[16] **viXra:1812.0208 [pdf]**
*submitted on 2018-12-11 16:27:17*

**Authors:** Kenneth A. Watanabe

**Comments:** 9 Pages.

A twin prime is defined as a pair of prime numbers (p1,p2) such that p1 + 2 = p2. The Twin Prime Conjecture states that there are an infinite number of twin primes. The first mention of the Twin Prime Conjecture was in 1849, when de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case where k = 1 is the Twin Prime Conjecture. In this document, I derive a function that corresponds to the number of twin primes less than n for large values of n. This equation is identical to that used to prove the Goldbach Conjecture. Then by proof by induction, it is shown that as n increases indefinitely, the function also increases indefinitely thus proving the Twin Prime Conjecture. Using the same methodology, de Polignac’s conjecture is also shown to be true.

**Category:** Number Theory

[15] **viXra:1812.0182 [pdf]**
*submitted on 2018-12-10 10:58:25*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 4 Pages. Submitted to the journal Research In Number Theory. Comments welcome.

In this paper, we use the Fermat's Last Theorem (FLT) to give a proof of the ABC conjecture. We suppose that FLT is false ====> we arrive that the ABC conjecture is false. Then taking the negation of the last statement, we obtain: ABC conjecture is true ====> FLT is true. But, as FLT is true, then we deduce that the ABC conjecture is true.

**Category:** Number Theory

[14] **viXra:1812.0154 [pdf]**
*submitted on 2018-12-08 16:12:41*

**Authors:** M. Sghiar

**Comments:** 7 Pages. french version © Copyright 2018 by M. Sghiar. All rights reserved. Respond to the author by email at: msghiar21@gmail.com

I show here that if $ x \in \mathbb{N}^*$ then $1 \in \mathcal{O}_S (x)= \{ S^n(x), n \in \mathbb{N}^* \} $ where $ \mathcal{O}_S (x)$ is the orbit of the function S defined on $\mathbb{R}^+$ by $S(x)= \frac{x}{2} + (x+\frac{1}{2}) sin^2(x\frac{\pi}{2})$, and I deduce the proof of the Syracuse conjecture.

**Category:** Number Theory

[13] **viXra:1812.0150 [pdf]**
*replaced on 2018-12-11 10:03:58*

**Authors:** Phil Aaron Bloom

**Comments:** 2 Pages.

An open problem is proving FLT simply for each $n\in\mathbb{N}, n>2$. Our \emph{direct proof} (not by way of contradiction) of FLT is based on our algebraic identity, denoted, {for convenience}, as $(r)^n+(s)^n=(t)^n$ with $r,s,t>0$ as functions of variables. We infer that $\{(r,s,t)|r,s,t\in\mathbb{N},(r)^n+(s)^n+(t)^n\}=\{(x,y,z)|x,y,z\in\mathbb{N},(x)^n+y^n=z^n\}$ for $n\in\mathbb{N}, n>2$. In addition, we show, for integral values of $n>2$, that $\{(r,s,t)|r,s,t\in\mathbb{N},(r)^n+(s)^n=t^n\}=\varnothing$. Hence, for $n\in\mathbb{N},n>2$, it is true that $\{(x,y,z)|x,y,z\in\mathbb{N},x^n+y^n=z^n\}=\varnothing$.

**Category:** Number Theory

[12] **viXra:1812.0133 [pdf]**
*submitted on 2018-12-07 13:32:39*

**Authors:** James Edwin Rock

**Comments:** 5 Pages. Copyright 2018 James Edwin Rock Create Commons Attribution-ShareAlike 4.0 International License

Collatz sequences are formed by applying the Collatz algorithm to a positive integer. If it is even repeatedly divide by two until it is odd, then multiply by three and add one to get an even number and vice versa. Eventually you get back to one. The Collatz Structure is created, which contains all positive integers exactly once. The terms of the Collatz Structure are joined together via the Collatz algorithm. Thus, every positive integer forms a Collatz sequence with unique terms terminating in the number one.

**Category:** Number Theory

[11] **viXra:1812.0130 [pdf]**
*submitted on 2018-12-07 21:04:48*

**Authors:** Zhang Tianshu

**Comments:** 12 Pages.

Since there are infinitely many consecutive satisfactory values of ε to enable A+B=C satisfying C>(rad(A, B, C))1+ε, thus the author uses a representative equality, namely 1+2N(2N-2)=(2N-1)2 satisfying (2N-1)2>[rad(1, 2N(2N-2), (2N-1)2)]1+ε, and that first let ε equal a value near the greater end of the infinitely many consecutive satisfactory values to prove the ABC conjecture; again let ε equal a value near the smaller end to negate the ABC conjecture. This shows that the ABC conjecture is in the ambiguity in which case of ε>0.

**Category:** Number Theory

[10] **viXra:1812.0112 [pdf]**
*submitted on 2018-12-06 09:38:05*

**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

[9] **viXra:1812.0107 [pdf]**
*submitted on 2018-12-06 14:23:59*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 8 Pages. Comments welcome.

In this paper, we give the elliptic curve (E) given by the equation:
y^2=x^3+px+q
with $p,q \in Z$ not null simultaneous. We study a part of the conditions verified by $(p,q)$ so that it exists (x,y) \in Z^2 the coordinates of a point of the elliptic curve (E) given by the equation above.

**Category:** Number Theory

[8] **viXra:1812.0074 [pdf]**
*replaced on 2018-12-10 15:13:28*

**Authors:** Stephen Marshall

**Comments:** 8 Pages.

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zero’s only at the negative even integers and complex numbers with real part 1/2
The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000).
It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann hypothesis implies significant results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series:
ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...
The Riemann hypothesis asserts that all interesting solutions of the equation:
ζ(s) = 0
lie on a certain vertical straight line.
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
Hn = 1 + 1/2+1/3+1/4+⋯+ 1/n = ∑_(n=1)^n▒1/n
Harmonic numbers have been studied since early times and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function.
The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.

**Category:** Number Theory

[7] **viXra:1812.0071 [pdf]**
*submitted on 2018-12-04 22:22:02*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

Using the standard wiki definition of the Collatz conjecture, we map a positive number to imply that a divisor of two implies either an even numbered result (unchanged) or an odd numbered result (changed to the number multiplied by three plus one) to imply the final result of one. This is the shortest known confirmation of the conjecture, and in mathematical logic.

**Category:** Number Theory

[6] **viXra:1812.0040 [pdf]**
*submitted on 2018-12-04 02:42:56*

**Authors:** Toshiro Takami

**Comments:** 3 Pages.

40.5+(n^2+1)/4 (n is positive integers)
This prime number production formula product integers and non-integers alternately, but picking up an integer portion yields a prime number up to the 40th number, and even after an integer that is not a prime number, many are prime numbers. Even in the region where prime numbers are prime, many are made prime.

**Category:** Number Theory

[5] **viXra:1812.0022 [pdf]**
*replaced on 2018-12-02 07:09:24*

**Authors:** Pankaj Mani, Frm, CQF manipankaj9@gmailcom, @mathstud1

**Comments:** 12 Pages.

Riemann Hypothesis is TRUE if we look at the Functional Equation satisfied by the Riemann Zeta function
upon analytical continuation in Game Perspective way as visualized by David Hilbert. It uses technical game
theoretical concepts e.g. Nash Equilibrium to confidently assert that Riemann Hypothesis has to be True.
Needs to be looked at the Foundational Principles underlying Mathematics. In other words,it’s the game of
arranging Zeros in the complex plane using the functional equation.

**Category:** Number Theory

[4] **viXra:1812.0020 [pdf]**
*submitted on 2018-12-01 15:11:29*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. Comments welcome.

In this paper, we assume that Beal conjecture is true, we give a complete proof of the ABC conjecture. We consider that Beal conjecture is false $\Longrightarrow$ we arrive that the ABC conjecture is false. Then taking the negation of the last statement, we obtain: ABC conjecture is true $\Longrightarrow$ Beal conjecture is true. But, if the Beal conjecture is true, then we deduce that the ABC conjecture is true

**Category:** Number Theory

[3] **viXra:1812.0019 [pdf]**
*replaced on 2018-12-08 04:52:18*

**Authors:** Kamal Barghout

**Comments:** 12 Pages. The material in this article is copyrighted. Please obtain authorization from the author before the use of any part of the manuscript

A probabilistic proof of the Collatz conjecture is described via identifying a sequential permutation of even natural numbers by divisions by 2 that follows a recurrent pattern of the form x,1,x,1…, where x represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to divisions by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1 of high counts. Considering Collatz function producing random numbers and over sufficient iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming the only cycle of the function is 1-4-2-1.

**Category:** Number Theory

[2] **viXra:1812.0018 [pdf]**
*submitted on 2018-12-01 17:00:45*

**Authors:** Khalid Ibrahim

**Comments:** 91 Pages.

In this paper, not only did we disprove the Riemann Hypothesis (RH) but also we showed that zeros of the Riemann zeta function $\zeta (s)$ can be found arbitrary close to the line $\Re (s) =1$. Our method to reach this conclusion is based on analyzing the fine behavior of the partial sum of the Dirichlet series with the Mobius function $M (s) = \sum_n \mu (n) /n^s$ defined over $p_r$ rough numbers (i.e. numbers that have only prime factors greater than or equal to $p_r$). Two methods to analyze the partial sum fine behavior are presented and compared. The first one is based on establishing a connection between the Dirichlet series with the Mobius function $M (s) $ and a functional representation of the zeta function $\zeta (s)$ in terms of its partial Euler product. Complex analysis methods (specifically, Fourier and Laplace transforms) were then used to analyze the fine behavior of partial sum of the Dirichlet series. The second method to estimate the fine behavior of partial sum was based on integration methods to add the different co-prime partial sum terms with prime numbers greater than or equal to $p_r$. Comparing the results of these two methods leads to a contradiction when we assume that $\zeta (s)$ has no zeros for $\Re (s) > c$ and $c <1$.

**Category:** Number Theory

[1] **viXra:1812.0002 [pdf]**
*submitted on 2018-12-01 05:01:55*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

In this article, the author first classify A, B and C according to their respective odevity, and thereby get rid of two kinds which belong not to AX+BY=CZ. Then, affirm the existence of AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, prove AX+BY≠CZ in which case A, B and C have not any common prime factor by the mathematical induction with the aid of the distinct odd-even relation on the premise whereby even number 2W-1HZ as symmetric center of positive odd numbers concerned after divide the inequality in four. Finally, reach 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