[32] **viXra:1801.0416 [pdf]**
*submitted on 2018-01-30 10:29:06*

**Authors:** Timothy W. Jones

**Comments:** 3 Pages. This suggests an entirely different angle on traditional number theory.

A phenomenon is described for analytic number theory. The purpose is to coordinate number theory and to give it a specific goal of modeling the phenomenon.

**Category:** Number Theory

[31] **viXra:1801.0381 [pdf]**
*submitted on 2018-01-27 16:15:50*

**Authors:** Zeolla Gabriel martin

**Comments:** 5 Pages.

This article develops a formula for calculating the simple prime numbers-7 and the simple composite numbers-7 of the Golden Pattern.

**Category:** Number Theory

[30] **viXra:1801.0365 [pdf]**
*submitted on 2018-01-26 05:18:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present seven sequences of Poulet numbers selected by some properties of their digits product: (1) - (5) Poulet numbers for which the product of their digits is equal to (1) q^2 – 1, where q prime; (2) q^2 – 9, where q prime; (3) 9*q^2 – 9, where q prime; (4) 2^n, where n natural; (5) Q – 1, where Q is also a Poulet number and (6) – (7) Poulet numbers divisible by 5 for which the product of their digits taken without the last one is equal to (6) q^2 – 1, where q prime; (7) Q – 1, where Q is also a Poulet number. Finally, I conjecture that all these seven sequences have an infinity of terms.

**Category:** Number Theory

[29] **viXra:1801.0341 [pdf]**
*replaced on 2018-09-05 03:42:22*

**Authors:** Predrag Terzic

**Comments:** 10 Pages.

Theorems and conjectures about prime numbers .

**Category:** Number Theory

[28] **viXra:1801.0311 [pdf]**
*submitted on 2018-01-24 11:27:10*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I presented seven sequences of numbers of the form 2*k*P – (30 + 290*n)*k – 315, where P is Poulet number, and I conjectured that two of them have all the terms odd abundant numbers and the other five have an infinity of terms odd abundant numbers. Because it is known that all the smaller numbers of amicable pairs are abundant numbers (see A002025 in OEIS), in this paper I revert the relation from above and I conjecture that all Poulet numbers P divisible by 5 can be written as P = (A + 315 + (30 + 290*n)*k)/(2*k), where A is a smaller of an amicable pair and n and k naturals. For example: 645 = (12285 + 315 + 30*10)/(2*10); also 1105 = (12285 + 315 + 2060*84)/(2*84) or 1105 = (69615 + 315 + 320*37)/(2*37). Note that for the first 17 such Poulet numbers there exist at least a combination [n, k] for A = 12285, the first smaller of an amicable pair divisible by 5!

**Category:** Number Theory

[27] **viXra:1801.0310 [pdf]**
*submitted on 2018-01-23 17:13:34*

**Authors:** Jean BENICHOU

**Comments:** 1 Page. jean.benichou@icloud.com

All curves defined by x^n + y^n = z^n should intersect the circle x^2 + y^2 but are contained in it and no common point exists when x, y, z, n are integers. This contradiction forbid x^n + y^n = z^n for n>2.

**Category:** Number Theory

[26] **viXra:1801.0297 [pdf]**
*submitted on 2018-01-23 08:51:22*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present seven sequences of numbers of the form 2*k*P – (30 + 290*n)*k – 315, where P is Poulet number and n and k naturals; I conjecture that two of them have all the terms odd abundant numbers (corresponding to [P, n] = [645, 0] and [1105, 1]) and the other five (corresponding to [P, n] = [11305, 4], [16705, 13], [11305, 25], [10585, 28] and [16705, 34]) have an infinity of terms odd abundant numbers.

**Category:** Number Theory

[25] **viXra:1801.0295 [pdf]**
*submitted on 2018-01-23 09:57:45*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I present three sequences of numbers of the form (4*k + 2)*P + n*(2002*k + 1001), where P is Poulet number, k natural and n integer (corresponding to [P, n] = [41041, -1], [101101, 5] and [401401, 35]); I conjecture that they have all the terms odd abundant numbers.

**Category:** Number Theory

[24] **viXra:1801.0256 [pdf]**
*submitted on 2018-01-20 12:23:10*

**Authors:** Ilija Barukčić

**Comments:** 14 pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by: Journal of Applied Mathematics and Physics Vol.6 No.4, April 25, 2018

Abstract
Objective: Accumulating evidence indicates that zero divided by zero equal one. Still it is not clear what number theory is saying about this.
Methods: To explore relationship between the problem of the division of zero by zero and number theory, a systematic approach is used while analyzing the relationship between number theory and independence.
Result: The theorems developed in this publication support the thesis that zero divided by zero equals one. It is possible to define the law of independence under conditions of number theory.
Conclusion: The findings of this study suggest that zero divided by zero equals one.
Keywords
Zero, One, Zero divide by zero, Independence, Number theory

**Category:** Number Theory

[23] **viXra:1801.0219 [pdf]**
*submitted on 2018-01-17 16:29:46*

**Authors:** Haofeng Zhang

**Comments:** 6 Pages.

This paper proves that equation 4/(x+y+z)=1/x+1/y+1/z has no positive integer solutions
using the method of solving third order equation.

**Category:** Number Theory

[22] **viXra:1801.0193 [pdf]**
*submitted on 2018-01-16 07:00:17*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

In this note we briefly explore the Ramanujan's cubic continued fraction.

**Category:** Number Theory

[21] **viXra:1801.0190 [pdf]**
*submitted on 2018-01-16 07:44:06*

**Authors:** Ryujin Choe

**Comments:** 13 Pages.

Proof of Goldbach's conjecture and twin prime conjecture

**Category:** Number Theory

[20] **viXra:1801.0187 [pdf]**
*replaced on 2018-07-13 01:45:53*

**Authors:** Juan Moreno Borrallo

**Comments:** 18 Pages.

In this paper it is proposed and proved a conjecture of existence of a prime number on the arithmetic progression
S_{a,b}=\left\{ ab+1,ab+2,ab+3,...,ab+(b-1)\right\}
As corollaries of this proof, they are proved many classical prime number’s conjectures and theorems, but mainly Bertrand's theorem, and Oppermann's, Legendre’s, Brocard’s, and Andrica’s conjectures.
It is also defined a new maximum interval between any natural number and the nearest prime number.
Finally, it is stated a corollary which implies some advance on the conjecture of the existence of infinite prime numbers of the form n^{2}+1.

**Category:** Number Theory

[19] **viXra:1801.0182 [pdf]**
*replaced on 2018-03-17 12:52:43*

**Authors:** Haofeng Zhang

**Comments:** 30 Pages.

In this paper for equation Axm+Byn=Czk , where m,n,k > 2, x,y,z > 2, A,B,C≥1 and gcd(Ax,By,Cz)=1, the author proves there are no positive integer solutions for this equation using “Order reducing method for equations” that the author invented for solving high order equations, in which let the equation become two equations, through comparing the two roots to prove there are no positive integer solutions for this equation under the assumption of no positive integer solutions for Ax^3+By^3=Cz^3 when Ax^m-i+By^n-i>Cz^k-i.

**Category:** Number Theory

[18] **viXra:1801.0165 [pdf]**
*submitted on 2018-01-15 00:42:03*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) All numbers of the form 2*k*645 – (345 + 30*(k – 1)), where k natural, are odd abundant numbers; the sequence of these numbers is 945, 2205, 3465, 4725, 5985, 7245, 8505, 9765...(II) All numbers of the form 2*k*1905 – (345 + 30*(k – 1)), where k natural, are odd abundant numbers; the sequence of these numbers is 3465, 7245, 11025, 14805, 18585, 22365, 26145, 29925...(III) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P – (345 + 30*(k – 1)), where k natural, are odd abundant numbers.

**Category:** Number Theory

[17] **viXra:1801.0164 [pdf]**
*submitted on 2018-01-15 03:28:23*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) All numbers of the form 2*k*41041 – 1001*k, where k odd, are odd abundant numbers; the sequence of these numbers is 81081, 243243, 405405, 567567, 729729, 891891, 1054053, 1216215...(II) All numbers of the form 2*k*101101 + 5005*k, where k odd, are odd abundant numbers; the sequence of these numbers is 207207, 621621, 1036035, 1450449, 1864863, 2279277, 2693691, 3108105...(III) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P – 1001*k, where k odd, are odd abundant numbers; (IV) There exist an infinity of Poulet numbers P such that all the numbers 2*k*P + 5005*k, where k odd, are odd abundant numbers.

**Category:** Number Theory

[16] **viXra:1801.0161 [pdf]**
*submitted on 2018-01-14 05:43:59*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: there exist palindromic abundant numbers P such that n = P – q^2 + 1 is an abundant number for any q prime, q ≥ 5 (of course, for q^2 < P + 1). The first such P is the first palindromic abundant number 66 (with corresponding [q, n] = [5, 42], [7, 18]. Another such palindromic abundant numbers are 222, 252, 282, 414, 444, 474, 606, 636, 666. Up to 666, the palindromic abundant numbers 88, 272, 464, 616 don’t have this property. Questions: are there infinite many such palindromic abundant numbers? What other sets of integers have this property beside palindromic abundant numbers?

**Category:** Number Theory

[15] **viXra:1801.0153 [pdf]**
*submitted on 2018-01-14 03:08:57*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: there exist Poulet numbers P such that n = P – q^2 is an abundant number for any q prime, q ≥ 5 (of course, for q^2 < P). The first such P is 1105 (with corresponding [q, n] = [5, 1080], [7, 1056], [11, 984], [13, 936], [17, 816], [19, 744], [23, 576], [29, 264], [31, 144]). Another such Poulet numbers are 1387, 1729, 2047, 2701, 2821. Up to 2821, the Poulet numbers 341, 561, 645, 1905, 2465 don’t have this property. Questions: are there infinite many such Poulet numbers? What other sets of integers have this property beside Poulet numbers?

**Category:** Number Theory

[14] **viXra:1801.0151 [pdf]**
*submitted on 2018-01-13 08:02:40*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of Poulet numbers P such that D + R(D), where R(D) is the number obtained reversing the digits of D which is the number obtained concatenating the prime factors of P, is a palindromic number (example: such a Poulet number is P = 12801; the prime factors of 12801 are 3, 17 and 251, then D = 317251 and D + R(D) = 317251 + 152713 = 469964, a palindromic number); (II) There is no a number obtained concatenating the prime factors of a Poulet number to be a Lychrel number.

**Category:** Number Theory

[13] **viXra:1801.0140 [pdf]**
*replaced on 2018-02-13 07:03:38*

**Authors:** Timothy W. Jones

**Comments:** 5 Pages. Proofs are added for two results used in earlier drafts.

We prove that partial sums of zeta(n>=2) are not given by any single decimal in a number base given by a denominator of their terms. This result, applied to all partials, shows that partials are excluded from an ever greater number of rational values. The limit of the partials is zeta(n) and the limit of the exclusions leaves only irrational numbers.

**Category:** Number Theory

[12] **viXra:1801.0138 [pdf]**
*submitted on 2018-01-12 11:09:28*

**Authors:** Preininger Helmut

**Comments:** 11 Pages.

This paper is an appendix of Natural Squarefree Numbers: Statistical Properties [PR04]. In this appendix we calculate the probability of c is squarefree, where c=a*b, a is an element of the set X and b is an element of the set Y.

**Category:** Number Theory

[11] **viXra:1801.0118 [pdf]**
*submitted on 2018-01-10 11:13:58*

**Authors:** MENDZINA ESSOMBA François

**Comments:** 03 Pages.

a new continuous fractions...

**Category:** Number Theory

[10] **viXra:1801.0093 [pdf]**
*submitted on 2018-01-08 09:25:55*

**Authors:** Zeolla Gabriel Martin

**Comments:** 10 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three, composite numbers and all twin prime numbers greater than three. The conditioning (n) will be the key to make the formula work.

**Category:** Number Theory

[9] **viXra:1801.0087 [pdf]**
*submitted on 2018-01-07 17:15:09*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: the number n = p – q, where p and q are Poulet numbers, needs very few iterations of “reverse and add” to reach a palindrome. For instance, taking q = 1729 and p = 999986341201, it can be seen that only 3 iterations are needed to reach a palindrome: n = 999986341201 – 1729 = 999986339472 and we have: 999986339472 + 274933689999 = 1274920029471; 1274920029471 + 1749200294721 = 3024120324192 and 3024120324192 + 2914230214203 = 5938350538395, a palindromic number. So, relying on this, I conjecture that there exist an infinity of n, even considering q and p successive, that need just one such iteration to reach a palindrome (see sequence A015976 in OEIS for these numbers) and I also conjecture that there is no a difference between two Poulet numbers to be a Lychrel number.

**Category:** Number Theory

[8] **viXra:1801.0082 [pdf]**
*submitted on 2018-01-08 02:20:51*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: the number n = p^2 – q^2, where p and q are primes, needs very few iterations of “reverse and add” to reach a palindrome. For instance, taking q = 563 and p = 104723, it can be seen that only 3 iterations are needed to reach a palindrome: n = 104723^2 – 563^6 = 10966589760 and we have: 10966589760 + 6798566901 = 17765156661; 17765156661 + 16665156771 = 34430313432 and 34430313432 + 23431303443 = 57861616875, a palindromic number. So, relying on this, I conjecture that there exist an infinity of n, even considering q and p successive, that need just one such iteration to reach a palindrome (see sequence A015976 in OEIS for these numbers) and I also conjecture that there is no a difference between two squares of primes to be a Lychrel number.

**Category:** Number Theory

[7] **viXra:1801.0080 [pdf]**
*submitted on 2018-01-07 05:12:33*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of Poulet numbers P such that (P + 4*196) + R(P + 4*196), where R(n) is the number obtained reversing the digits of n, is a palindromic number; note that I wrote 4*196 instead 784 because 196 is a number known to be related with palindromes: is the first Lychrel number, which gives the name to the “196-algorithm”; (II) For every Poulet number P there exist an infinity of primes q such that the number (P + 16*q^2) + R(P + 16*q^2) is a palindrome. The three sequences (presumed infinite by the conjectures above) mentioned in title of the paper are: (1) Palindromes of the form (P + 4*196) + R(P + 4*196), where P is a Poulet number; (2) Palindromes of the form (P + 16*q^2) + R(P + 16*q^2), where P is a Poulet number and q the least prime for which is obtained such a palindrome; (3) Palindromes of the form (1729 + 16*q^2) + R(1729 + 16*q^2), where q is prime (1729 is a well known Poulet number).

**Category:** Number Theory

[6] **viXra:1801.0078 [pdf]**
*submitted on 2018-01-07 07:13:35*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of squares of primes p^2 such that (p^2 + 4*196) + R(p^2 + 4*196), where R(n) is the number obtained reversing the digits of n, is a palindromic number; note that I wrote 4*196 instead 784 because 196 is a number known to be related with palindromes: is the first Lychrel number, which gives the name to the “196-algorithm”; (II) For every square of odd prime p^2 there exist an infinity of primes q such that the number (p^2 + 16*q^2) + R(p^2 + 16*q^2) is a palindrome. The three sequences (presumed infinite by the conjectures above) mentioned in title of the paper are: (1) Palindromes of the form (p^2 + 4*196) + R(p^2 + 4*196), where p^2 is a square of prime; (2) Palindromes of the form (p^2 + 16*q^2) + R(p^2 + 16*q^2), where p^2 is a square of prime and q the least prime for which is obtained such a palindrome; (3) Palindromes of the form (13^2 + 16*q^2) + R(13^2 + 16*q^2), where q is prime.

**Category:** Number Theory

[5] **viXra:1801.0070 [pdf]**
*submitted on 2018-01-06 07:15:45*

**Authors:** Timothy W. Jones

**Comments:** 2 Pages. It might help to read "Visualizing Zeta(n>1) and Proving Its Irrationality" by the same author.

In a universe with meteorites on concentric circles equally spaced, spaceships can avoid collisions by every smaller increments of their trajectories. Using this idea, a story conveys the sense that Zeta increments avoid all meteorites and thus converge to an irrational number.

**Category:** Number Theory

[4] **viXra:1801.0068 [pdf]**
*replaced on 2018-03-19 09:17:05*

**Authors:** Haofeng Zhang

**Comments:** 23 Pages.

In this paper the author gives a simplest elementary mathematics method to solve the famous Fermat's Last Theorem (FLT), in which let this equation become a one unknown number equation, in order to solve this equation the author invented a method called "Order reducing method for equations" where the second order root compares to one order root and with some necessary techniques the author successfully proved Fermat's Last Theorem.

**Category:** Number Theory

[3] **viXra:1801.0065 [pdf]**
*submitted on 2018-01-05 06:35:44*

**Authors:** Zeolla Gabriel martin

**Comments:** 27 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three and composite numbers divisible by numbers greater than three. This paper develops formulas to break down the prime numbers and the composite numbers in their reductions, these formulas based on equalities allow to regroup them according to congruence characteristics.

**Category:** Number Theory

[2] **viXra:1801.0064 [pdf]**
*submitted on 2018-01-05 06:38:16*

**Authors:** Zeolla Gabriel martin

**Comments:** 7 Pages.

This paper develops the divisibility of the so-called Simple Primes numbers (1 to 9), the discovery of a pattern to infinity, the demonstration of the Inharmonics that are 2,3,5,7 and the harmony of 1. The discovery of infinite harmony represented in fractal numbers and patterns. This is a family before the prime numbers.

**Category:** Number Theory

[1] **viXra:1801.0063 [pdf]**
*submitted on 2018-01-05 06:43:09*

**Authors:** Zeolla Gabriel martin

**Comments:** 8 Pages.

This paper develops a modified an old and well-known expression for calculating and obtaining all prime numbers greater than three and composite numbers divisible by numbers greater than 3. The key for this formula to work correctly is in the equalities and inequalities. These equalities and inequalities are created from the uncovering of the patterns of the composite numbers. The composite numbers follow very clear and determining patterns, making it possible to find them through a formula.

**Category:** Number Theory