Number Theory

1801 Submissions

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

The Positive Integer Solutions of Equation 4/(x+y+z)=1/x+1/y+1/z

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

[26] viXra:1801.0217 [pdf] replaced on 2018-01-18 09:49:18

A Ridiculously-Simple Direct Proof of FLT

Authors: Philip Aaron Bloom
Comments: 1 Page.

For any given positive integral value of n, we devise an algebraic identity that holds for positive integral Fermat triples equal to (z,y,x) for which z^n-y^n=x^n holds. This identity does not hold for positive integral Fermat triples when the value of n is greater or equal to three. Consequently, our argument constitutes a direct proof that z^n-y^n=x^n does not hold for positive integral values of (z,y,x) when the value of n is greater or equal to three.
Category: Number Theory

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

Ramanujan's Cubic Continued Fraction , Experimental Mathematics , Maple Experiment , Identify Command

Authors: Edgar Valdebenito
Comments: 2 Pages.

In this note we briefly explore the Ramanujan's cubic continued fraction.
Category: Number Theory

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

Theorem for Distribution of Prime Pairs

Authors: Ryujin Choe
Comments: 13 Pages.

Proof of Goldbach's conjecture and twin prime conjecture
Category: Number Theory

[23] viXra:1801.0187 [pdf] submitted on 2018-01-16 14:33:02

On the Existence of Prime Numbers in Arithmetic Progressions

Authors: Juan Moreno Borrallo
Comments: 6 Pages.

In this paper it is proposed a conjecture of existence of prime numbers on a particular arithmetic progression, and demonstrated a particular case.
Category: Number Theory

[22] viXra:1801.0182 [pdf] submitted on 2018-01-17 06:15:54

The Positive Integer Solutions of Equation Ax^m+By^n=Cz^k

Authors: Haofeng Zhang
Comments: 30 Pages.

In this paper for equation Ax^m+By^n=Cz^k , where m,n,k > 2, x,y,z > 1, A,B,C≥1 and gcd(Ax,By,Cz)=1, the author proved 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.
Category: Number Theory

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

Odd Abundant Numbers of the Form 2∙k∙P-(345+30∙(k-1)) Where P Are Poulet Numbers

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

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

Odd Abundant Numbers of the Forms 2∙k∙P-1001∙k and 2∙k∙P+5005∙k Where P Are Poulet Numbers

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

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

Palindromic Abundant Numbers P for Which P-Q^2+1 is an Abundant Number for Any Q Prime Greater Than 3

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

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

Poulet Numbers P for Which P-Q^2 is an Abundant Number for Any Q Prime Greater Than 3

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

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

Palindromes Obtained Concatenating the Prime Factors of a Poulet Number and Adding to the Number Obtained Its Reversal

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

[16] viXra:1801.0140 [pdf] replaced on 2018-01-18 15:58:09

A Simple Proof that Zeta(2) is Irrational

Authors: Timothy W. Jones
Comments: 6 Pages. One typo corrected.

We prove that partial sums of zeta(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(2) and the limit of the exclusions leaves only irrational numbers.
Category: Number Theory

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

Natural Squarefree Numbers: Statistical Properties II

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

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

François Mendzina Essomba Continuous Fraction's

Authors: MENDZINA ESSOMBA François
Comments: 03 Pages.

a new continuous fractions...
Category: Number Theory

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

Expression to Get Prime Numbers and Twin Prime Numbers.

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

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

Number P-Q Where P and Q Poulet Numbers Needs Very Few Iterations of “reverse and Add” to Reach a Palindrome

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

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

Number P^2-Q^2 Where P and Q Primes Needs Very Few Iterations of “reverse and Add” to Reach a Palindrome

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

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

Three Sequences of Palindromes Obtained from Poulet Numbers

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

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

Three Sequences of Palindromes Obtained from Squares of Primes

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

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

The Irrationality of Zeta(n>1): A Proof by Story

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

[7] viXra:1801.0068 [pdf] replaced on 2018-01-09 10:47:39

The Simplest Elementary Mathematics Proving Method of Fermat's Last Theorem

Authors: Haofeng Zhang
Comments: 18 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

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

Prime Numbers and Composite Numbers Congruent to 1,4,7,2,5,8 (Mod 9)

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

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

Golden Pattern

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

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

Formula for Prime Numbers and Composite Numbers.

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

[3] viXra:1801.0052 [pdf] replaced on 2018-01-18 09:56:09

A Trivially Simple Proof of Fermat's Last Theorem

Authors: Philip A. Bloom
Comments: 1 Page.

For any given positive integral n, we devise an algebraic identity that we show as holding for positive integral Fermat triples equal to (x,y,z) for which x^n+y^n=z^n holds (per our assumption). However, our algebraic identity does not hold for values of n greater or equal to three, thereby contradicting this assumption. Consequently, no positive integral (x,y,z) exists when n has values greater or equal to three.
Category: Number Theory

[2] viXra:1801.0006 [pdf] submitted on 2018-01-01 20:57:30

Reversibility in Number Theory

Authors: A. Polorovskii
Comments: 6 Pages.

Let |l| ⊂ ℵ0. In [19], the authors extended manifolds. We show that F ≤ M. In this context, the results of [10] are highly relevant. It is not yet known whether there exists a Fourier additive polytope, although [10] does address the issue of uniqueness.
Category: Number Theory

[1] viXra:1801.0001 [pdf] replaced on 2018-01-02 13:45:30

Positivity of li Coefficients for N>10^24

Authors: Leonhard Schuster
Comments: 13 Pages.

In this paper, we prove the positivity of Li coefficients for n>10^24. We investigate the Riemann Zeta function, in the form (s-1)zeta(s), under the transformation s = 1/(1-z). We apply a generalised Poisson-Jensen formula to show that Riemann Zeta function has only a finite number of zeros not lying the critical line, and that the Li coefficients are positive for n>10^24. This implicitly proves the validity of Riemann Hypothesis.
Category: Number Theory