**Previous months:**

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2014 - 1401(20) - 1402(10) - 1403(23) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(52) - 1409(47) - 1410(17) - 1411(18) - 1412(18)

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

Any replacements are listed further down

[1031] **viXra:1511.0270 [pdf]**
*submitted on 2015-11-28 06:51:19*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I observe that many Poulet numbers P having a prime factor q of the form 30*n + 23, where n positive integer, can be written as P = m*(q^2 – q) + q^2, where m positive integer, and I conjecture that any Poulet number P having 23 as a prime factor can be written as P = 506*m + 529, where m positive integer.

**Category:** Number Theory

[1030] **viXra:1511.0229 [pdf]**
*submitted on 2015-11-23 22:19:44*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist, for any p prime, p greater than or equal to 7, an infinity of positive integers n such that the number n*p^2 – n*p + p – 2 is prime.

**Category:** Number Theory

[1029] **viXra:1511.0227 [pdf]**
*submitted on 2015-11-23 13:55:19*

**Authors:** Hitesh Jain

**Comments:** 5 Pages.

We obtained two interesting congruence relations related to Wilson’s theorem.

**Category:** Number Theory

[1028] **viXra:1511.0226 [pdf]**
*submitted on 2015-11-23 14:26:13*

**Authors:** David Brown

**Comments:** 4 Pages.

According to the Clay Mathematics Institute, “The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.” Furthermore, if you can write out a valid mathematical proof of the Riemann hypothesis and get it published in a refereed mathematical journal then the Clay Mathematics Institute will, after due deliberation, give you a prize of one million U.S. dollars. The Riemann hypothesis has a generalization to Dirichlet L-functions, among others. What might the Riemann hypothesis and medical predictions have in common? Experience suggests that both are difficult. It might be that accurate prediction of outcomes is mathematically and empirically intractable in almost all interesting cases. Stephen Wolfram’s Principle of Computational Equivalence states that “Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication.” This brief communication offers two conjectures concerning the generalized Riemann hypothesis for Dirichlet L-functions. In addition to medical doctors and number theory, this brief communication makes reference to Abraham Lincoln and a set of dogs with cardinality one.

**Category:** Number Theory

[1027] **viXra:1511.0218 [pdf]**
*submitted on 2015-11-23 03:45:24*

**Authors:** Dhananjay P. Mehendale

**Comments:** 4 pages.

Prime numbers are infinite since the time when Euclid gave his one of the most beautiful proof of this fact! Prime number theorem (PNT) reestablishes this fact and further it also gives estimate about the count of primes less than or equal to x. PNT states that as x tends to infinity the count of primes up to x tends to x divided by the natural logarithm of x. Twin primes are those primes p for which p+2 is also a prime number. The well known twin prime conjecture (TPC) states that twin primes are (also) infinite. Related to twin primes further conjectures that can be made by extending the thought along the line of TPC, are as follows: Prime numbers p for which p+2n is also prime are (also) infinite for all n, where n = 1(TPC), 2, 3, …, k, …. In this paper we provide a simple argument in support of all twin prime conjectures.

**Category:** Number Theory

[1026] **viXra:1511.0193 [pdf]**
*submitted on 2015-11-20 10:44:03*

**Authors:** Safaa Moallim

**Comments:** 10 Pages.

In this paper, I introduce two concepts, first is number influence strength, and second is count of influenced multiples less than x. going from there, we can calculate how many composites is less than x stepping to how many primes is less than x.

**Category:** Number Theory

[1025] **viXra:1511.0140 [pdf]**
*submitted on 2015-11-16 23:08:46*

**Authors:** Hajime Mashima

**Comments:** 1 Page.

Prime number is infinite. This proof is similar to the Euclid's theorem(300 B.C.).

**Category:** Number Theory

[1024] **viXra:1511.0102 [pdf]**
*submitted on 2015-11-12 15:42:14*

**Authors:** Kunle Adegoke

**Comments:** 21 Pages.

In this paper, direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. The high point of the paper is the discovery of certain combinations of Euler sums that are reducible to Riemann zeta values

**Category:** Number Theory

[1023] **viXra:1511.0031 [pdf]**
*submitted on 2015-11-03 11:56:46*

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

**Comments:** 188 Pages.

The authors in this book introduce a new class of natural neutrsophic numbers using MOD intervals. These natural MOD neutrosophic numbers behave in a different way for the product of two natural neutrosophic numbers can be neutrosophic zero divisors or idempotents or nilpotents. Several open problems are suggested in this book.

**Category:** Number Theory

[1022] **viXra:1510.0519 [pdf]**
*submitted on 2015-10-31 15:25:45*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 2 Pages.

We give a proof of Pierre de Fermat's Last Theorem using that Beal Conjecture is true (A. Ben Hadj Salem, viXra.org, 2015).

**Category:** Number Theory

[1021] **viXra:1510.0493 [pdf]**
*submitted on 2015-10-28 20:17:19*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 8 Pages.

Ramanujan’s Ternary Quadratic Form represents a series of numbers that satisfy a tripartite quadratic relation. In the present work, we examine the sequence of numbers generated by such forms and other related forms obtained by varying the coefficients and exponents to other values. Chaotic characterization using standard techniques such as Lyapunov Exponents, Kolmogorov Entropy, Fractal Dimensions, Phase Portraits and Distance plots is performed. It is seen that the Ramanujans Form as well as the related forms, when expressed as time series exhibit chaotic ehaviour. Finally, we conclude by stating that the form to series mapping outlined in the present work enables the generation of chaotic signals without the need for excessive system complexity and memory, and we note that such chaotic signals can be used as the basis for carriers in secure communication systems.

**Category:** Number Theory

[1020] **viXra:1510.0491 [pdf]**
*submitted on 2015-10-28 20:19:51*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 3 Pages.

This paper discusses the origins, arithmetic operations and conversion operations involving a non-zero ternary number system, comprising of the numbers 1, 8 and 9. This number system, christened the “OEN number system”, presents itself as a strong contender for coding and encrypting applications due to the absence of zero, which makes it impossible to use conventional coding/ encrypting algorithms, thus providing a higher degree of security.

**Category:** Number Theory

[1019] **viXra:1510.0475 [pdf]**
*submitted on 2015-10-28 23:22:39*

**Authors:** Stephen Crowley

**Comments:** 6 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated and it's relation to optimal control theory, the calculus of variations, and viscosity solutions theory are explored as a way to prove the existence and uniqueness of a solution to an infinite dimensional system of transcendental equations which has been shown to be equivalent to the Riemann Hypothesis

**Category:** Number Theory

[1018] **viXra:1510.0425 [pdf]**
*submitted on 2015-10-27 15:46:51*

**Authors:** Victor Sorokine

**Comments:** 3 Pages.

Preuve élémentaire du FLT en 15 lines ne comprend que quatre opérations:
1) 1 х 1 = 1,
2) a + 1 > a,
3) La solution de l'équation a+x=n est x=n-a, et une déclaration:
4) le système {A+B-C>0, A+B-C=0} est incompatible. Et TOUT! (I.e. : Si A, B, C sont naturels et A^n+B^n=C^n, alors A+B=C ! No errors!)

**Category:** Number Theory

[1017] **viXra:1510.0372 [pdf]**
*submitted on 2015-10-23 19:12:53*

**Authors:** Alfredo Olmos Hernández

**Comments:** 7 Pages.

Beal's conjecture is studied, which arises from investigations Andrew Beal, on Fermat's Last Theorem in 1993.
Beal's conjecture, proposed to the equation a^x+b^y=c^z where A, B, C, x, y, z positive integers x, y, z> 2 so that the equation has a solution A, B, and C must have a common prime factor.
Given the vain attempts to find a counterexample to the conjecture (which has been proven by the help of modular arithmetic, for all values of the six variables to a value of 1000) values for the six variables. To advance the theory of numbers.
Given the relationship that has Beal's conjecture with Fermat's last theorem; it is considered important to number theory, demonstration of this conjecture.
To solve Beal's conjecture, using Fermat's last theorem and the remainder theorem is proposed.

**Category:** Number Theory

[1016] **viXra:1510.0348 [pdf]**
*submitted on 2015-10-21 13:19:24*

**Authors:** Victor Sorokine

**Comments:** 1 Page.

Aucune conclusion, preuve élémentaire du DTF en 15 lignes ne comprend que quatre opérations arithmétiques: 1) 1 х 1 = 1,
2) a + 1 > a,
3) La solution de l'équation a+x=n est x=n-a, et une déclaration:
4) le système {A+B-C>0, A+B-C=0} est contradictoir.
Et TOUT!
P.S. 8000 personnes (et 2 professeurs d'université) ont regardé la preuve sur les forums de discussion. Personne n'a détecté une erreur.

**Category:** Number Theory

[1015] **viXra:1510.0339 [pdf]**
*submitted on 2015-10-21 03:45:24*

**Authors:** Zeraoulia Elhadj

**Comments:** 6 Pages.

In this paper, we define a positive integers sub-sequence that gives twin primes. Then we show that the twin prime conjecture is true by using simple mathematical manipulations.

**Category:** Number Theory

[502] **viXra:1511.0227 [pdf]**
*replaced on 2015-11-23 22:21:31*

**Authors:** Hitesh Jain

**Comments:** 5 Pages.

We obtained two interesting congruence relations related to Wilson’s theorem.

**Category:** Number Theory

[501] **viXra:1511.0226 [pdf]**
*replaced on 2015-11-27 05:22:41*

**Authors:** David Brown

**Comments:** 3 Pages.

According to the Clay Mathematics Institute, “The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann's 1859 paper, it asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.” Furthermore, if you can write out a valid mathematical proof of the Riemann hypothesis and get it published in a refereed mathematical journal then the Clay Mathematics Institute will, after due deliberation, give you a prize of one million U.S. dollars. The Riemann hypothesis has a generalization to Dirichlet L-functions, among others. What might the Riemann hypothesis and medical predictions have in common? Experience suggests that both are difficult. It might be that accurate prediction of outcomes is mathematically and empirically intractable in almost all interesting cases. Stephen Wolfram’s Principle of Computational Equivalence states that “Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication.” This brief communication offers two conjectures concerning the generalized Riemann hypothesis for Dirichlet L-functions. In addition to medical doctors and number theory, this brief communication makes reference to Abraham Lincoln and a set of dogs with cardinality one.

**Category:** Number Theory

[500] **viXra:1511.0102 [pdf]**
*replaced on 2015-11-17 20:25:56*

**Authors:** Kunle Adegoke

**Comments:** 25 Pages. Added more theorems and examples

In this paper, direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. The high point of the paper is the discovery of certain combinations of Euler sums that are reducible to Riemann zeta values

**Category:** Number Theory

[499] **viXra:1511.0102 [pdf]**
*replaced on 2015-11-13 15:37:58*

**Authors:** Kunle Adegoke

**Comments:** 21 Pages. Corrected typos

In this paper, direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. The high point of the paper is the discovery of certain combinations of Euler sums that are reducible to Riemann zeta values

**Category:** Number Theory

[498] **viXra:1511.0102 [pdf]**
*replaced on 2015-11-13 13:57:51*

**Authors:** Kunle Adegoke

**Comments:** 21 Pages. Corrected typos

**Category:** Number Theory

[497] **viXra:1510.0475 [pdf]**
*replaced on 2015-11-21 16:33:31*

**Authors:** Stephen Crowley

**Comments:** 10 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated. The França-LeClair exact equation for the Riemann zeros ϑ(t_)+S(t_)=(n-3/2)π (where ϑ(t) is the Riemann-Siegel theta function and S(t) is the argument of the Riemann zeta function on the line with real part 1/2) proves that Riemann's original hypothesis originally stated in 1859 if it can be shown that there is a unique well-defined solution for each integer n⩾1. An expression for S(t) as an infinite sum over an orthogonal basis of Chebyshev polynomials which relates to Li's criteria for the Riemann hypothesis is also explored. An extension of the Berry-Keating Hamiltonian is also suggested where the infinite sequence of roots are considered as an unbounded set of viscosity solutions satisfying the Hamilton-Jacobi equations. It is also noted that the Chebyshev polynomials used in the Li coefficients are orthogonal with respect to sqrt(1-t^2)∀t∈[-1,-1], the multiplicative inverse of which appears in the Lagrangian of the prototypical example of finding a real-valued function on the interval [t_0,t_1] whose graph has a length which is as short as possible and satisfies the boundary conditions f(t_0)=x_0 and f(t_1)=x_1.

**Category:** Number Theory

[496] **viXra:1510.0475 [pdf]**
*replaced on 2015-11-10 18:05:36*

**Authors:** Stephen Crowley

**Comments:** 9 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated. The França-LeClair exact equation for the Riemann zeros ϑ(t)+S(t)=(n-3/2)π (where ϑ(t) is the Riemann-Siegel theta function and S(t) is the argument of the Riemann zeta function on the line with real part 1/2) proves that Riemann's original hypothesis originally stated in 1859 if it can be shown that there is a unique well-defined solution for each integer n⩾1. An expression for S(t) as an infinite sum over an orthogonal basis of Chebyshev polynomials which relates to Li's criteria for the Riemann hypothesis is also explored. An extension of the Berry-Keating Hamiltonian is also suggested where the infinite sequence of roots are considered as an unbounded set of viscosity solutions satisfying the Hamilton-Jacobi equations. It is also noted that the Chebyshev polynomials are orthogonal with respect to sqrt(1-t^2)∀t∈[-1,-1], the multiplicative inverse of which appears in the Lagrangian of the prototypical example of finding a real-valued function on the interval $[t_0,t_1]$ whose graph has a length which is as short as possible and satisfies the boundary conditions $f(t_0)=x_0$ and $f(t_1)=x_1$.

**Category:** Number Theory

[495] **viXra:1510.0475 [pdf]**
*replaced on 2015-11-08 17:19:36*

**Authors:** Stephen Crowley

**Comments:** 8 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated. The França-LeClair exact equation for the Riemann zeros
ϑ(t_n)+S(t_n) =(n-3/2)π proves the Riemann Hypothesis if it can be shown that there is uniquel well-defined solution for each intger n⩾1. An expression for S(t) as an infinite sum over an orthogonal basis of Chebyshev polynomials is also explored. An extension of the Berry-Keating Hamiltonian is also suggested where the infinite sequence of roots are considered as an unbounded set of visicosity solutions satisfying the Hamilton-Jacobi equations.

**Category:** Number Theory

[494] **viXra:1510.0475 [pdf]**
*replaced on 2015-11-04 17:47:29*

**Authors:** Stephen Crowley

**Comments:** 8 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated. The variational iteration method, based on the Banach fixed point theorem, which generate a rapdily convergent series expansion, is suggested as a way to calculate analytic solutions to the França-LeClair exact equation for the Riemann zeros ϑ(t_n)+S(t_n) =(n-3/2)π, whose uniquely existing solution for each n is equivalent to the Riemann Hypothesis. An extension of the Berry-Keating Hamiltonian is also suggested.

**Category:** Number Theory

[493] **viXra:1510.0475 [pdf]**
*replaced on 2015-10-30 01:54:27*

**Authors:** Stephen Crowley

**Comments:** 5 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated and it's relation to optimal control theory, the calculus of variations, and viscosity solutions theory are explored as a way to prove the existence and uniqueness of a solution to an infinite dimensional system of transcendental equations which has been shown to be equivalent to the Riemann Hypothesis. Some very striking and remarkable graphs are also included.

**Category:** Number Theory

[492] **viXra:1510.0475 [pdf]**
*replaced on 2015-10-29 17:45:02*

**Authors:** Stephen Crowley

**Comments:** 8 Pages.

The integral of the logarithmic derivative of the Hardy Z function is calculated and it’s relation to optimal control theory, the calculus of variations, and viscosity solutions theory are explored as a way to prove the existence and uniqueness of a solution to an infinite dimensional system of transcendental equations which has been shown to be equivalent to the Riemann Hypothesis. Some very striking and remarkable graphs are also included.

**Category:** Number Theory