**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(3)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)

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2013 - 1301(5) - 1302(10) - 1303(16) - 1304(15) - 1305(12) - 1306(12) - 1307(25) - 1308(11) - 1309(8) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(27) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(52) - 1409(47) - 1410(17) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(35) - 1504(23) - 1505(19) - 1506(13) - 1507(16) - 1508(15) - 1509(15) - 1510(13) - 1511(9) - 1512(26)

2016 - 1601(14) - 1602(18) - 1603(78) - 1604(56) - 1605(29) - 1606(20) - 1607(1)

Any replacements are listed further down

[1272] **viXra:1607.0003 [pdf]**
*submitted on 2016-07-01 03:27:48*

**Authors:** Pingyuan Zhou

**Comments:** 13 Pages. If so-called strong Goldbach number sequence introduced in this paper is acceptable then the existence of strong Goldbach number sequence will imply both Goldbach and twin prime conjectures.

Abstract: In this note, we present a new and direct appraoch to prove the Goldbach conjecture that if the existence of the limit of

**Category:** Number Theory

[1271] **viXra:1606.0345 [pdf]**
*submitted on 2016-06-30 11:00:58*

**Authors:** Scott B. Guthery

**Comments:** 8 Pages.

An algorithm for recursively generating the sequence of solutions of a prime constellation is described. The algorithm is based on an polynomial equation formed from the first n elements of the constellation. A root of this equation is the next element of the sequence.

**Category:** Number Theory

[1270] **viXra:1606.0327 [pdf]**
*submitted on 2016-06-29 07:46:25*

**Authors:** Mohamed Ouiter

**Comments:** 7 Pages.

From the definition that a number is Prime if he admits that two dividers and any number not prime can it write as a product of prime factors, I was able to identify a simple law that governs the distribution of prime numbers.

**Category:** Number Theory

[1269] **viXra:1606.0310 [pdf]**
*submitted on 2016-06-28 09:09:00*

**Authors:** Peter Bissonnet

**Comments:** 14 Pages.

When a subset of sequential integers is arranged in a specific way, there appears a paired set of slanted straight lines along which prime numbers seem to naturally arrange themselves in a repeated fashion. This arrangement can further be observed to be a two dimensional surface applicable to the cylinder. If this arrangement is on a piece of paper, then one can fold the paper in the form of a cylinder, and the paired set of slanted straight lines unite at the page edges to form a double helix winding down the cylinder. Paired primes are thus seen to be an association between primes residing on these two paired helixes, and further analysis shows that there appears to be two types of paired primes. Is the prime number double helix nature’s secret for jump starting life (DNA) and thereby defeating entropy (at least in the initial stage) by creating order from order instead of order from disorder? The prime number trends examined exhibit right handed chirality; left handed chirality is obtained by symmetry considerations.

**Category:** Number Theory

[1268] **viXra:1606.0306 [pdf]**
*submitted on 2016-06-28 09:30:16*

**Authors:** Peter Bissonnet

**Comments:** 2 Pages. This is a Russian journal named "Almanac of Modern Science and Education" and can be found at the following link: http://www.gramota.net/eng/editions/5.html

This very short paper illustrates a mathematical method which illustrates the possibility of reducing a prime product (N = PQ) from two independent variables, namely the two original prime numbers P and Q, to only one independent variable which determines both prime numbers.

**Category:** Number Theory

[1267] **viXra:1606.0302 [pdf]**
*submitted on 2016-06-27 21:24:29*

**Authors:** T.Nakashima

**Comments:** 1 Page.

reration of recurring decimal and primitive root

**Category:** Number Theory

[1266] **viXra:1606.0271 [pdf]**
*submitted on 2016-06-25 20:21:56*

**Authors:** T.Nakashima

**Comments:** 1 Page.

Conway’s problem is correct.

**Category:** Number Theory

[1265] **viXra:1606.0264 [pdf]**
*submitted on 2016-06-24 20:45:44*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

Near m,the destance of primes is lower order than logm.This is the Legendre’s conjecture.

**Category:** Number Theory

[1264] **viXra:1606.0251 [pdf]**
*submitted on 2016-06-23 21:35:51*

**Authors:** T.Nakashima

**Comments:** 1 Page.

The abc conjecture is unprovable.

**Category:** Number Theory

[1263] **viXra:1606.0184 [pdf]**
*submitted on 2016-06-17 22:38:57*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** Number Theory

[1262] **viXra:1606.0173 [pdf]**
*submitted on 2016-06-17 08:50:55*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 16 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Holistic Beauty Primality Tree Of Any Set, Universal Growth Of Any Given Set’.

**Category:** Number Theory

[1261] **viXra:1606.0154 [pdf]**
*submitted on 2016-06-15 07:31:44*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 14 Pages.

In this research investigation, the author has presented a theory of ‘The Universal
Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1260] **viXra:1606.0144 [pdf]**
*submitted on 2016-06-15 00:52:10*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

A form of the exponential Mangoldt function is derived using indicator functions. The function's relationship to other important number theoretic functions are derived and discussed.

**Category:** Number Theory

[1259] **viXra:1606.0136 [pdf]**
*submitted on 2016-06-14 00:55:44*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 13 Pages.

In this research investigation, the author has presented a theory of ‘The Universal Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1258] **viXra:1606.0135 [pdf]**
*submitted on 2016-06-14 00:58:08*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 13 Pages.

In this research investigation, the author has presented a theory of ‘The Universal Irreducible Any Field Generating Metric’.

**Category:** Number Theory

[1257] **viXra:1606.0118 [pdf]**
*submitted on 2016-06-13 03:17:38*

**Authors:** Reuven Tint

**Comments:** 6 Pages. written Russian

Let us prove that invariant identity is used for the proof of the FLT and Beal conjecture.

**Category:** Number Theory

[1256] **viXra:1606.0109 [pdf]**
*submitted on 2016-06-11 15:07:52*

**Authors:** William R. Blickos

**Comments:** 12 Pages.

This paper provides a method using periodic functions to check for primality, count factors, list factors,
calculate the exact prime distribution, and determine the Nth prime. It describes this method in a straightforward
manner, from one equation to the next, using graphs between each key step to help quickly visualize the
reasoning and result of each maneuver. It concludes with brief afterthoughts and ongoing questions about the
technique.

**Category:** Number Theory

[1255] **viXra:1606.0108 [pdf]**
*submitted on 2016-06-11 15:09:58*

**Authors:** William R. Blickos

**Comments:** 4 Pages.

A Trigonometric Series is crafted that gives the number of factors of x for all x. It is then manipulated to list all those factors. It is then used to give the exact distribution of the primes. It is then changed into a recursive sequence that generates the nth prime. Finally, a Product Polynomial Series is suggested as an improved alternative to the original. There is a series of questions in the second section regarding the functions.

**Category:** Number Theory

[1254] **viXra:1606.0088 [pdf]**
*submitted on 2016-06-09 13:43:39*

**Authors:** Hitesh Jain

**Comments:** 4 Pages.

To find out necessary conditions for any number to be expressed as difference of 4th powers of 2 natural numbers and maximum number of ways.

**Category:** Number Theory

[1253] **viXra:1606.0072 [pdf]**
*submitted on 2016-06-07 11:02:32*

**Authors:** William R. Blickos

**Comments:** 7 Pages.

Surfaces representing the primes and composites, the upper twin prime, the difference between squares, and the relations among them, are used to select input for a quadratic in such a way as to always generate a Twin Prime. Because each input generates a unique Twin, and because the input set is infinite, there are infinitely many twin primes.

**Category:** Number Theory

[1252] **viXra:1606.0065 [pdf]**
*submitted on 2016-06-06 21:27:48*

**Authors:** Igor Turkanov

**Comments:** 29 Pages.

This theorem is based on the study of holomorphy functions and on the fact that near the singularity point of the imaginary part of some rational function can accept an arbitrary preassigned value.

**Category:** Number Theory

[1251] **viXra:1605.0311 [pdf]**
*submitted on 2016-05-31 10:35:22*

**Authors:** José de Jesús Camacho Medina

**Comments:** 6 Pages.

This article disseminates a series of formulas that generate primes numbers as a product of the investigations of the author since the year 2011. This document describes general patterns on the entities primales and conceived an order in its distribution.

**Category:** Number Theory

[1250] **viXra:1605.0310 [pdf]**
*submitted on 2016-05-31 12:07:57*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 47 Pages. In French. Paper submitted to the Journal Algebra and Number Theory.

The paper gives a complete proof of the conjecture of BEAL followed by some numerical examples.

**Category:** Number Theory

[1249] **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

[1248] **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

[1247] **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

[1246] **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

[1245] **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

[1244] **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

[1243] **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

[1242] **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

[1241] **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

[1240] **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

[1239] **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

[1238] **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

[1237] **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

[1236] **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

[1235] **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

[569] **viXra:1606.0144 [pdf]**
*replaced on 2016-06-24 21:17:25*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

A form of the exponential Mangoldt function is derived using indicator functions. The function's relationship to other important number theoretic functions are derived and discussed.

**Category:** Number Theory

[568] **viXra:1606.0118 [pdf]**
*replaced on 2016-06-13 05:09:24*

**Authors:** Reuven Tint

**Comments:** 12 Pages. Original written Russian

Let us prove that invariant identity is used for the proof of the FLT and Beal conjecture.

**Category:** Number Theory

[567] **viXra:1605.0229 [pdf]**
*replaced on 2016-06-07 15:21:17*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

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.My Visual Proof by Automorphism asserts Symmetry. If 1/2 10^13 is inside (0,1) then 1/2 10^100 is inside (0,1). In other words if there is Symmetry about the real axis and Symmetry about the critical line then there is Symmetry between 1/2 10^13 is inside (0,1) and 1/2 10^100 which is inside (0,1).

**Category:** Number Theory

[566] **viXra:1605.0195 [pdf]**
*replaced on 2016-06-03 03:24:22*

**Authors:** A. A. Frempong

**Comments:** 9 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. Five similar versions of the proof are presented, using a single page for each version. 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 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. 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. For the fourth and fifth versions of the proof, ratio terms were used to begin the construction of the proof. It is also exemplified that if some of the lengths are not positive integers but positive radicals, the derived necessary condition for c^n = a^n + b^n to hold is applicable. Each 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

[565] **viXra:1605.0195 [pdf]**
*replaced on 2016-06-01 00:31:28*

**Authors:** A. A. Frempong

**Comments:** 9 Pages. Copyright © 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. Five similar versions of the proof are presented, using a single page for each version. 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 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. 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. For the fourth and fifth versions of the proof, ratio terms were used to begin the construction of the proof. Each 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

[564] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-29 00:16:35*

**Authors:** A. A. Frempong

**Comments:** 8 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. Four similar versions of the proof are presented, using a single page for each version. 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 (n an integer), the general equation, c^n = a^n + b^n does not hold. For the first three versions, one applies a polar coordinate system as follows. 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. For the fourth version of the proof, ratio terms were used to begin the construction of the proof. Each 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

[563] **viXra:1605.0195 [pdf]**
*replaced on 2016-05-25 03:37:01*

**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 (n an integer), 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

[562] **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

[561] **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

[560] **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