:1609.0058 [pdf]**
*submitted on 2016-09-05 11:49:34*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

This is the positive answer of Gilbreath's conjecture

**Category:** Number Theory

[1307] **viXra:1609.0052 [pdf]**
*submitted on 2016-09-04 16:05:23*

**Authors:** Aleksandr Tsybin

**Comments:** 3 Pages.

This problem is devoted a huge number of articles and books. So it does
not make sense to list them. I wrote this note 10 years ago and since then
a lot of time I tried to find the error in the reasoning and I can not this to
do. I’ll be glad if someone will be finds a mistake and even more will be
happy if an error will be not found.

**Category:** Number Theory

[1306] **viXra:1609.0048 [pdf]**
*submitted on 2016-09-05 06:28:40*

**Authors:** Predrag Terzic

**Comments:** 5 Pages.

Polynomial time compositeness tests for generalized Fermat numbers are introduced .

**Category:** Number Theory

[1305] **viXra:1609.0046 [pdf]**
*submitted on 2016-09-04 16:01:51*

**Authors:** Aleksandr Tsybin

**Comments:** 14 Pages.

For a positive integer n I construct an n × n matrix of special shape,
whose determinant equals the n-th prime number, and whose entries
are equal to 1,-1 or 0. Specific calculations which I have carried out
so far, allowed me to construct such matrices for all n up to 63.
These calculations are based on my own method for quick
calculations of determinants of special matrices along with a
variation on the Sieve of Eratosthenes.

**Category:** Number Theory

[1304] **viXra:1609.0030 [pdf]**
*submitted on 2016-09-03 03:04:18*

**Authors:** Alexander K

**Comments:** 2 Pages.

Fermat last theorem, fermat-catalan conjecture, beal conjecture proof

**Category:** Number Theory

[1303] **viXra:1609.0025 [pdf]**
*submitted on 2016-09-02 07:36:57*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

The even Goldbach conjecture states that any even integer greater than four may be expressed as the sum of two odd primes. The odd Goldbach conjecture states that any odd integer greater than seven must be expressible as a sum of three odd primes. These conjectures remain unverified. In this paper we explore the possible constraints that exist on the smallest possible counterexample of the even Goldbach conjecture. We prove that the odd numbers immediately flanking the smallest counterexample of the even Goldbach conjecture are themselves expressible as the sum of three odd primes and are therefore consistent with the odd Golbach conjecture.

**Category:** Number Theory

[1302] **viXra:1609.0012 [pdf]**
*submitted on 2016-09-01 00:00:52*

**Authors:** D. D. Somashekara, S. L. Shalini, K. N. Vidya

**Comments:** 15 Pages.

In this paper, we give an alternate and simple proofs for Sear’s three term 3 φ 2 transformation formula, Jackson’s 3 φ 2 transformation formula and for a nonterminating form of the q-Saalschütz sum by using q exponential operator techniques. We also give an alternate proof for a nonterminating form of the q-Vandermonde sum. We also obtain some interesting special cases of all the three identities, some of which are analogous to the identities stated by Ramanujan in his lost notebook.

**Category:** Number Theory

[1301] **viXra:1608.0449 [pdf]**
*submitted on 2016-08-31 17:53:09*

**Authors:** Joe Chizmarik

**Comments:** 2 Pages. This is a proof by contradiction.

We first prove a weak form of Fermat's Last Theorem; this unique lemma is key to the entire proof. A corollary and lemma follow inter-relating Pythagorean and Fermat solutions. Finally, we prove Fermat's Last Theorem.

**Category:** Number Theory

[1300] **viXra:1608.0439 [pdf]**
*submitted on 2016-08-30 21:46:42*

**Authors:** Watcharakiete Wongcharoenbhorn

**Comments:** 4 Pages. English

We study on the cycle in the Collatz conjecture and there is something surprise us. Our goal is to show that there is no Collatz cycle

**Category:** Number Theory

[1299] **viXra:1608.0429 [pdf]**
*submitted on 2016-08-31 09:51:31*

**Authors:** Gyeongmin Yang

**Comments:** 5 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[1298] **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

[1297] **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

[1296] **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

[1295] **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

[1294] **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

[1293] **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

[1292] **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

[1291] **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

[1290] **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

[1289] **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

[1288] **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

[1287] **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

[1286] **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

[1285] **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

[1284] **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

[1283] **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

[1282] **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

[1281] **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

[1280] **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

[1279] **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

[1278] **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

[1277] **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

[1276] **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

[1275] **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

[1274] **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

[1273] **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

[1272] **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

[1271] **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

[1270] **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

[606] **viXra:1610.0065 [pdf]**
*replaced on 2016-10-10 23:28:04*

**Authors:** Bing He

**Comments:** 22 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. Some of these generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[605] **viXra:1610.0016 [pdf]**
*replaced on 2016-10-26 05:46:31*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.29253.65761

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search has been used to find over 1000 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[604] **viXra:1609.0425 [pdf]**
*replaced on 2016-10-27 10:44:46*

**Authors:** Philip Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided.

**Category:** Number Theory

[603] **viXra:1609.0398 [pdf]**
*replaced on 2016-10-26 15:46:30*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentale des nombres premiers , ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture à savoir la Sommation et la primalité des ses éléments, ...et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[602] **viXra:1609.0383 [pdf]**
*replaced on 2016-10-01 23:01:09*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[601] **viXra:1609.0263 [pdf]**
*replaced on 2016-10-10 20:15:27*

**Authors:** A. A. Frempong

**Comments:** 6 Pages. Copyright © by A. A. Frempong

Honorable Pierre de Fermat could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Three similar versions of the proof are presented, using a single page for each version. The approach used in each proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary conditions in the solution for n = 2. The necessary conditions in the solutions for n = 2. will guide one to determine if there are solutions when n > 2.. For the first two versions, the proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1; and for the third version, on (a^2 + b^2)/c^2 = 1, with n = 2, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary conditions in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. For the first version of the proof, the proof began with reference to a right triangle. The second version of the proof began with ratio terms without any reference to a geometric figure. The third version occupies about half of a page. The third version of the proof began without any reference to a geometric figure or ratio terms. The second and third versions confirmed the proof in the first version. Each proof version is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, would be $107,250,000.

**Category:** Number Theory

[600] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-21 20:45:45*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple, (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system

**Category:** Number Theory

[599] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-18 23:32:38*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple, (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system

**Category:** Number Theory

[598] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-16 01:43:10*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a Pythagorean triple, a, b, c, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[597] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-13 13:35:35*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be
made in the system

**Category:** Number Theory

[596] **viXra:1609.0080 [pdf]**
*replaced on 2016-09-16 01:53:45*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1 for a Pythagorean triple, a, b, c, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[595] **viXra:1609.0080 [pdf]**
*replaced on 2016-09-13 21:17:38*

**Authors:** A. A. Frempong

**Comments:** 2 Pages. Copyright © by A. A. Frempong

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[594] **viXra:1609.0030 [pdf]**
*replaced on 2016-09-12 11:35:51*

**Authors:** Alexander K

**Comments:** 1 Page.

Beal conjecture, fermat last theorem

**Category:** Number Theory

[593] **viXra:1609.0030 [pdf]**
*replaced on 2016-09-11 16:47:25*

**Authors:** Alexander K

**Comments:** 2 Pages.

Beal conjecture, fermat last theorem

**Category:** Number Theory

[592] **viXra:1608.0429 [pdf]**
*replaced on 2016-11-18 21:18:34*

**Authors:** Gyeongmin Yang

**Comments:** 4 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[591] **viXra:1608.0375 [pdf]**
*replaced on 2016-08-30 23:38:11*

**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

[590] **viXra:1608.0100 [pdf]**
*replaced on 2016-11-30 02:33:10*

**Authors:** Choe Ryong Gil

**Comments:** 27 pages, 4 tables

In this paper we consider the Riemann hypothesis (RH) by the Euler function and the primorial numbers. We find a new sufficient condition for the RH from well known Robin theorem and prove that the condition holds unconditionally.

**Category:** Number Theory

[589] **viXra:1607.0360 [pdf]**
*replaced on 2016-10-25 13:27:33*

**Authors:** Reuven Tint

**Comments:** 7 Pages.

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

[588] **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