[11] **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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **viXra:1510.0100 [pdf]**
*submitted on 2015-10-12 07:01:38*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present two formulae which seems to conduct to primes or products of very few prime factors, both of them inspired by the prime decomposition of the first absolute Fermat pseudoprime, the number 561.

**Category:** Number Theory

[4] **viXra:1510.0089 [pdf]**
*replaced on 2015-10-15 20:36:26*

**Authors:** Mr Romdhane DHIFAOUI

**Comments:** 2 Pages.

Proof of Fermat's last theorem with conventional means.

**Category:** Number Theory

[3] **viXra:1510.0088 [pdf]**
*replaced on 2015-10-15 20:38:44*

**Authors:** Mr Romdhane DHIFAOUI

**Comments:** 4 Pages.

Another form of Fermat's last theorem : I prove that the Fermat's last theorem consist in finding 3 integers (x, y, and z) such as
〖(x+z)〗^n +(〖y+z)〗^n = 〖(x+ y+z)〗^n
From the Pythagorean triple we obtain a square equals the sum of three squares
If c2 = a2 + b2, and where d is the complement of c to (a + b) was
(c-d) 2 = (a-d) 2 + (b-d) 2 + d2.
From each even integer we obtain at least a Pythagorean triple
The surface of the Pythagorean triangle
Any number s = ( w^(3 )- w )/4 is the surface of a Pythagorean triangle
w^2+ 〖 (( w^2-1 )/2)〗^2=〖 (( w^2+1 )/2)〗^2

**Category:** Number Theory

[2] **viXra:1510.0085 [pdf]**
*replaced on 2015-10-15 20:34:15*

**Authors:** Mr Romdhane DHIFAOUI

**Comments:** 4 Pages.

Proof of Fermat's last theorem with conventional means.

**Category:** Number Theory

[1] **viXra:1510.0020 [pdf]**
*replaced on 2018-12-23 11:36:42*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 63 Pages. Final version of the proof. In French. Submitted to the Journal of the European Mathematical Society. Comments welcome.

In 1997, Andrew Beal announced the following conjecture: \textit{Let $A, B,C, m,n$, and $l$ be positive integers with $m,n,l > 2$. If $A^m + B^n = C^l$ then $A, B,$ and $C$ have a common factor.} We begin to construct the polynomial $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ with $p,q$ integers depending of $A^m,B^n$ and $C^l$. We resolve $x^3-px+q=0$ and we obtain the three roots $x_1,x_2,x_3$ as functions of $p,q$ and a parameter $\theta$. Since $A^m,B^n,-C^l$ are the only roots of $x^3-px+q=0$, we discuss the conditions that $x_1,x_2,x_3$ are integers and have or not a common factor. Three numerical examples are given.

**Category:** Number Theory