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2016 - 1601(14) - 1602(18) - 1603(78) - 1604(56) - 1605(29)

Any replacements are listed further down

[1253] **viXra:1605.0247 [pdf]**
*submitted on 2016-05-24 00:51:23*

**Authors:** Hajime Mashima

**Comments:** 4 Pages.

This is the expectation that "two or more of the natural number 4/n will be represented by the sum of the three unit fractions".

**Category:** Number Theory

[1252] **viXra:1605.0245 [pdf]**
*submitted on 2016-05-23 13:23:29*

**Authors:** Terubumi Honjou

**Comments:** 4 Pages.

The new equation by the elementary particle pulsation hypothesis.
Arrow of time turns. In response to it, mass m changes with +1, 0, -1.
Arrow of time turns.
It is the same as the arrow of time of the Schrodinger equation.
Arrow of time turns.
It is the same as a material wave of the elementary particle pulsation hypothesis.
Current physics is always +mc²>0. Elementary particle mass m does not change.
A pulsation principle suggests a super-high-speed change of elementary particle mass m.

**Category:** Number Theory

[1251] **viXra:1605.0229 [pdf]**
*submitted on 2016-05-22 07:48:37*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

As non-trivial Riemann zero's grow larger on the 1/2 critical line so does the distance grow proportionally between 0 and 1. Therefore, a non-trivial Riemann zero will not be outside the critical strip or off of the critical line.

**Category:** Number Theory

[1250] **viXra:1605.0224 [pdf]**
*submitted on 2016-05-22 03:27:18*

**Authors:** V.I.Saenko

**Comments:** 4 Pages.

It is proved that the point $(x=q, y=t )$ does not lie on a cubic curve $y^2 t^{p-2}-x^3 q^{p-3}=1 $ at any $(t,q) \in Q $ and prime $ p=2 i+1 \geq 3 $ . Therefore $ t^p-q^p=1$ is unsolvable in rational numbers and the Fermat's last theorem holds true.

**Category:** Number Theory

[1249] **viXra:1605.0221 [pdf]**
*submitted on 2016-05-21 11:18:19*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The purpose of this paper is to share a simple way to solve a Sodoku in three steps. (P = NP ).

**Category:** Number Theory

[1248] **viXra:1605.0214 [pdf]**
*submitted on 2016-05-20 17:06:37*

**Authors:** T.Nakashima

**Comments:** 3 Pages.

Odd perfect number can’t exist.

**Category:** Number Theory

[1247] **viXra:1605.0201 [pdf]**
*submitted on 2016-05-19 07:29:23*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

Proof (article) for the Lehman expectation by Mr.Dobranju became the topic.
I think. A prime number and an illustration of commentary of the fusion of the physics make up for prism interpretation. (Mr.Dobranju)
Various wavelengths of the material wave appear as light (rainbow) of each color in the horizon (three-dimensional space).
The summary of the article is called 0 points of prism interpretation.
Light is branched into the light of various colors by a prism and is projected on a straight line.
The figure of prime number, physics fusion by the elementary particle pulsation principle is similar to 0 points of prism interpretation.
A material wave and the point of intersection with the horizon (straight line of Lehman) are zero points of the Lehman expectation.
A prime number and elementary particle pulsation principle hypothesis.(2).
(Fusion of a prime number and the physics. )

**Category:** Number Theory

[1246] **viXra:1605.0195 [pdf]**
*submitted on 2016-05-18 18:45:13*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, .the general equation, c^n = a^n + b^n holds, followed by showing that if
n > 2, the general equation, c^n = a^n + b^n does not hold. Let a,, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also let the acute angle between the hypotenuse and the horizontal be denoted by theta. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin space for it in his paper.

**Category:** Number Theory

[1245] **viXra:1605.0177 [pdf]**
*submitted on 2016-05-16 10:59:18*

**Authors:** T.Nakashima

**Comments:** 3 Pages.

Some type of odd perfect number can't exist.Our result is already
known.

**Category:** Number Theory

[1244] **viXra:1605.0164 [pdf]**
*submitted on 2016-05-14 20:42:59*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

I introduce a congruence that restates the characterization of primes that have 2 as a quadratic residue, non-residue.

**Category:** Number Theory

[1243] **viXra:1605.0162 [pdf]**
*submitted on 2016-05-15 02:34:43*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

This short paper presents a new form of the Liouville function. To the author's best knowledge, this relationship is previously unknown.

**Category:** Number Theory

[1242] **viXra:1605.0160 [pdf]**
*submitted on 2016-05-14 14:20:12*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

In this short paper I present a closed form formula for the right half of Pascal's triangle.

**Category:** Number Theory

[1241] **viXra:1605.0140 [pdf]**
*submitted on 2016-05-13 11:27:31*

**Authors:** José de Jesús Camacho Medina

**Comments:** 2 Pages.

This article disseminates a new mathematical concept designed by the author in the year 2015 to which entitled: "Magic Dart", this mathematical entity has a property exquisite not before discovered by the mathematical community, which will be of great pleasure for those who are interested in the educational and recreational mathematics.
One of the activities that are carried out by the author in his daily work is the teaching practice and it is there where attempts to combine the objectives entrusted to him in the delivery of the classes with the implementation of the Recreational Mathematics, always seeking to expand and disclose in an entertaining way the mathematical knowledge that are manifest in a classroom.

**Category:** Number Theory

[1240] **viXra:1605.0134 [pdf]**
*submitted on 2016-05-12 19:39:00*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

Online posting to YOUTUBE.
Einstein continued looking for four-dimensional space.
I discovered 4-dimensional space.
I upload it in 2010/09/02.
Contributor excellent Hikari368.
It is 215533 access (as of 2016.5.12).
As the key which realized a general theory, Einstein of later years continued looking for four-dimensional space, but was not able to carry it out. An elementary particle pulsation principle discovered the four-dimensional space. The elementary particle pulsates in the four-dimensional space.

**Category:** Number Theory

[1239] **viXra:1605.0131 [pdf]**
*submitted on 2016-05-13 01:10:28*

**Authors:** Terubumi Honjou

**Comments:** 11 Pages.

Online posting to YOUTUBE.
When the mystery of the prime number is removed, the elementary particle pulsation principle will be recognized as the theory of all things.
I assume real number axis, cross axle an imaginary number axis with a vertical axis and insert a figure of elementary particle pulse motion principle energy wave pattern in the complex number coordinate which developed the non-self-evident zero point of a prime number and the zeta function.
I assumed the straight line that 0 points formed a line the horizon which showed the vacuum space of the figure of elementary particle pulsation principle energy wave pattern . I turn a prime number and a figure of that Mr. Sugimoto made which showed 0 points 90 degrees. Non-self-evident straight line and gap 1/2 with the imaginary number axis where 0 points form a line of the zeta function show energy density (zero point energy) of the vacuum space.
I set the top (a mountain and valley) of the wave pattern of the elementary particle pulsation with the prime number on the real number axis. Furthermore, I set it as if a wave pattern passes on a non-self-evident zero point.
I assumed a prime number an orbital pole in a period and assumed a point zero eigenvalue.
The wave (peculiar empty ¬ interval) of the material wave pulsating every 0 points accompanies it. This wave pattern shows elementary particle pulsation, and a minus number particle, a wave pattern do a horizontal state with a wave in the grain ¬ child whom the energy of the place centered on in the heap of wave patterns, the valley of the wave pattern and repeat particle (+mc²), wave (0)¬, conversion (pulsation) of minus number particle (-mc²) super in (Planck time) for a short time.

**Category:** Number Theory

[1238] **viXra:1605.0123 [pdf]**
*submitted on 2016-05-11 23:17:44*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have spotted an error of Euler's proof, so that the used infinite descent is impossible in his proof

**Category:** Number Theory

[1237] **viXra:1605.0120 [pdf]**
*submitted on 2016-05-11 13:47:07*

**Authors:** Terubumi Honjou

**Comments:** 20 Pages.

The author announced the elementary particle pulsation hypothesis in the Physical Society of Japan of 1980.
To date, I study an elementary particle pulsation hypothesis.
The author contributed the article that let a prime number and physics fuse. youtube (December, 2012).
This page is the gravity in the elementary particle pulsation hypothesis and an article about the electromagnetic force.

**Category:** Number Theory

[1236] **viXra:1605.0113 [pdf]**
*submitted on 2016-05-11 04:59:52*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[1235] **viXra:1605.0105 [pdf]**
*submitted on 2016-05-10 13:36:32*

**Authors:** Terubumi Honjo

**Comments:** 20 Pages.

The author announced the elementary particle pulsation hypothesis in the Physical Society of Japan of 1980.
To date, I study an elementary particle pulsation hypothesis.
The author contributed the article that let a prime number and physics fuse. youtube (December, 2012).
This page is the gravity in the elementary particle pulsation hypothesis and an article about the electromagnetic force.

**Category:** Number Theory

[1234] **viXra:1605.0104 [pdf]**
*submitted on 2016-05-10 13:58:07*

**Authors:** José de Jesús Camacho Medina

**Comments:** 4 Pages.

This article educational and recreational mathematics, broadcasts a new category of numbers designed by the author in the year 2014 to which I have called "Numbers Fresnillenses" which permeates this title to a sense of belonging because I have is a native from the city of Fresnillo Zacatecas Mexico. This new classification of numbers you have an exquisite property not previously discovered by the mathematical community; it will be a great pleasure for fans of the numbers.

**Category:** Number Theory

[1233] **viXra:1605.0103 [pdf]**
*submitted on 2016-05-10 14:09:27*

**Authors:** José de Jesús Camacho Medina

**Comments:** 11 Pages.

This article offers a solution in relation to the distribution of the primes, in this research we provide simple formulas and unpublished with a new approach that allow us to assimilate and conclude that the entities primales are sorted as regular as possible. We provide a new vision for addressing what that since ancient times has been a real challenge for the minds linked to the world mathematician, we deliver the reader a key to unravel the structure and behavior of the primes without open the door to the complexity.

**Category:** Number Theory

[1232] **viXra:1605.0095 [pdf]**
*submitted on 2016-05-10 06:59:15*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

1) A hypothesis of a prime number and the physics fusion.
The prime number is energy, mcc and a concentric circle. The center is 1/2.
2). The prime number is related deeply with Planck's constant, (h).
3). Natural number is 1* integer. The energy is h X integer.
4). I do it with integer, 1 = 1h.
In m, a radius is a circle of nh. The prime number can express a radius with the circle of the prime number.
5). The equation of the prime number. I accord with energy nh and the prime number of the prime number stairs n turn. (expectation).
6). The prime number distance of the prime number stairs n turn accords with distance of the nuclear resonance energy nh neighborhood. (expectation).

**Category:** Number Theory

[1231] **viXra:1605.0082 [pdf]**
*submitted on 2016-05-09 02:55:23*

**Authors:** Jaykov Foukzon

**Comments:** 23 Pages.

In this paper possible completion of the Robinson non-Archimedean field *R constructed by Dedekind sections. As interesting example I show how, a
few simple ideas from non-archimedean analysis on the pseudo-ring *R_d gives a short clear nonstandard reconstruction for the Euler’s original proof of the Goldbach-Euler theorem.

**Category:** Number Theory

[1230] **viXra:1605.0066 [pdf]**
*submitted on 2016-05-06 02:22:44*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

1)Equations of Euler product multiplied by PI, the equation for the area of a circle.
2) All these equations become concentric.
3) Is the Riemann Zeta function can be represented as a sum of an infinite number of simple sin, cos waves by Fourier transform.
4) And sine curve and cosine curve in circular motion, all in concentric circles that share the same center line.
5) Non-trivial Riemann zero was zero and area of a circle of sin and cosine curve that is equivalent to the Riemann Zeta-function
To be equivalent.
6) And the area of a circle of sine curve and cosine curve to zero all become concentric axis of rotation (1/2 straight).
7) This is Riemann's "line other than zero point does not exist" would prove that. And Honjo expected.

**Category:** Number Theory

[1229] **viXra:1605.0061 [pdf]**
*submitted on 2016-05-05 06:55:33*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[1228] **viXra:1605.0056 [pdf]**
*submitted on 2016-05-04 11:34:05*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

A fusion of prime number and quantum physics.(1)
This figure shows a prime number of the Lehman expectation and the relations with 0 points.
The zeta function is expressed as the sum of simple sin function and cosine function.
The complicated period function is expressed for the sum of sin function and cosine function that are the mathematical expression of a simple wave.

**Category:** Number Theory

[1227] **viXra:1605.0010 [pdf]**
*submitted on 2016-05-01 10:14:30*

**Authors:** Reuven Tint

**Comments:** 3 Pages.

Proposed short proof of the fallacy of the ABC assumption
conjecture on the finiteness of the number of "exceptional" triples for r = 2
( "Pythagorean" equation) and other equations, and provides a number of examples.

**Category:** Number Theory

[1226] **viXra:1605.0009 [pdf]**
*submitted on 2016-05-01 10:16:44*

**Authors:** Reuven Tint

**Comments:** 3 Pages.

The proof of the fallacy of the assumption of the ABC conjecture. Russian language.

**Category:** Number Theory

[1225] **viXra:1605.0008 [pdf]**
*submitted on 2016-05-01 11:42:34*

**Authors:** Terubumi Honjou

**Comments:** 15 Pages.

The hypothesis of the pulsation principle leads a prime number and physical relations.
The horizon and the pulsation wave pattern of the figure of fusion include profound, enormous physical concepts.
The geometry of all things.
A figure of prime number, physical fusion.
I illustrate an uncertainty principle for the first time.
I solve the mystery of the double slit experiment.

**Category:** Number Theory

[1224] **viXra:1604.0391 [pdf]**
*submitted on 2016-04-30 19:01:30*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

1) The function of the prime number, the wave pattern of the zeta function are equivalent with the friendship of the sin wave of the having many kinds, cos wave by Fourier transform.
2) There are a sin wave, a cos wave and the point of intersection (0 points) with the axis on an axis. The point of intersection that left the axis (straight line) does not exist.
3) The top of the pulsation wave pattern has deep relation to prime number, mass, quark, 0 points, ...
4) Circle that is a trace of the circular motion is a quantum-mechanical autocoupling operator, an L meat operator.
5) Circle to assume a prime number a radius is a trace of the tops of the material wave of the pulsation principle, and Japanese yen and the point of intersection with the axis are zero points.
(6) As for the product indication equation of the oiler, a radius is a circle of 1 integral multiple.
(7) As for the quantum mechanics, a radius is a circle of the integral multiples of "h".
(8) As for the mass of the quantum-mechanical mass, a radius is a circle of the integral multiples of "m".
9) Circle of the prime number, all the circular center have a radius on 1/2 line. (Lehman expectation)
10) The top (prime number, mass) of the pulsation wave pattern becomes the straight line by レッジェ trace graph.
11) As for the レッジェ trace graph, square of the mass becomes the straight line.
12) 1/2h is the important fixed number to often come up in a quantum-mechanical equation.
13) 1/2 of the Lehman expectation is the straight line that is the mystery that 0 points form a line of the infinite unit.
14) A sine wave by the Fourier transform, a cosine wave and the point of intersection of the 1/2 straight line are 0 points.
15) The eddy of the solution (material wave) of the Schrodinger equation is equivalent with circular motion.

**Category:** Number Theory

[1223] **viXra:1604.0386 [pdf]**
*submitted on 2016-04-30 12:49:08*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

1) The function of the prime number, the wave pattern of the zeta function are equivalent with the friendship of the sin wave of the having many kinds, cos wave by Fourier transform.
2) There are a sin wave, a cos wave and the point of intersection (0 points) with the axis on an axis. The point of intersection that left the axis (straight line) does not exist.
3) The top of the pulsation wave pattern has deep relation to prime number, mass, quark, 0 points, ...
4) Circle that is a trace of the circular motion is a quantum-mechanical autocoupling operator, an L meat operator.
5) Circle to assume a prime number a radius is a trace of the tops of the material wave of the pulsation principle, and Japanese yen and the point of intersection with the axis are zero points.
(6) As for the product indication equation of the oiler, a radius is a circle of 1 integral multiple.
(7) As for the quantum mechanics, a radius is a circle of the integral multiples of "h".
(8) As for the mass of the quantum-mechanical mass, a radius is a circle of the integral multiples of "m".
9) Circle of the prime number, all the circular center have a radius on 1/2 line. (Lehman expectation)
10) The top (prime number, mass) of the pulsation wave pattern becomes the straight line by レッジェ trace graph.
11) As for the レッジェ trace graph, square of the mass becomes the straight line.
12) 1/2h is the important fixed number to often come up in a quantum-mechanical equation.
13) 1/2 of the Lehman expectation is the straight line that is the mystery that 0 points form a line of the infinite unit.
14) A sine wave by the Fourier transform, a cosine wave and the point of intersection of the 1/2 straight line are 0 points.
15) The eddy of the solution (material wave) of the Schrodinger equation is equivalent with circular motion.

**Category:** Number Theory

[1222] **viXra:1604.0357 [pdf]**
*submitted on 2016-04-26 20:35:55*

**Authors:** Terubumi Honjou

**Comments:** 9 Pages.

0 points and the distribution map of the prime number of the zeta function are expressed by a complex number coordinate.
The figure of elementary particle pulsation principle energy wave pattern is expressed by a complex number coordinate.
The figure of fusion synchronized a straight line and the horizon of the figure of elementary particle pulsation principle energy wave pattern where 0 points formed a line and fused with neither.
Four dimensions of lower domains express space on the horizon, and the prime number in the top of the material wave pulsates by a turn of the four-dimensional space as a top of the waves.
There are all the non-self-evident zero points of the zeta function on the horizon (three-dimensional space) of the figure of elementary particle pulsation principle energy wave pattern and is real part 1/2.
It fuses in a complex number coordinate and a figure of elementary particle pulsation principle energy wave pattern (complex number coordinate) that 0 points of a prime number and the zeta function present.

**Category:** Number Theory

[1221] **viXra:1604.0345 [pdf]**
*submitted on 2016-04-26 03:04:31*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

A humble attempt is made at proving the twin prime conjecture. An argument involving a form derived from a set of characteristic equations and a parity argument is used in the proof.

**Category:** Number Theory

[1220] **viXra:1604.0344 [pdf]**
*submitted on 2016-04-25 23:39:15*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

Use of the harmonic numbers to create congruencies is discussed. Interesting relations to known congruencies are shown.

**Category:** Number Theory

[1219] **viXra:1604.0342 [pdf]**
*submitted on 2016-04-25 08:41:45*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we diagnose the critical line.

**Category:** Number Theory

[1218] **viXra:1604.0337 [pdf]**
*submitted on 2016-04-24 19:45:00*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

A prime number was the top of the material wave in the theory physics and, in "a challenge to Lehman expectation which I announced in YOUTUBE for 2,012 years ," expected it so that a material wave and the point of intersection of the figure of pulsation energy wave pattern were non-self-evident zero points of the Lehman expectation.
I tried a prime number and the conversion of the equation indicating the connection with π (Circle) that Euler discovered recently.
As a result of the right side sprinkling π to a denominator, molecules of the left side of a go board of the product formula of π 2, and having converted it into the equation of the area of Japanese yen, a radius got an equation of Circle of the prime number. This suggests that expectation of 2012 saying that it is a prime number on the top of the material wave of the figure of pulsation energy wave pattern was right.

**Category:** Number Theory

[1217] **viXra:1604.0327 [pdf]**
*submitted on 2016-04-24 08:28:51*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we analyze the behavior of prime numbers.

**Category:** Number Theory

[1216] **viXra:1604.0324 [pdf]**
*submitted on 2016-04-23 18:09:26*

**Authors:** Anthony J. Browne

**Comments:** 3 Pages.

A different approach involving roots and leading to the basis of the AKS test are introduced and discussed.

**Category:** Number Theory

[1215] **viXra:1604.0321 [pdf]**
*submitted on 2016-04-23 11:20:06*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of Characteristic equations are discussed and several number theoretic functions are derived.

**Category:** Number Theory

[1214] **viXra:1604.0316 [pdf]**
*submitted on 2016-04-23 06:05:21*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The Tijdeman–Zagier conjecture, also known as Beal's conjecture, is a conjecture in number theory: –
If A^x+B^y=C^z,
Where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor. Equivalently,
There are no solutions to the above equation in positive integers A, B, C, x, y, z with A, B, and C being pairwise coprime and all of x, y, z being greater than 2.

**Category:** Number Theory

[1213] **viXra:1604.0315 [pdf]**
*submitted on 2016-04-23 06:09:00*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The purpose of this paper is to show how a Pseudo Random pattern appears in Pi.The reason there are no repeating numbers in Pi is because there is a Pseudo Random process in Pi. The Pseudo Random Process causes no repeating numbers in Pi. As in the prime numbers A+B +/- 1=C there is a Pseudo Random Process in Pi. In Pi, characteristics of the Pseudo Random Process can be seen by taking the digits in Pi and doing a progression which starting with 3 take its square root. Then take the next two digits, add them up and take the square root. After progressing the patterns appear. At the 6th &7th series,11th & 12th series and 16th&17th series.

**Category:** Number Theory

[1212] **viXra:1604.0295 [pdf]**
*submitted on 2016-04-20 13:24:30*

**Authors:** Ilija Barukčić

**Comments:** 7 Pages. Copyright © 2016 by Ilija Barukčić, Jever, Germany.

Unfortunately, however, the relation between a finite and an infinite is not always so straightfor-ward. The infinite and the finite mutually related as sheer others are inseparable. A related point is that while the infinite is determined in its own self by the other of itself, the finite, the finite itself is determined by its own infinite. Each of both is thus far the unity of its own other and itself. The inseparability of the infinite and the finite does not mean that a transition of the finite into the infinite and vice versa is not possible. In the finite, as this negation of the infinite, we have the sat-isfaction that determinateness, alteration, limitation et cetera are not vanished, are not sublated. The finite is a finite only in its relation to its own infinite, and the infinite is only infinite in its rela-tion to its own finite. As will become apparent, the infinite as the empty beyond the finite is bur-dened by the fact that determinateness, alteration, limitation et cetera are vanished. The relation between the finite and the infinite finds its mathematical formulation in the division of one by zero. As we will see, it is +1/+0=+oo.

**Category:** Number Theory

[1211] **viXra:1604.0259 [pdf]**
*submitted on 2016-04-17 14:22:16*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

In the simplest terms here is a counterexample to Fermat's Last Theorem and s solution to Beal's Conjecture. Dr. Andrew Wiles proved Fermat's Last Theorem but I think my solution below is an example for n=3 if allowed. It also satisfies Beal's conjecture and is a counterexample to Fermat’s Last Theorem.

**Category:** Number Theory

[1210] **viXra:1604.0258 [pdf]**
*submitted on 2016-04-17 14:25:44*

**Authors:** Ricardo Gil

**Comments:** 11 Pages.

The purpose of this paper is to show that prime numbers are structured in a Pseudo Random manner. Like the Fibonacci or the Lucas sequence, the prime number sequence is a sequence in which 2 primes when added together (+ or -1) makes the next prime. The sum of the two primes, A+B(+or-1)=C dictates the next prime number in the sequence. Goldbach's conjecture is that every even integer is the sum of two primes, A+B(+or-1)=C is two primes +or-1 make up another prime and dictates the gap between the primes. Progressing along the prime number line is similar to the Fibonacci sequence and the Lucas sequence. In a sense the A+B(+or-1)=C is a sequence but for prime numbers. In the Pseudo Random Prime Number Sequence or A+B(+or -1)=c, 5+3-1=7 and 7+5-1=11. The "A" side progresses or dictates the progression and in the progression or sequence if 5 were used in 5+5+1=11 instead of 7+5-11 it would be out of order in the progression sequence.

**Category:** Number Theory

[1209] **viXra:1604.0257 [pdf]**
*submitted on 2016-04-17 14:29:48*

**Authors:** Ricardo Gil

**Comments:** 8 Pages.

The purpose of this paper is to show how Riemann Zeros and Prime Numbers synchronize at N+6 and why there are no Riemann Zeros smaller than 14.

**Category:** Number Theory

[1208] **viXra:1604.0255 [pdf]**
*submitted on 2016-04-17 16:02:09*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The purpose of this paper is to provide algorithm that is 4 lines of code and that finds P & Q when N is given. It will work for RSA-1024 & RSA-2018 if the computer can float large numbers in PyCharm or Python.

**Category:** Number Theory

[1207] **viXra:1604.0243 [pdf]**
*submitted on 2016-04-15 23:18:19*

**Authors:** Marius Coman

**Comments:** 150 Pages. Published by Education Publishing, USA. Copyright 2016 by Marius Coman.

The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (S sequences) but also on “Smarandache-Coman sequences of primes” (SC sequences), defined by the author as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation”: the SC sequences presented in this book are related, of course, to concatenation, but in three different ways: the S sequence is obtained by the method of concatenation but the operation applied on its terms is some other arithmetical operation; the S sequence is not obtained by the method of concatenation but the operation applied on its terms is concatenation, or both S sequence and SC sequence are using the method of concatenation. Part Two of this book, “Sequences of primes obtained by the method of concatenation”, brings together 51 articles which aim, using the mentioned method, to highlight sequences of numbers that are rich in primes or are liable to lead to large primes. The method of concatenation is applied to different classes of numbers, e.g. squares of primes, Poulet numbers, triangular numbers, reversible primes, twin primes, repdigits, factorials, primorials, in order to obtain sequences, possible infinite, of primes. Part Two of this book also contains a paper which lists a number of 33 sequences of primes obtained by the method of concatenation, sequences presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”.

**Category:** Number Theory

[1206] **viXra:1604.0241 [pdf]**
*submitted on 2016-04-15 10:06:17*

**Authors:** F. Portela

**Comments:** 8 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to a possibly important conclusion.

**Category:** Number Theory

[1205] **viXra:1604.0227 [pdf]**
*submitted on 2016-04-13 19:31:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) There exist an infinity of primes obtained concatenating two consecutive primorial numbers and adding 1 to the resulted number; example: concatenating the tenth and eleventh primorials then adding 1 is obtained the prime 6469693230200560490131; (II) There exist an infinity of primes obtained concatenating two consecutive primorial numbers and subtracting 1 from the resulted number; example: concatenating the ninth and tenth primorials then subtracting 1 is obtained the prime 2230928706469693229.

**Category:** Number Theory

[1204] **viXra:1604.0226 [pdf]**
*submitted on 2016-04-13 19:32:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: let [p, q] be a pair of sexy primes (q = p + 6); then: (I) there exist an infinity of primes obtained concatenating 30*p with 30*q and adding 1 to the resulted number; example: for [p, q] = [23, 29], the number 690871 is prime; (II) there exist an infinity of primes obtained concatenating 30*p with 30*q and subtracting 1 from the resulted number; example: for [p, q] = [23, 29], the number 690869 is prime; (III) there exist an infinity of pairs of twin primes obtained concatenating 30*p with 30*q and adding/subtracting 1 from the resulted number; example: for [p, q] = [101, 107], the numbers 30303209 and 30303211 are primes.

**Category:** Number Theory

[1203] **viXra:1604.0219 [pdf]**
*submitted on 2016-04-13 11:31:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: (I) There exist an infinity of primes q obtained deconcatenating to the right with 1 the Poulet numbers of the form 30*k + 1 then subtracting 1 (example: from P = 997465414921 is obtained q = 99746541491); (II) There exist an infinity of primes q obtained deconcatenating to the right with 1 the Poulet numbers of the form 30*k + 1 then adding 1 (example: from P = 996881835961 is obtained q = 99688183597); (III) There exist an infinity of primes q obtained deconcatenating to the right with 01 the Poulet numbers of the form 300*k + 1 then subtracting 1 (example: from P = 999666754801 is obtained q = 9996667547).

**Category:** Number Theory

[1202] **viXra:1604.0218 [pdf]**
*submitted on 2016-04-13 11:33:58*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) For any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the factorial numbers and adding 1 to the resulted number; (II) for any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the factorial numbers and subtracting 1 from the resulted number. It is worth noting the pair of twin primes having 49 digits each obtained for k = 9: (5502622159812088949850305428800254892961651752959,
5502622159812088949850305428800254892961651752961).

**Category:** Number Theory

[1201] **viXra:1604.0217 [pdf]**
*submitted on 2016-04-13 11:35:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) For any k positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the fibonorial numbers and adding 1 to the resulted number; (II) for any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the fibonorial numbers and subtracting 1 from the resulted number. It is known that fibonorial numbers are defined as the products of nonzero Fibonacci numbers.

**Category:** Number Theory

[1200] **viXra:1604.0216 [pdf]**
*submitted on 2016-04-13 08:42:42*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

In this note we show some solutions of the equation 4xz=4+y*y , and a relation with the constant pi

**Category:** Number Theory

[1199] **viXra:1604.0201 [pdf]**
*submitted on 2016-04-12 07:14:59*

**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

[1198] **viXra:1604.0200 [pdf]**
*submitted on 2016-04-12 07:24:12*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we diagnose the critical line.

**Category:** Number Theory

[1197] **viXra:1604.0189 [pdf]**
*submitted on 2016-04-11 20:36:15*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Numerical verification is possible through nominative determinism (visibility theory). Adding learning (adaptive learning) to the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory

[1196] **viXra:1604.0181 [pdf]**
*submitted on 2016-04-12 02:23:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: for any positive integer n > 1 there exist a sequence having an infinity of prime terms p, obtained concatenating to the right with 1 the terms of the sequence of concatenated n-th powers. For n = 2 the primes p are obtained concatenating with 1 to the right the terms of the Smarandache concatenated squares sequence; for n = 3 the primes p are obtained concatenating with 1 to the right the terms of the Smarandache concatenated cubic sequence.

**Category:** Number Theory

[1195] **viXra:1604.0180 [pdf]**
*submitted on 2016-04-12 02:24:28*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I state the following two conjectures: (I) If p is a prime which admits deconcatenation in two primes p1 and p2, both of the form 6*k – 1, then there exist an infinity of primes q obtained concatenating q1 with q2, where q1 = 30*n – p1, q2 = 30*n – p2 and n positive integer; (II) If p is a prime which admits deconcatenation in two primes p1 and p2, both of the form 6*k + 1, then there exist an infinity of primes q obtained concatenating q1 with q2, where q1 = 30*n + p1, q2 = 30*n + p2 and n positive integer.

**Category:** Number Theory

[1194] **viXra:1604.0179 [pdf]**
*submitted on 2016-04-12 02:25:49*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following four conjectures. Let q be the number obtained concatenating to the right with 1 the numbers p – 1, where p primes of the form 30*k + 11; then: (I) there exist an infinity of primes q; (II) there exist an infinity of semiprimes q = q1*q2, such that q2 + q1 - 1 is prime. Let q be the number obtained concatenating to the right with 1 the numbers p + 1, where p primes of the form 30*k + 11; then: (III) there exist an infinity of primes q; (IV) there exist an infinity of semiprimes q = q1*q2, such that q2 – q1 + 1 is prime.

**Category:** Number Theory

[1193] **viXra:1604.0171 [pdf]**
*submitted on 2016-04-10 18:40:43*

**Authors:** Zhang Tianshu

**Comments:** 14 Pages.

In this article, the author gave a specific example to negate the ABC conjecture once and for all.

**Category:** Number Theory

[1192] **viXra:1604.0169 [pdf]**
*submitted on 2016-04-10 13:18:17*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: For any positive integer n, n > 1, there exist a sequence having an infinity of prime terms obtained concatenating n consecutive numbers and then the resulting number, to the right, with 1. Examples: for n = 2, the sequence obtained this way contains the primes 10111, 15161, 18191, 21221 (...); for n = 9, the sequence obtained this way contains the primes 1234567891, 910111213141516171, 2021222324252627281, 2930313233343536371 (...).

**Category:** Number Theory

[1191] **viXra:1604.0163 [pdf]**
*submitted on 2016-04-10 10:36:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: there exist an infinity of numbers q = (30*k + 7)*(60*k + 13) which admit a deconcatenation in two primes p1 and p2. Examples: for k = 2, q = 67*133 = 8911 which can be deconcatenated in p1 = 89 and p2 = 11; for k = 5, q = 157*313 = 49141 which can be deconcatenated in p1 = 491 and p2 = 41.

**Category:** Number Theory

[1190] **viXra:1604.0162 [pdf]**
*submitted on 2016-04-10 10:38:56*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: For any digit from 1 to 9 there exist a sequence with an infinity of prime terms obtained concatenating to the right with 1 the partial sums of the repdigits. Examples: for repunit numbers 1, 11, 111 (...), concatenating the sum S(3) = 1 + 11 + 111 = 123 to the right with 1 is obtained 1231, prime; for repdigit numbers 3, 33, 333, 3333 (...), concatenating the sum S(4) = 3 + 33 + 333 + 3333 = 3702 to the right with 1 is obtained 37021, prime.

**Category:** Number Theory

[1189] **viXra:1604.0161 [pdf]**
*submitted on 2016-04-10 06:06:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “multiples of 3” sequence (defined as the sequence obtained through the concatenation of multiples of 3, in reverse order); such prime is, for example, 13330272421181512963; (II) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “odd multiples of 3” sequence (defined as the sequence obtained through the concatenation of odd multiples of 3, in reverse order); such prime is, for example, 145393327211593.

**Category:** Number Theory

[1188] **viXra:1604.0160 [pdf]**
*submitted on 2016-04-10 06:07:45*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture three conjectures: (I) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “powers of 3” sequence (defined as the sequence obtained through the concatenation of powers of 3, in reverse order); such prime is, for example, 1243812793; (II) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “odd powers of 3” sequence (defined as the sequence obtained through the concatenation of odd powers of 3, in reverse order); such prime is, for example, 1243273; (III) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “even powers of 3” sequence (defined as the sequence obtained through the concatenation of even powers of 3, in reverse order); such prime is, for example, 14782969531441590496561729819.

**Category:** Number Theory

[1187] **viXra:1604.0158 [pdf]**
*submitted on 2016-04-10 02:09:11*

**Authors:** Marius Coman

**Comments:** 2 Pages.

I was studying the sequences of primes obtained applying concatenation to some well known classes of numbers, when I discovered that the second Poulet number, 561 (also the first Carmichael number, also a very interesting number – I wrote a paper dedicated to some of its properties), is also a triangular number. Continuing to look, I found, up to the triangular number T(817), if we note T(n) = n*(n + 1)/2 = 1 + 2 +...+ n, fifteen Poulet numbers. In this paper I state the conjecture that there exist an infinity of Poulet numbers which are also triangular numbers.

**Category:** Number Theory

[1186] **viXra:1604.0147 [pdf]**
*submitted on 2016-04-09 01:41:21*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: There exist an infinity of primes p obtained concatenating to the right with 1 the triangular numbers.

**Category:** Number Theory

[1185] **viXra:1604.0146 [pdf]**
*submitted on 2016-04-09 01:43:13*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: (I) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache reverse sequence; (II) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache back concatenated odd sequence; (III) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache back concatenated square sequence.

**Category:** Number Theory

[1184] **viXra:1604.0138 [pdf]**
*submitted on 2016-04-08 09:21:17*

**Authors:** Méhdi Pascal

**Comments:** 35 Pages. document en langue français

Ce papier contient deux petits résultats, le premier est sur une toute petite liaison qui lie les nombres parfaits avec les nombres de Carmichael. Le second résultat, est un simple exemple de traduction d’une méthode en fonction. A la fin de ce papier, je donne une introduction à la prochaine lettre qui montre que l’infinité des nombres premiers sous forme de n²+1 est lié à l’infinité des nombres premiers dans les deux progressions arithmétiques 4n+1 & n, par une simple identité asymptotique.

**Category:** Number Theory

[1183] **viXra:1604.0132 [pdf]**
*submitted on 2016-04-08 03:51:10*

**Authors:** Pankaj Mani, Frm, Cqf

**Comments:** 14 Pages.

In this paper, I try to look at Riemann Hypothesis from the Game Theoretical Perspective. As David Hilbert had visualized that advanced math is actually a game of symbols satisfying certain fixed rules. Indeed, here number theoretical system plays the Non-Cooperative game and more precisely the Game of Perfect Information.
Applying the technical Game Theoretic concepts, I have tried to show that Riemann Hypothesis is definitely true !
In case of any typos, please avoid them or else feel free to write to me. I shall correct them.
Author : Pankaj Mani,FRM,CQF
New Delhi, India
Email: manipankaj9@gmail.com

**Category:** Number Theory

[1182] **viXra:1604.0110 [pdf]**
*submitted on 2016-04-06 01:14:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: Let p be a prime of the form 30*k + 17; then there exist an infinity of primes q obtained concatenating p – 1 with 3; example: 677, 797, 827, 857, 887, 947 are primes (successive primes of the form 30*k + 17) and the numbers 6763, 7963, 8263, 8563, 8863, 9463 are also primes. As an incidental observation, many of the semiprimes x*y obtained in the way defined have one of the following two properties: (i) y – x + 1 is a prime of the form 13 + 30*k; (ii) y – x + 1 is a prime of the form 19 + 30*k.

**Category:** Number Theory

[1181] **viXra:1604.0105 [pdf]**
*submitted on 2016-04-05 08:25:12*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any term S(n) of the Smarandache consecutive numbers sequence (1, 12, 123, 1234, 12345, 123456, 1234567...) there exist an infinity of primes p such that the number q obtained concatenating S(n) both to the left and to the right with p is prime.

**Category:** Number Theory

[1180] **viXra:1604.0104 [pdf]**
*submitted on 2016-04-05 09:40:07*

**Authors:** Allen D. Allen

**Comments:** 5 Pages.

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving that there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a condition in which > changes to < so their transitive properties cannot be used. FLT implies that for an integral exponent n, the inequality changes over the interval with the minimum extent 1 ≤ n ≤ 3. For any exponent in the positive real numbers, a solution to Fermat’s equation occurs and inequality is replaced by equality at the instant when four distinct exponential curves collapse into two intersecting curves.

**Category:** Number Theory

[1179] **viXra:1604.0103 [pdf]**
*submitted on 2016-04-05 04:39:52*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q with n then with q + 6 is prime; (II) let n be defined as in Conjecture 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q + 6 with n then with q is prime; (III) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k + 1; there exist an infinity of primes q of the form 6*h - 1 such that the number r obtained concatenating q with n then with q + 6 is prime; (IV) let n be defined as in Conjecture 3; there exist an infinity of primes q of the form 6*h - 1 such that the number r obtained concatenating q + 6 with n then with q is prime.

**Category:** Number Theory

[1178] **viXra:1604.0101 [pdf]**
*submitted on 2016-04-04 16:43:39*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1 (e.g. n = 12345 for p = 5); there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q + 2 with n then with q is prime (e.g. for n = 12345 there exist q = 19 such that r = 211234519 is prime); (II) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q - 4 with n then with q is prime (e.g. for n = 12345 there exist q = 37 such that r = 331234537 is prime). I use the operator “]c[“ with the meaning “concatenated to”.

**Category:** Number Theory

[561] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-23 01:58:04*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat was truthful. He could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if n > 2, the general equation, c^n = a^n + b^n does not hold. Applying a polar coordinate system, let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. Three similar versions of the proof are presented. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper.

**Category:** Number Theory

[560] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-19 16:43:49*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on a single page. The proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1. One will first show that if n = 2, the general equation, c^n = a^n + b^n holds, followed by showing that if
n > 2, the general equation, c^n = a^n + b^n does not hold. Applying a polar coordinate system, let a, b and c be three relatively prime positive integers which are the lengths of the sides of a right triangle, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Also, let the acute angle between the hypotenuse and the horizontal be denoted by theta. The proof is very simple, and even high school students can learn it. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin space for it in his paper.

**Category:** Number Theory

[559] **viXra:1605.0123 [pdf]**
*replaced on 2016-05-15 23:40:15*

**Authors:** Quang Nguyen Van

**Comments:** 3 Pages.

We have spotted an error of Euler's proof, so that the used infinite descent is impossible in his proof

**Category:** Number Theory

[558] **viXra:1605.0113 [pdf]**
*replaced on 2016-05-21 11:04:35*

**Authors:** Anthony J. Browne

**Comments:** 6 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[557] **viXra:1605.0113 [pdf]**
*replaced on 2016-05-12 00:27:47*

**Authors:** Anthony J. Browne

**Comments:** 6 Pages.

An equivalent form of the Goldbach Conjecture is stated using manipulation of characteristic equations and simple logical arguments that lead to an equation which restates the conjecture. A new form of the number of unordered partitions of an even number into two primes is presented.

**Category:** Number Theory

[556] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-12 06:55:09*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[555] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-10 22:16:21*

**Authors:** T.Nakashima

**Comments:** 5 Pages.

This paper is all cycle case resolve.But sequence goes to infinity case I can not prove not possibillity.

**Category:** Number Theory

[554] **viXra:1605.0061 [pdf]**
*replaced on 2016-05-07 04:18:28*

**Authors:** T.Nakashima

**Comments:** 6 Pages.

**Category:** Number Theory

[553] **viXra:1604.0324 [pdf]**
*replaced on 2016-05-20 01:33:52*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

Approximations of square roots are discussed. A very close approximation to their decimal expansion is derived in the form of a simple fraction. Their relationship with the AKS test is also discussed.

**Category:** Number Theory

[552] **viXra:1604.0324 [pdf]**
*replaced on 2016-05-17 23:10:26*

**Authors:** Anthony J. Browne

**Comments:** 4 Pages.

Approximations of n^th roots are discussed. A close approximation to their decimal expansion is derived. Their relationship with the AKS test is also discussed.

**Category:** Number Theory

[551] **viXra:1604.0321 [pdf]**
*replaced on 2016-05-20 16:31:25*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Summing characteristic equations to find forms of theoretical functions in number theory will be discussed. Forms of many number theoretic functions will be derived. Although many may not be efficient in a computing sense for large numbers, the aim in this paper will simply be to explore what these forms are and show relationships between expressions.

**Category:** Number Theory

[550] **viXra:1604.0321 [pdf]**
*replaced on 2016-05-17 15:42:53*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of Characteristic equations are discussed and several number theoretic functions are derived.

**Category:** Number Theory

[549] **viXra:1604.0321 [pdf]**
*replaced on 2016-05-04 00:23:05*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of Characteristic equations are discussed and several number theoretic functions are derived.

**Category:** Number Theory

[548] **viXra:1604.0321 [pdf]**
*replaced on 2016-04-28 15:15:04*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

**Category:** Number Theory

[547] **viXra:1604.0321 [pdf]**
*replaced on 2016-04-23 23:06:35*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of characteristic equations are discussed. Several number theoretic functions are derived and different techniques are introduced and discussed.

**Category:** Number Theory

[546] **viXra:1604.0295 [pdf]**
*replaced on 2016-05-01 04:58:57*

**Authors:** Jan Pavo Barukčić, Ilija Barukčić

**Comments:** 10 Pages. (C) Jan Pavo Barukčić, Münster and Ilija Barukčić, Jever, Germany, 2016.

Unfortunately, however, the relation between a finite and an infinite is not always so straightfor-ward. The infinite and the finite mutually related as sheer others are inseparable. A related point is that while the infinite is determined in its own self by the other of itself, the finite, the finite itself is determined by its own infinite. Each of both is thus far the unity of its own other and itself. The inseparability of the infinite and the finite does not mean that a transition of the finite into the infinite and vice versa is not possible. In the finite, as this negation of the infinite, we have the sat-isfaction that determinateness, alteration, limitation et cetera are not vanished, are not sublated. The finite is a finite only in its relation to its own infinite, and the infinite is only infinite in its rela-tion to its own finite. As will become apparent, the infinite as the empty beyond the finite is bur-dened by the fact that determinateness, alteration, limitation et cetera are vanished. The relation between the finite and the infinite finds its mathematical formulation in the division of one by zero. As we will see, it is +1/+0=+oo.

**Category:** Number Theory

[545] **viXra:1604.0241 [pdf]**
*replaced on 2016-05-04 06:40:28*

**Authors:** F. Portela

**Comments:** 10 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to an important conclusion.

**Category:** Number Theory

[544] **viXra:1604.0241 [pdf]**
*replaced on 2016-05-01 14:23:35*

**Authors:** F. Portela

**Comments:** 10 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to an important conclusion.

**Category:** Number Theory

[543] **viXra:1604.0241 [pdf]**
*replaced on 2016-04-19 15:27:35*

**Authors:** F. Portela

**Comments:** 9 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to an important conclusion.

**Category:** Number Theory

[542] **viXra:1604.0241 [pdf]**
*replaced on 2016-04-17 17:32:39*

**Authors:** F. Portela

**Comments:** 8 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to a possibly important conclusion.

**Category:** Number Theory

[541] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-15 16:12:25*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Inspection of the conjecture shows that it is a phenomenological model. Thus, a solution could be found through regression analysis. Numerical verification is possible through nominative determinism/visibility theory. By adding adaptive learning (AL) to the model, the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory

[540] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-14 20:41:28*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Inspection of the conjecture shows that it is a phenomenological model. Thus, a solution could be found through regression analysis. Numerical verification is possible through nominative determinism/visibility theory. By adding adaptive learning (AL) to the model, the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory

[539] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-14 14:04:15*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Inspection of the conjecture shows that it is a phenomenological model. Thus, a solution could be found through regression analysis. Numerical verification is possible through nominative determinism/visibility theory. By adding adaptive learning (AL) to the model, the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory

[538] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-12 10:36:44*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Numerical verification is possible through nominative determinism (visibility theory). Adding learning (adaptive learning) to the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory