**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)

2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

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(51) - 1409(47) - 1410(17) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(33) - 1504(23) - 1505(18) - 1506(12) - 1507(16) - 1508(14) - 1509(15) - 1510(13) - 1511(9) - 1512(26)

2016 - 1601(14) - 1602(18) - 1603(77) - 1604(55) - 1605(28) - 1606(18) - 1607(21) - 1608(18) - 1609(24) - 1610(24) - 1611(12) - 1612(21)

2017 - 1701(11)

Any replacements are listed further down

[1390] **viXra:1701.0483 [pdf]**
*submitted on 2017-01-13 13:46:54*

**Authors:** Reuven Tint

**Comments:** 4 Pages. original papper in russian

Annotation. Are given in Section 1 the theorem and its proof, complementing the classical formulation of the ABC conjecture, and in Chapter 2 addressed the issue of communication with the elliptic curve Frey's "Great" Fermat's theorem.

**Category:** Number Theory

[1389] **viXra:1701.0482 [pdf]**
*submitted on 2017-01-13 09:00:42*

**Authors:** guilhem CICOLELLA

**Comments:** 4 Pages.

the only consecutives powers being 8 and 9 the probleme consisted in demonstrating that the quantities of primes numbers inferior to one billion depended on one single equation based on two different methods of calculation with congruent results,the ultimate purpose being to prove the existence of an algorithm capable of determining two intricate values more quickly than with computer(rapid mathematical system r.m.S)

**Category:** Number Theory

[1388] **viXra:1701.0478 [pdf]**
*submitted on 2017-01-12 13:25:43*

**Authors:** Tom Masterson

**Comments:** 1 Page. © 1965 by Tom Masterson

A number theory query related to Fermat's last theorem in higher dimensions.

**Category:** Number Theory

[1387] **viXra:1701.0475 [pdf]**
*submitted on 2017-01-12 10:27:06*

**Authors:** Nikolay Dementev

**Comments:** 5 Pages.

Based on the observation of randomly chosen primes it has been conjectured that the sum of digits that form any prime number should yield either even number or another prime number. The conjecture was successfully tested for the first 100 primes.

**Category:** Number Theory

[1386] **viXra:1701.0397 [pdf]**
*submitted on 2017-01-10 07:35:16*

**Authors:** Quang Nguyen Van

**Comments:** 1 Page.

We have found a solution of FLT for n = 3, so that FLT is wrong. In this paper, we give a counterexample ( the solution in integer for equation x^3 + y^3 = z^3 only. It is too large ( 18 digits).

**Category:** Number Theory

[1385] **viXra:1701.0329 [pdf]**
*submitted on 2017-01-08 11:02:17*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following conjecture: For any pair of consecutive primes [p1, p2], p2 > p1 > 43, p1 and p2 having the same number of digits, there exist a prime q, 5 < q < p1, such that the number n obtained concatenating (from the left to the right) q with p2, then with p1, then again with q is prime. Example: for [p1, p2] = [961748941, 961748947] there exist q = 19 such that n = 1996174894796174894119 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of consecutive primes with 9 digits are 19, 17, 107, 23, 131, 47, 83, 79, 61, 277, 163, 7, 41, 13, 181, 19, 7, 37, 29 and 23 (all twenty primes lower than 300!), the corresponding primes n obtained having 20 to 24 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.

**Category:** Number Theory

[1384] **viXra:1701.0320 [pdf]**
*submitted on 2017-01-07 12:05:30*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: For any pair of twin primes [p, p + 2], p > 5, there exist a prime q, 5 < q < p, such that the number n obtained concatenating (from the left to the right) q with p + 2, then with p, then again with q is prime. Example: for [p, p + 2] = [18408287, 18408289] there exist q = 37 such that n = 37184082891840828737 is prime. Note that the least values of q that satisfy this conjecture for twenty consecutive pairs of twins with 8 digits are 19, 7, 19, 11, 23, 23, 47, 7, 47, 17, 13, 17, 17, 37, 83, 19, 13, 13, 59 and 97 (all twenty primes lower than 100!), the corresponding primes n obtained having 20 digits! This method appears to be a good way to obtain big primes with a high degree of ease and certainty.

**Category:** Number Theory

[1383] **viXra:1701.0281 [pdf]**
*submitted on 2017-01-04 06:46:28*

**Authors:** Ryujin Choe

**Comments:** 1 Page.

Every even integer greater than 2 can be expressed as the sum of two primes

**Category:** Number Theory

[1382] **viXra:1701.0014 [pdf]**
*submitted on 2017-01-03 01:34:45*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[1381] **viXra:1701.0012 [pdf]**
*submitted on 2017-01-02 10:39:11*

**Authors:** Clive Jones

**Comments:** 2 Pages.

An exploration of prime-number summing grids

**Category:** Number Theory

[1380] **viXra:1701.0008 [pdf]**
*submitted on 2017-01-02 04:55:37*

**Authors:** Ryujin Choe

**Comments:** 2 Pages.

Twin primes are infinitely many

**Category:** Number Theory

[1379] **viXra:1612.0414 [pdf]**
*submitted on 2016-12-31 11:40:34*

**Authors:** Clive Jones

**Comments:** 2 Pages.

An exploration of prime-number summing grids

**Category:** Number Theory

[1378] **viXra:1612.0406 [pdf]**
*submitted on 2016-12-30 11:14:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of primes p = 30*h + j, where j can be 1, 7, 11, 13, 17, 19, 23 or 29, such that, concatenating to the left p with a number m, m < p, is obtained a number n having the property that the number of primes of the form 30*k + j up to n is equal to p. Example: such a number p is 67 = 30*2 + 7, because there are 67 primes of the form 30*k + 7 up to 3767 and 37 < 67. I also conjecture that there exist an infinity of primes q that don’t belong to the set above, i.e. doesn’t exist m, m < q, such that, concatenating to the left q with m, is obtained a number n having the property shown. Primes can be classified based on this criteria in two sets: primes p that have the shown property like 13, 17, 23, 31, 37, 41, 47, 59, 61, 67, 71, 73, 89, 103 (...) and primes q that don’t have it like 7, 11, 19, 29, 43, 53, 79, 83, 101 (...).

**Category:** Number Theory

[1377] **viXra:1612.0400 [pdf]**
*submitted on 2016-12-30 02:12:38*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p, p > 5, there exist q prime, q > p, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n, where n is the number obtained concatenating p with q, is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for p = 17 there exist q = 23 such that there are 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.

**Category:** Number Theory

[1376] **viXra:1612.0395 [pdf]**
*submitted on 2016-12-29 16:06:30*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of numbers n obtained concatenating two primes p and q, where p = 30*k + m1 and q = 30*h + m2, p < q, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 1723 obtained concatenating the primes p = 17 and q = 23, there exist 34 primes of the form 30*k + 17 up to 1723 and 34 primes of the form 30*k + 23 up to 1723.

**Category:** Number Theory

[1375] **viXra:1612.0387 [pdf]**
*submitted on 2016-12-28 20:35:01*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper we prove a simple theorem that is distantly related to the Even Goldbach conjecture and is weaker than Chen’s theorem regarding the expression of any even integer as the sum of a prime number and a semiprime number. We show that any even integer greater than six can be written as the sum of two odd integers coprime to one another and atleast one of them is a prime.

**Category:** Number Theory

[1374] **viXra:1612.0383 [pdf]**
*submitted on 2016-12-29 01:16:00*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that there exist an infinity of palindromes n for which the number of primes up to n of the form 30k + 7 is equal to the number of primes up to n of the form 30k + 11 and I found the first 40 terms of the sequence of n (I also found few larger terms, as 99599, 816618 or 1001001 up to which the number of primes from the two sets, equally for each, is 1154, 8159, respectively 9817).

**Category:** Number Theory

[1373] **viXra:1612.0294 [pdf]**
*submitted on 2016-12-18 23:45:17*

**Authors:** Zhang Tianshu

**Comments:** 21 Pages.

The ABC conjecture is both likely of the wrong and likely of the right in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we find directly a specific equality 1+2N (2N-2)=(2N-1)2 with N≥2, then set about analyzing limits of values of ε to discuss the right and the wrong of the ABC conjecture in which case satisfying 2N-1>(Rad(1, 2N(2N-2), 2N-1))1+ε . Thereby supply readers to make with a judgment concerning a truth or a falsehood which the ABC conjecture is.

**Category:** Number Theory

[1372] **viXra:1612.0278 [pdf]**
*submitted on 2016-12-17 11:51:55*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 52 pages. In French. Submitted to the journal Functiones et Approximatio, Commentarii Mathematici. Comments welcome.

En 1997, Andrew Beal \cite{B1} avait annonc\'e la conjecture suivante : \textit{Soient $A, B,C, m,n$, et $l$ des entiers positifs avec $m,n,l > 2$. Si $A^m + B^n = C^l$ alors $A, B,$ et $C$ ont un facteur en commun}. Nous consid\'erons le polyn\^ome $P(x)=(x-A^m)(x-B^n)(x+C^l)=x^3-px+q$ avec $p,q$ des entiers qui d\'ependent de $A^m,B^n$ et $C^l$. La r\'esolution de $x^3-px+q=0$ nous donne les trois racines $x_1,x_2,x_3$ comme fonctions de $p,q$ et d'un param\`etre $\theta$. Comme $A^m,B^n,-C^l$ sont les seules racines de $x^3-px+q=0$, nous discutons les conditions pour que $x_1,x_2,x_3$ soient des entiers. Quatre exemples num\'eriques sont pr\'esent\'es.
\\

**Category:** Number Theory

[1371] **viXra:1612.0262 [pdf]**
*submitted on 2016-12-16 09:29:19*

**Authors:** Marius Coman

**Comments:** 1 Page.

In my previous paper “Conjecture involving repunits, repdigits, repnumbers and also the primes of the form 30k + 11 and 30k + 13” I conjectured that there exist an infinity of repnumbers n (repunits, repdigits and numbers obtained concatenating not the unit or a digit but a number) for which the number of primes up to n of the form 30k + 11 is equal to the number of primes up to n of the form 30k + 13 and I found the first 18 terms of the sequence of n (I also found few larger terms, as 11111, 888888 and 11111111 up to which the number of primes from the two sets, equally for each, is 167, 8816, respectively 91687). In this paper I extend the search to first 40 terms of the sequence.

**Category:** Number Theory

[1370] **viXra:1612.0260 [pdf]**
*submitted on 2016-12-15 16:20:52*

**Authors:** Marius Coman

**Comments:** 1 Page.

In my previous paper “Conjecture on semiprimes n = p*q related to the number of primes up to n” I was wondering if there exist a class of numbers n for which the number of primes up to n of the form 30k + 1, 30k + 7, 30k + 11, 30k + 13, 30k + 17, 30k + 19, 30k + 23 and 30k + 29 is equal in each of these eight sets. I didn’t yet find such a class, but I observed that around the repdigits, repunits and repnumbers (numbers obtained concatenating not the unit or a digit but a number) the distribution of primes in these eight sets tends to draw closer and I made a conjecture about it.

**Category:** Number Theory

[1369] **viXra:1612.0257 [pdf]**
*submitted on 2016-12-15 10:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of semiprimes n = p*q, where p = 30*k + m1 and q = 30*h + m2, m1 and m2 distinct, having one from the values 1, 7, 11, 13, 17, 19, 23, 29, such that the number of primes congruent to m1 (mod 30) up to n is equal to the number of primes congruent to m2 (mod 30) up to n. Example: for n = 91 = 7*13, there exist 3 primes of the form 30*k + 7 up to 91 (7, 37 and 67) and 3 primes of the form 30*k + 13 up to 91 (13, 43 and 73).

**Category:** Number Theory

[1368] **viXra:1612.0253 [pdf]**
*submitted on 2016-12-15 06:24:20*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I conjecture that: (I) for any prime p of the form 6*k + 1 there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 10)/3; (II) for any prime p of the form 6*k – 1, p ≥ 11, there are obtained at least n primes concatenating p to the left with the (p – 1) integers lesser than p, where n ≥ (p - 8)/3.

**Category:** Number Theory

[1367] **viXra:1612.0223 [pdf]**
*submitted on 2016-12-11 17:29:09*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.

**Category:** Number Theory

[1366] **viXra:1612.0200 [pdf]**
*submitted on 2016-12-11 02:20:30*

**Authors:** Simon Plouffe

**Comments:** 28 Pages.

A presentation is made on the numerical world of mathematics. Round table on the numerical data.
Une présentation du numérique à Nantes, table ronde organisée par ADN ouest au Lycée Clémenceau

**Category:** Number Theory

[1365] **viXra:1612.0142 [pdf]**
*submitted on 2016-12-09 02:54:12*

**Authors:** Brian Ekanyu

**Comments:** 6 Pages.

This paper proves an identity for generating a special kind of Pythagorean quadruples by conjecturing that the shortest is defined by a=1,2,3,4...... and b=a+1, c=ab and d=c+1. It also shows that a+d=b+c and that the surface area to volume ratio of these Pythagorean boxes is given by 4/a where a is the length of the shortest edge(side).

**Category:** Number Theory

[1364] **viXra:1612.0140 [pdf]**
*submitted on 2016-12-09 03:46:53*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I conjectured that for any largest prime factor of a Poulet number p1 with two prime factors exists a series with infinite many Poulet numbers p2 formed this way: p2 mod (p1 - d) = d, where d is the largest prime factor of p1 (see the sequence A214305 in OEIS). In this paper I conjecture that for any least prime factor of an odd Harshad number h1 with two prime factors, not divisible by 3, exists a series with infinite many Harshad numbers h2 formed this way: h2 mod (h1 - d) = d, where d is the least prime factor of p1.

**Category:** Number Theory

[1363] **viXra:1612.0138 [pdf]**
*submitted on 2016-12-08 15:52:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) For any prime p, p > 5, there exist n positive integer such that the sum of the digits of the number p*2^n is divisible by p; (II) For any prime p, p > 5, there exist an infinity of positive integers m such that the sum of the digits of the number p*2^m is prime.

**Category:** Number Theory

[1362] **viXra:1612.0101 [pdf]**
*submitted on 2016-12-07 11:18:19*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that for any pair of sexy primes (p, p + 6) there exist a prime q = p + 6*n, where n > 1, such that the number p*(p + 6)*(p + 6*n) is a Harshad number.

**Category:** Number Theory

[1361] **viXra:1612.0072 [pdf]**
*submitted on 2016-12-07 05:45:46*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that for any prime p of the form 6*k + 1 there exist an infinity of Harshad numbers of the form p*q1*q2, where q1 and q2 are distinct primes, q1 = p + 6*m and q2 = p + 6*n.

**Category:** Number Theory

[1360] **viXra:1612.0042 [pdf]**
*submitted on 2016-12-03 10:57:30*

**Authors:** Safaa Abdallah Moallim

**Comments:** 5 Pages.

In this paper we prove that there exist infinitely many twin prime numbers by studying n when 6n±1 are primes. By studying n we show that for every n that generates a twin prime number, there has to be m>n that generates a twin prime number too.

**Category:** Number Theory

[1359] **viXra:1611.0410 [pdf]**
*submitted on 2016-11-30 07:48:39*

**Authors:** Zhang Tianshu

**Comments:** 18 Pages.

The ABC conjecture seemingly is difficult to carry conviction in the face of satisfactory many primes and satisfactory many odd numbers of 6K±1 from operational results of computer programs. So we select and adopt a specific equality 1+2N(2N-2)=(2N-1)2 with N≥2 satisfying 2N-1>(Rad(2N-2))1+ ε. Then, proceed from the analysis of the limits of values of ε to find its certain particular values, thereby finally negate the ABC conjecture once and for all.

**Category:** Number Theory

[1358] **viXra:1611.0390 [pdf]**
*submitted on 2016-11-29 03:29:40*

**Authors:** Robert Deloin

**Comments:** 13 Pages.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate infinitely many primes.
The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new
approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences.
As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and
the binary Goldbach conjecture.
The method is also used to prove that there are infinitely many primorial and factorial primes.

**Category:** Number Theory

[1357] **viXra:1611.0373 [pdf]**
*submitted on 2016-11-27 08:39:53*

**Authors:** Victor Christianto

**Comments:** 4 Pages. This paper will be submitted to Annals of Mathematics

In this paper we will give an outline of proof of Fermat’s Last Theorem using a graphical method. Although an exact proof can be given using differential calculus, we choose to use a more intuitive graphical method.

**Category:** Number Theory

[1356] **viXra:1611.0224 [pdf]**
*submitted on 2016-11-14 18:05:57*

**Authors:** Jonas Kaiser

**Comments:** 11 Pages.

The sieve of Collatz is a new algorithm to trace back the non-linear Collatz problem to a linear cross out algorithm. Until now it is unproved.

**Category:** Number Theory

[1355] **viXra:1611.0178 [pdf]**
*submitted on 2016-11-12 09:51:56*

**Authors:** Aaron Chau

**Comments:** 3 Pages.

十分幸运，本文应用的是永不改变的定律（多与少），而不再是重复那类受局限的定理。
感谢数学的美妙，因为多与少的个数区别永远会造成二个质数的距离= 2。简述，= 2。

**Category:** Number Theory

[1354] **viXra:1611.0176 [pdf]**
*submitted on 2016-11-12 04:58:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of even numbers n for which n^2 is a Harshad-Coman number and I also make a classification in four classes of all the even numbers.

**Category:** Number Theory

[1353] **viXra:1611.0175 [pdf]**
*submitted on 2016-11-12 05:01:08*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I defined the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer. In this paper I conjecture that there exist an infinity of odd numbers n for which n^2 is a Harshad-Coman number and I also make a classification in three classes of all the odd numbers greater than 1.

**Category:** Number Theory

[1352] **viXra:1611.0172 [pdf]**
*submitted on 2016-11-11 15:58:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) If P is both a Poulet number and a Harshad number, than the number P – 1 is also a Harshad number; (II) If P is a Poulet number divisible by 5 under the condition that the sum of the digits of P – 1 is not divisible by 5 than P – 1 is a Harshad number; (III) There exist an infinity of Harshad numbers of the form P – 1, where P is a Poulet number.

**Category:** Number Theory

[1351] **viXra:1611.0171 [pdf]**
*submitted on 2016-11-11 16:00:16*

**Authors:** Marius Coman

**Comments:** 2 Pages.

OEIS defines the notion of Harshad numbers as the numbers n with the property that n/s(n), where s(n) is the sum of the digits of n, is integer (see the sequence A005349). In this paper I define the notion of Harshad-Coman numbers as the numbers n with the property that (n – 1)/(s(n) – 1), where s(n) is the sum of the digits of n, is integer and I make the conjecture that there exist an infinity of Poulet numbers which are also Harshad-Coman numbers.

**Category:** Number Theory

[1350] **viXra:1611.0120 [pdf]**
*submitted on 2016-11-09 07:22:21*

**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

[1349] **viXra:1611.0089 [pdf]**
*submitted on 2016-11-07 11:29:42*

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

**Comments:** 10 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutanis problem (after Shizuo
Kakutani) and so on. Several various generalization of the Collatz conjecture
has been carried. In this paper a new generalization of the Collatz conjecture
called as the 3n ± p conjecture; where p is a prime is proposed. It functions on
3n + p and 3n - p, and for any starting number n, its sequence eventually enters
a finite cycle and there are finitely many such cycles. The 3n ± 1 conjecture, is
a special case of the 3n ± p conjecture when p is 1.

**Category:** Number Theory

[1348] **viXra:1611.0085 [pdf]**
*submitted on 2016-11-07 06:46:24*

**Authors:** Predrag Terzic

**Comments:** 32 Pages.

Some theorems and conjectures concerning prime numbers .

**Category:** Number Theory

[1347] **viXra:1610.0356 [pdf]**
*submitted on 2016-10-29 14:52:21*

**Authors:** Caitherine Gormaund

**Comments:** 2 Pages.

In which the Collatz Conjecture is proven using fairly simple mathematics.

**Category:** Number Theory

[1346] **viXra:1610.0349 [pdf]**
*submitted on 2016-10-28 13:23:48*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper we offer the some details and particulars about some famous conjectures in relative to consecutive primes.

**Category:** Number Theory

[1345] **viXra:1610.0313 [pdf]**
*submitted on 2016-10-26 05:42:56*

**Authors:** Jared Beal

**Comments:** 14 Pages.

This paper describes an algorithm for finding all the prime numbers. It also describes how this pattern among primes can be used to show the ratio of primes to not primes in an infinite set of X integers. It can also be used to show that the ratio of twin primes to not twin primes in an infinite set of X integers is always going to be greater than zero.

**Category:** Number Theory

[1344] **viXra:1610.0284 [pdf]**
*submitted on 2016-10-24 03:05:49*

**Authors:** Reuven Tint

**Comments:** Updates: 4.3.2 - 4.3.5.. page 7

Аннотация. Предложен вариант решения гипотезы Била с помощью прямого доказательства» Великой» теоремы Ферма элементарными методами. Новыми являются «инвариантное тождество « (ключевое слово) и полученные нами приведенные в тексте работы тождества, позволившие напрямую решить ВТФ и гипотезу Била,и ряд других. Предложены также новая формулировка теорем ( п.2.1.4.), ,доказательства для n= 1,2,3,..n>2 и x,y,z>2.

**Category:** Number Theory

[1343] **viXra:1610.0276 [pdf]**
*submitted on 2016-10-24 00:02:00*

**Authors:** John Smith

**Comments:** 19 Pages.

Riemann's prime-counting function R(x) looks good for every value of x we can compute, but in the light of Littlewood's result its superiority over li(x) is illusory: Ingram (1938) pointed out that 'for special values of x (as large as we please), the one approximation will deviate as widely as the other from the true value'. This note introduces a type of prime-counting function that is always better than li(x)...

**Category:** Number Theory

[1342] **viXra:1610.0275 [pdf]**
*submitted on 2016-10-23 13:15:42*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Аннотация. Интерес к названной в заглавии проблеме вызван следующими соображениями:
1) Возьмем, к примеру, «пифагорово» уравнение, все взаимно простые решения которого опре-
деляются формулами A= a^2- b^2 и B=2ab. Но если мы выберем A≠a^2- b^2 и B≠2ab как гипо-
тетически «верные» решения этого уравнения, то, наверное, можно будет доказать, что, в этом
случае, «пифагорово» уравнение не существует. Но оно действительно не существует для гипотетически выбранных «верных» решений.
2) Уравнение A^N+B^N = C^N и уравнение эллиптической кривой Фрея (как будет показано ниже для предложенного варианта их решения) не совместны.
3) Поэтому, как представляется, выглядит не совсем убедительной связь между уравнением
эллиптической кривой Фрея и соответствующим уравнением Ферма.
4) Приведено приложение.

**Category:** Number Theory

[1341] **viXra:1610.0274 [pdf]**
*submitted on 2016-10-23 13:19:39*

**Authors:** Reuven Tint

**Comments:** 2 Pages.

Annotation. Interest in the title problem is caused by the following considerations:
1) Take, for example, "Pythagoras' equation, all of which are relatively prime solutions determined
Delyan formulas A= a^2- b^2 and B=2ab. But if we choose A≠a^2- b^2 and B≠2ab both hypo-
Tethyan "correct" solutions of this equation, then perhaps it will be possible to prove that, in this
case, "Pythagoras" equation exists. But it really does not exist for the selected hypothetically "true" solutions.
2) The equation A^N+B^N = C^N and the equation of the elliptic curve Frey (as will be shown below for the proposed options to solve them) are not compatible.
3) Therefore, it seems, it does not look quite convincing relationship between the equation
elliptic curve Frey Farm and the corresponding equation.
4) Supplement.

**Category:** Number Theory

[1340] **viXra:1610.0272 [pdf]**
*submitted on 2016-10-23 13:58:45*

**Authors:** Luca Nascimbene

**Comments:** 13 Pages.

In this paper the author continue the works [6] [11] [12] and present a proposal for a demonstration on the Riemann Hypothesis and the conjecture on the multiplicity of non-trivial zeros of the Riemann s zeta.

**Category:** Number Theory

[1339] **viXra:1610.0253 [pdf]**
*submitted on 2016-10-21 18:17:51*

**Authors:** Filippos Nikolaidis

**Comments:** 10 Pages. fil_nikolaidis@yahoo.com

The present study is an effort for giving some evidence that the goldbach conjecture is not true, by showing that not all even natural numbers greater than two can be expressed as a sum of two primes. This conclusion can be drawn by showing that prime numbers are not enough –in population- so that, when added in couples, to give all the even numbers.

**Category:** Number Theory

[1338] **viXra:1610.0183 [pdf]**
*submitted on 2016-10-17 05:37:47*

**Authors:** Edward Szaraniec

**Comments:** 5 Pages.

Equation constituting the Beal conjecture is rearranged and squared, then rearranged
again and raised to power 4. The result, standing as an equivalent having the same
property, is emerging as a singular primitive Pythagorean equation with no solution.
So, the conjecture is proved. General line of proving the Pythagorean equation is
observed as a moving spirit.

**Category:** Number Theory

[1337] **viXra:1610.0172 [pdf]**
*submitted on 2016-10-16 05:13:25*

**Authors:** Mugur B. Răuţ

**Comments:** 5 Pages.

In this paper we propose another proof for Fermat’s Last Theorem (FLT). We found a simpler approach through Pythagorean Theorem, so our demonstration would be close to the times FLT was formulated. On the other hand it seems the Pythagoras’ Theorem was the inspiration for FLT. It resulted one of the most difficult mathematical problem of all times, as it was considered. Pythagorean triples existence seems to support the claims of the previous phrase.

**Category:** Number Theory

[1336] **viXra:1610.0106 [pdf]**
*submitted on 2016-10-10 03:35:21*

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

**Comments:** 9 Pages.

The Collatz conjecture is an open conjecture in mathematics named so after Lothar Collatz who proposed it in 1937. It is also known as 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), Kakutani's problem (after Shizuo Kakutani) and so on.
In this paper a new conjecture called as the 3n-1 conjecture which is akin to the Collatz conjecture is proposed. It functions on 3n -1, for any starting number n, its sequence eventually reaches either 1, 5 or 17. The 3n-1 conjecture is compared with the Collatz conjecture.

**Category:** Number Theory

[1335] **viXra:1610.0099 [pdf]**
*submitted on 2016-10-08 17:28:15*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve de la conjecture de Sophie Germain en utilisant le theoreme de densité de Chebotarev ,le principe d' inclusion d'exclusion de Moivre ,la formule de Mertens . en 13 pages nous donnons une preuve convaincante

In this paper We give Sophie Germain 's conjecture proof by using Chebotarev density theorem, principle inclusion -exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1334] **viXra:1610.0083 [pdf]**
*submitted on 2016-10-07 06:34:33*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

ζ(s)=1/(((1/(2))/log(2)))+ 1/(((1/(3))/log(3)))+ 1/(((1/(4))/log(4)))+1/(((1/(5))/log(5))) is a form of Riemann Zeta Function and it shows an approximate relationship between the Riemann Zeta Function and Prime Numbers.

**Category:** Number Theory

[1333] **viXra:1610.0082 [pdf]**
*submitted on 2016-10-07 06:37:51*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

The classical Distribution of Primes Equation can be modified to make an Nth Prime Equation which generates the Nth Prime.

**Category:** Number Theory

[1332] **viXra:1610.0065 [pdf]**
*submitted on 2016-10-05 09:48:06*

**Authors:** Bing He

**Comments:** 14 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. These generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[1331] **viXra:1610.0042 [pdf]**
*submitted on 2016-10-04 12:01:31*

**Authors:** Idriss Olivier Bado

**Comments:** Dans ce présent document nous donnons la preuve du théorème de Mertens en utilisant le théorème de densité de Chebotarev ,principle d'inclusion - exclusion de Moivre,formule de Mertens en 15 pages nous donnons une élégante preuve

In this paper we give the proof of Sophie Germain's conjecture by using Chebotarev density theorem, the principle inclusion-exclusion of Moivre, Mertens formula

**Category:** Number Theory

[1330] **viXra:1610.0034 [pdf]**
*submitted on 2016-10-03 19:56:15*

**Authors:** Chunxuan Jiang

**Comments:** 6 Pages.

using complex hyperbolic function we prove Fermat last theorem

**Category:** Number Theory

[1329] **viXra:1610.0033 [pdf]**
*submitted on 2016-10-03 20:01:14*

**Authors:** Chunxuan Jiang

**Comments:** 5 Pages.

using trogonometric function we prove Fermat last theorem

**Category:** Number Theory

[1328] **viXra:1610.0024 [pdf]**
*submitted on 2016-10-03 09:06:13*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

(1/2 Part)>1.002 (1.002, 2.16, 4.008 & 6.012) Generate Riemann Non Trivial Zero’s Off Of Critical Line. A Riemann Non Trivial Zero off the Critical Line occurs between 1 /2 or .50 and Gamma 0.577215664901532860606512090 08240243104 215 93 359399.When (1/2 Part) = (1.002 , 2.16, 4.008 & 6.012) Riemann Non Trivial Zero’s Are Off .001 To The Rt. Of The Critical Line & When (1/2 Part)= (1 / 2) A Riemann Non Trivial Zero’s Will Be On Critical Line.

**Category:** Number Theory

[1327] **viXra:1610.0016 [pdf]**
*submitted on 2016-10-02 14:25:25*

**Authors:** Philip E Gibbs

**Comments:** 14 Pages.

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search is used to find over 300 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[1326] **viXra:1610.0009 [pdf]**
*submitted on 2016-10-01 19:37:45*

**Authors:** Liujingru

**Comments:** 4 Pages.

This work reveals the intrinsic relationship of numbers with the conception of “prime multiple” to prove the “hypothesis of twin primes”. Based on this proof, “Goldbach conjecture” is proved with the “Odd-Gaussian Corresponding”. The nature of “prime number” can be thus obtained.Paper is using the axiom Ⅶ twice. For the first time: high high more than nonsingular group, according to the axiom Ⅶ get there will be a (high + high group). Second: high + high group) will be (prime number + prime)

**Category:** Number Theory

[1325] **viXra:1610.0008 [pdf]**
*submitted on 2016-10-01 20:19:40*

**Authors:** 刘静儒

**Comments:** 4 Pages.

通过“素数的倍数”这一概念，揭示了数的内在关系，论证了“孪生素数猜想”，并在此基础上给出了“奇高组”的定义，并结合“高斯对应”，论文只是两次运用公理Ⅶ。第一次：奇高组多于非奇高组，根据公理Ⅶ得到必有这样的结果：（奇高组+奇高组）。第二次：（奇高组+奇高组）必有这样的结果：（素数+素数），这就证明了“哥德巴赫猜想”。

**Category:** Number Theory

[1324] **viXra:1610.0001 [pdf]**
*submitted on 2016-10-01 01:46:45*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Let us consider positive integers which have a common prime factor as a kind, then the positive half line of the number axis consists of infinite many recurring line segments of same permutations of c kinds of integers’ points, where c≥1. In this article we proved Grimm’s conjecture by stepwise change symbols of each kind of composite numbers’ points at the number axis, so as to form consecutive composite numbers’ points under the qualification of proven Legendre-Zhang conjecture as the true.

**Category:** Number Theory

[1323] **viXra:1609.0425 [pdf]**
*submitted on 2016-09-29 11:39:24*

**Authors:** Philip E Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided

**Category:** Number Theory

[1322] **viXra:1609.0398 [pdf]**
*submitted on 2016-09-27 14:41:12*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand ...

**Category:** Number Theory

[1321] **viXra:1609.0384 [pdf]**
*submitted on 2016-09-26 21:46:39*

**Authors:** Bing He, Long Li

**Comments:** 16 Pages.

In this paper we give a finite field analogue of one of the Appell series and obtain some transformation and reduction formulae and the generating functions for the Appell series over finite fields.

**Category:** Number Theory

[1320] **viXra:1609.0383 [pdf]**
*submitted on 2016-09-26 23:16:52*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[1319] **viXra:1609.0377 [pdf]**
*submitted on 2016-09-26 11:05:09*

**Authors:** Brekouk

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[1318] **viXra:1609.0374 [pdf]**
*submitted on 2016-09-26 10:09:55*

**Authors:** Wei Ren

**Comments:** 17 Pages.

Collatz conjecture (or 3x+1 problem) is out for about 80 years. The
verification of Collatz conjecture has reached to the number about
60bits until now. In this paper, we propose new algorithms that can
verify whether the number that is about 100000bits (30000 digits)
can return 1 after 3*x+1 and x/2 computations. This is the largest
number that has been verified currently. The proposed algorithm
changes numerical computation to bit computation, so that extremely
large numbers (without upper bound) becomes possible to be verified.
We discovered that $2^{100000}-1$ can return to 1 after 481603 times
of 3*x+1 computation, and 863323 times of x/2 computation.

**Category:** Number Theory

[1317] **viXra:1609.0373 [pdf]**
*submitted on 2016-09-26 10:14:45*

**Authors:** Wei Ren

**Comments:** 22 Pages.

Collatz conjecture (or 3x+1 problem) has not been proved to be true
or false for about 80 years. The exploration on this problem seems
to ask for introducing a totally new method. In this paper, a
mathematical induction method is proposed, whose proof can lead to
the proof of the conjecture. According to the induction, a new
representation (for dynamics) called ``code'' is introduced, to
represent the occurred $3*x+1$ and $x/2$ computations during the
process from starting number to the first transformed number that is
less than the starting number. In a code $3*x+1$ is represented by 1
and $x/2$ is represented by 0. We find that code is a building block
of the original dynamics from starting number to 1, and thus is more
primitive for modeling quantitative properties. Some properties only
exist in dynamics represented by code, but not in original dynamics.
We discover and prove some inherent laws of code formally. Code as a
whole is prefix-free, and has a unified form. Every code can be
divided into code segments and each segment has a form $\{10\}^{p
\geq 0}0^{q \geq 1}$. Besides, $p$ can be computed by judging
whether $x \in[0]_2$, $x\in[1]_4$, or computed by $t=(x-3)/4$,
without any concrete computation of $3*x+1$ or $x/2$. Especially,
starting numbers in certain residue class have the same code, and
their code has a short length. That is, $CODE(x \in [1]_4)=100,$
$CODE((x-3)/4 \in [0]_4)=101000,$ $CODE((x-3)/4 \in
[2]_8)=10100100,$ $CODE((x-3)/4 \in [5]_8)=10101000,$ $CODE((x-3)/4
\in [1]_{32})=10101001000,$ $CODE((x-3)/4\in [3]_{32})=10101010000,$
$CODE((x-3)/4\in [14]_{32})=10100101000.$ The experiment results
again confirm above discoveries. We also give a conjecture on $x \in
[3]_4$ and an approach to the proof of Collatz conjecture. Those
discoveries support the proposed induction and are helpful to the
final proof of Collatz conjecture.

**Category:** Number Theory

[1316] **viXra:1609.0358 [pdf]**
*submitted on 2016-09-25 11:28:38*

**Authors:** N.Prosh

**Comments:** 6 Pages.

About prime numbers and new way of find prime numbers

**Category:** Number Theory

[1315] **viXra:1609.0353 [pdf]**
*submitted on 2016-09-25 09:09:01*

**Authors:** Brekouk

**Comments:** 12 Pages.

ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentales des nombres premiers , et quatre autres théorèmes plus quatre lemmes ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture , et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[1314] **viXra:1609.0263 [pdf]**
*submitted on 2016-09-18 00:13:23*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Three similar versions of the proof are presented, using a single page for each version. The approach used in each proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary conditions in the solution for n = 2. The necessary conditions in the solutions for n = 2. will guide one to determine if there are solutions when n > 2.. For the first two versions, the proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1; and for the third version, on (a^2 + b^2)/c^2 = 1, with n = 2, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary conditions in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. For the first version of the proof, the proof began with reference to a right triangle. The second version of the proof began with ratio terms without any reference to a geometric figure. The third version occupies about half of a page. The third version of the proof began without any reference to a geometric figure or ratio terms. The second and third versions confirmed the proof in the first version. Each proof version is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, would be $107,250,000.

**Category:** Number Theory

[1313] **viXra:1609.0258 [pdf]**
*submitted on 2016-09-17 09:37:47*

**Authors:** Junnichi Fujii

**Comments:** 2 Pages.

The definition in time in the present-day physics is insufficient. Several problems which are to reconsider a definition in time and concern in time can be settled.

**Category:** Number Theory

[1312] **viXra:1609.0157 [pdf]**
*submitted on 2016-09-13 00:19:51*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[1311] **viXra:1609.0123 [pdf]**
*submitted on 2016-09-09 13:09:01*

**Authors:** T.Nakashima

**Comments:** 6 Pages.

First, we prove the relation of the sum of the mobius function and Riemann Hypothesis. This relationship is well known. I prove next section, without no tool we prove Riemann Hypothesis about mobius function. This is very chalenging attempt.

**Category:** Number Theory

[1310] **viXra:1609.0121 [pdf]**
*submitted on 2016-09-09 13:54:25*

**Authors:** Bijoy Rahman Arif

**Comments:** 5 Pages.

In this paper, we are going to prove Oppermann’s conjecture which states there are at least one prime presents between first and second halves of two consecutive pronic numbers greater than one. Subsequently, we are going to prove the logarithmic sum of primes between two pronic numbers increase highest magnitude of log(4).

**Category:** Number Theory

[1309] **viXra:1609.0115 [pdf]**
*submitted on 2016-09-09 08:08:37*

**Authors:** Bijoy Rahman Arif

**Comments:** 4 Pages.

In this paper, we are going to find the number of primes between consecutive squares. We are going to prove a special case: Brocard’s conjecture which states between the square of two consecutive primes greater than 2 at least four primes will present. Subsequently, we will approximate the number of primes between consecutive square

**Category:** Number Theory

[1308] **viXra:1609.0112 [pdf]**
*submitted on 2016-09-09 06:28:05*

**Authors:** Bijoy Rahman Arif

**Comments:** 3 Pages.

In this paper, we are going to prove a famous problem concerning prime numbers. Legendre’s conjecture states that there is always a prime p with n^2 < p < (n+1)^2, if n > 0. In 1912, Landau called this problem along with other three problems “unattackable at the presesnt state of mathematics.” Our approach to solve this problem is very simple. We will find a lower bound of the difference of second Chebyshev functions using a better Moiver-Stirling approximation and finally, we transfer it to the difference of first Chebyshev functions. The final difference is always greater than zero will prove Legendre’s conjecture.

**Category:** Number Theory

[1307] **viXra:1609.0080 [pdf]**
*submitted on 2016-09-06 23:20:51*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[1306] **viXra:1609.0059 [pdf]**
*submitted on 2016-09-05 11:45:39*

**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

[1305] **viXra:1609.0058 [pdf]**
*submitted on 2016-09-05 11:49:34*

**Authors:** T.Nakashima

**Comments:** 2 Pages.

This is the positive answer of Gilbreath's conjecture

**Category:** Number Theory

[1304] **viXra:1609.0052 [pdf]**
*submitted on 2016-09-04 16:05:23*

**Authors:** Aleksandr Tsybin

**Comments:** 3 Pages.

This problem is devoted a huge number of articles and books. So it does
not make sense to list them. I wrote this note 10 years ago and since then
a lot of time I tried to find the error in the reasoning and I can not this to
do. I’ll be glad if someone will be finds a mistake and even more will be
happy if an error will be not found.

**Category:** Number Theory

[1303] **viXra:1609.0048 [pdf]**
*submitted on 2016-09-05 06:28:40*

**Authors:** Predrag Terzic

**Comments:** 5 Pages.

Polynomial time compositeness tests for generalized Fermat numbers are introduced .

**Category:** Number Theory

[1302] **viXra:1609.0046 [pdf]**
*submitted on 2016-09-04 16:01:51*

**Authors:** Aleksandr Tsybin

**Comments:** 14 Pages.

For a positive integer n I construct an n × n matrix of special shape,
whose determinant equals the n-th prime number, and whose entries
are equal to 1,-1 or 0. Specific calculations which I have carried out
so far, allowed me to construct such matrices for all n up to 63.
These calculations are based on my own method for quick
calculations of determinants of special matrices along with a
variation on the Sieve of Eratosthenes.

**Category:** Number Theory

[1301] **viXra:1609.0025 [pdf]**
*submitted on 2016-09-02 07:36:57*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

The even Goldbach conjecture states that any even integer greater than four may be expressed as the sum of two odd primes. The odd Goldbach conjecture states that any odd integer greater than seven must be expressible as a sum of three odd primes. These conjectures remain unverified. In this paper we explore the possible constraints that exist on the smallest possible counterexample of the even Goldbach conjecture. We prove that the odd numbers immediately flanking the smallest counterexample of the even Goldbach conjecture are themselves expressible as the sum of three odd primes and are therefore consistent with the odd Golbach conjecture.

**Category:** Number Theory

[1300] **viXra:1609.0012 [pdf]**
*submitted on 2016-09-01 00:00:52*

**Authors:** D. D. Somashekara, S. L. Shalini, K. N. Vidya

**Comments:** 15 Pages.

In this paper, we give an alternate and simple proofs for Sear’s three term 3 φ 2 transformation formula, Jackson’s 3 φ 2 transformation formula and for a nonterminating form of the q-Saalschütz sum by using q exponential operator techniques. We also give an alternate proof for a nonterminating form of the q-Vandermonde sum. We also obtain some interesting special cases of all the three identities, some of which are analogous to the identities stated by Ramanujan in his lost notebook.

**Category:** Number Theory

[1299] **viXra:1608.0449 [pdf]**
*submitted on 2016-08-31 17:53:09*

**Authors:** Joe Chizmarik

**Comments:** 2 Pages. This is a proof by contradiction.

We first prove a weak form of Fermat's Last Theorem; this unique lemma is key to the entire proof. A corollary and lemma follow inter-relating Pythagorean and Fermat solutions. Finally, we prove Fermat's Last Theorem.

**Category:** Number Theory

[1298] **viXra:1608.0439 [pdf]**
*submitted on 2016-08-30 21:46:42*

**Authors:** Watcharakiete Wongcharoenbhorn

**Comments:** 4 Pages. English

We study on the cycle in the Collatz conjecture and there is something surprise us. Our goal is to show that there is no Collatz cycle

**Category:** Number Theory

[1297] **viXra:1608.0429 [pdf]**
*submitted on 2016-08-31 09:51:31*

**Authors:** Gyeongmin Yang

**Comments:** 5 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory

[595] **viXra:1701.0014 [pdf]**
*replaced on 2017-01-12 06:19:40*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This is a two page attempt using simple concepts

**Category:** Number Theory

[594] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-24 13:07:51*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages. typographical error on the abstract

ABSTRACT
Riemann Hypothesis states that all the non-trivial zeros of the zeta function ζ(s) have real part equal to 1⁄2. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[593] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-20 03:29:10*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of the zeta funtion ζ(s) are real in the critical strip, 0 ≤ σ ≤ 1; or equivalently, if ζ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[592] **viXra:1612.0296 [pdf]**
*replaced on 2016-12-19 05:18:45*

**Authors:** Armando M. Evangelista Jr.

**Comments:** 4 Pages.

In Riemann’s 1859 paper he conjecture that all the zeros of ξ(s) are real in the critical strip 0 ≤ σ ≤ 1, or equivalently, if ξ(s) is a complex quantity in the said strip, then it has no zero. It is the purpose of this present work to prove that the Riemann Hypothesis is true.

**Category:** Number Theory

[591] **viXra:1612.0223 [pdf]**
*replaced on 2016-12-15 14:31:17*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

The even Goldbach conjecture suggests that every even integer greater than four may be written as the sum of two odd primes. This conjecture remains unproven. We explore whether two probable primes satisfying the Fermat’s little theorem can potentially exist for every even integer greater than four. Our results suggest that there are no obvious constraints on this possibility.

**Category:** Number Theory

[590] **viXra:1612.0042 [pdf]**
*replaced on 2016-12-19 03:14:49*

**Authors:** Safa Abdallah Moallim

**Comments:** 8 Pages.

In this paper we prove that there exist infinitely many twin
prime numbers by studying n when 6n ± 1 are primes. By studying n we
show that for every n that generates a twin prime number, there has to be
m > n that generates a twin prime number too.

**Category:** Number Theory

[589] **viXra:1611.0390 [pdf]**
*replaced on 2016-12-08 03:13:44*

**Authors:** Robert Deloin

**Comments:** 10 Pages. This is version 2 with important changes.

Bunyakovsky's conjecture states that under special conditions, polynomial integer functions of degree greater than one generate innitely
many primes.
The main contribution of this paper is to introduce a new approach that enables to prove Bunyakovsky's conjecture. The key idea of this new
approach is that there exists a general method to solve this problem by using only arithmetic progressions and congruences.
As consequences of Bunyakovsky's proven conjecture, three Landau's problems are resolved: the n^2+1 problem, the twin primes conjecture and
the binary Goldbach conjecture.
The method is also used to prove that there are infinitely many primorial and factorial primes.

**Category:** Number Theory

[588] **viXra:1610.0065 [pdf]**
*replaced on 2016-10-10 23:28:04*

**Authors:** Bing He

**Comments:** 22 Pages.

In this paper we give a finite field analogue of the Lauricella hypergeometric series and
obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. Some of these generalize some known results of Li \emph{et al} as well as several other well-known results.

**Category:** Number Theory

[587] **viXra:1610.0016 [pdf]**
*replaced on 2016-10-26 05:46:31*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.29253.65761

A rational Diophantine m-tuple is a set of m distinct positive rational numbers such that the product of any two is one less than a rational number squared. A computational search has been used to find over 1000 examples of rational Diophantine sextuples of low height which are then analysed in terms of algebraic relationships between entries. Three examples of near-septuples are found where a rational Diophantine quintuple can be extended to sextuples in two different ways so that the combination fails to be a rational Diophantine septuple only in one pair.

**Category:** Number Theory

[586] **viXra:1609.0425 [pdf]**
*replaced on 2016-10-27 10:44:46*

**Authors:** Philip Gibbs

**Comments:** 13 Pages.

A polynomial equation in six variables is given that generalises the definition of regular rational Diophantine triples, quadruples and quintuples to regular rational Diophantine sextuples. The definition can be used to extend a rational Diophantine quintuple to a weak rational Diophantine sextuple. In some cases a regular sextuple is a full rational Diophantine sextuple. Ten examples of this are provided.

**Category:** Number Theory

[585] **viXra:1609.0398 [pdf]**
*replaced on 2016-10-26 15:46:30*

**Authors:** BERKOUK Mohamed

**Comments:** 12 Pages.

Ceci est une démonstration de la conjecture de C.Goldbach émise en 1742 , aussi bien la faible que la forte , elle repose essentiellement sur le théorème fondamentale des nombres premiers , ...la démarche consiste à démontrer pour chaque pair ou impair l’existence d’au moins un couplet ou un triplet dont les éléments sont premiers qui répondent aux deux énoncés de la conjecture à savoir la Sommation et la primalité des ses éléments, ...et que plus ce nombre pair ou impair est grand , plus le nombre de couplets ou triplets premiers est grand .

**Category:** Number Theory

[584] **viXra:1609.0383 [pdf]**
*replaced on 2016-10-01 23:01:09*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on a single page; and the proof has been specialized to prove Fermat's last theorem, on half of a page. The approach used in the proof is exemplified by the following system. If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one would first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solutions for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2, will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2, identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y. One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. Two proof versions are covered. The first version begins with only the terms in the given equation, but the second version begins with the introduction of ratio terms which are subsequently and "miraculously" eliminated to allow the introduction of a much needed term for the necessary condition for c^z = a^x + b^y to have solutions or to be true. Each proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[583] **viXra:1609.0263 [pdf]**
*replaced on 2016-10-10 20:15:27*

**Authors:** A. A. Frempong

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

Honorable Pierre de Fermat could have squeezed the proof of his last theorem into a page margin. Fermat's last theorem has been proved on a single page. Three similar versions of the proof are presented, using a single page for each version. The approach used in each proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary conditions in the solution for n = 2. The necessary conditions in the solutions for n = 2. will guide one to determine if there are solutions when n > 2.. For the first two versions, the proof is based on the Pythagorean identity (sin x)^2 + (cos x)^2 = 1; and for the third version, on (a^2 + b^2)/c^2 = 1, with n = 2, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary conditions in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. For the first version of the proof, the proof began with reference to a right triangle. The second version of the proof began with ratio terms without any reference to a geometric figure. The third version occupies about half of a page. The third version of the proof began without any reference to a geometric figure or ratio terms. The second and third versions confirmed the proof in the first version. Each proof version is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a single page proof (considering the advantages) using inverse proportion, would be $107,250,000.

**Category:** Number Theory

[582] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-21 20:45:45*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple, (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system

**Category:** Number Theory

[581] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-18 23:32:38*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a primitive Pythagorean triple, (a, b, c). It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system

**Category:** Number Theory

[580] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-16 01:43:10*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 for a Pythagorean triple, a, b, c, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system.

**Category:** Number Theory

[579] **viXra:1609.0157 [pdf]**
*replaced on 2016-09-13 13:35:35*

**Authors:** A. A. Frempong

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

Beal conjecture has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^z = a^x + b^y when x, y, z > 2, one should first determine why there are solutions when x, y, z = 2, and note the necessary condition in the solution for x, y, z = 2. The necessary condition in the solutions for x, y, z = 2 will guide one to determine if there are solutions when x, y, z > 2. The proof in this paper is based on the identity (a^2 + b^2 )/c^2 = 1 where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the x, y, z = 2 identity excludes all other x, y, z-values, x, y, z > 2 from satisfying the equation c^z = a^x + b^y . One will first show that if x, y, z = 2, c^z = a^x + b^y holds, noting the necessary condition in the solution; followed by showing that if x, y, z > 2 ( x, y, z integers), c^z = a^x + b^y has no solutions. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be
made in the system

**Category:** Number Theory

[578] **viXra:1609.0080 [pdf]**
*replaced on 2016-09-16 01:53:45*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1 for a Pythagorean triple, a, b, c, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[577] **viXra:1609.0080 [pdf]**
*replaced on 2016-09-13 21:17:38*

**Authors:** A. A. Frempong

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

Fermat's last theorem has been proved on half of a page. The approach used in the proof is exemplified by the following system: If a system functions properly and one wants to determine if the same system will function properly with changes in the system, one will first determine the necessary conditions which allow the system to function properly, and then guided by the necessary conditions, one will determine if the changes will allow the system to function properly. So also, if one wants to prove that there are no solutions for the equation c^n = a^n + b^n when n > 2, one should first determine why there are solutions when n = 2, and note the necessary condition in the solution for n = 2. The necessary condition in the solutions for n = 2 will guide one to determine if there are solutions when n > 2. The proof in this paper is based on the identity (a^2 + b^2)/c^2 = 1, where a, b, and c are relatively prime positive integers. It is shown by contradiction that the uniqueness of the n = 2 identity excludes all other n-values, n > 2, from satisfying the equation c^n = a^n + b^n. One will first show that if n = 2 , c^n = a^n + b^n holds, noting the necessary condition in the solution; followed by showing that if n > 2 (n an integer), c^n = a^n + b^n does not hold. The proof began without reference to any geometric figure or ratio terms. The proof is very simple, and even high school students can learn it. The approach used in the proof has applications in science, engineering, medicine, research, business, and any properly working system when desired changes are to be made in the system. Perhaps, the proof in this paper is the proof that Fermat wished there were enough margin for it in his paper. With respect to prizes, if the prize for a 150-page proof were $715,000, then the prize for a half-page proof (considering the advantages) using inverse proportion, would be $214,500,000.

**Category:** Number Theory

[576] **viXra:1608.0429 [pdf]**
*replaced on 2016-11-18 21:18:34*

**Authors:** Gyeongmin Yang

**Comments:** 4 Pages.

This article is based on how to look for a closed-form expression related to the odd zeta function values and explained what meaning of the expansion of the Euler zigzag numbers is.

**Category:** Number Theory