Number Theory

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2018 - 1801(21)

Recent submissions

Any replacements are listed further down

[1679] 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

[1678] 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

[1677] 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

[1676] 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

[1675] 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

[1674] viXra:1801.0140 [pdf] submitted on 2018-01-12 09:11:07

A Simple Proof that Zeta(2) is Irrational

Authors: Timothy W. Jones
Comments: 8 Pages. This does generalize to all zeta(n>1), but the necessary inequality for the general case is difficult.

We prove that a partial sum of zeta(2)-1=z is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that there are an infinite number of partial sums in one interval of the form [.(x-1),.x] where .x is single decimal in a number base of the denominators of the terms of z. We show that z is contained in an open interval inside [.(x-1),.x]. As all possible rational values of z are in these intervals, z must be irrational.
Category: Number Theory

[1673] 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

[1672] 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

[1671] 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

[1670] 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

[1669] 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

[1668] 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

[1667] 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

[1666] 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

[1665] viXra:1801.0068 [pdf] submitted on 2018-01-06 09:26:17

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

Authors: Haofeng Zhang
Comments: 13 Pages.

Abstract : 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 su ccessfully proved that there are no positive integer solutions for x^n+y^n=z^n which means FLT has been proved by elementary mathematics.
Category: Number Theory

[1664] 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

[1663] 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

[1662] 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

[1661] viXra:1801.0052 [pdf] submitted on 2018-01-05 21:55:43

A Trivially Simple Proof of Fermat's Last Theorem

Authors: Philip A. Bloom
Comments: Pages.

We formulate an algebraic identity that has positive integral Fermat triples equal to (x,y,z) of x^n+y^n=z^n. We assume that x^n+y^n=z^n has positive integral Fermat triples for n greater or equal to three, and we derive a contradiction. Hence, for any given n greater or equal to three, there is a null set of positive integral (x,y,z).
Category: Number Theory

[1660] 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

[1659] viXra:1801.0001 [pdf] submitted on 2018-01-01 12:02:19

Positivity of li Coefficients for N > 10^24

Authors: Leonhard Schuster
Comments: 13 Pages.

We investigate Riemann's Zeta function, as $(s-1)\zeta(s)$, under the M{\"o}bius transformation $s = \frac{1}{1-z}$ which maps the half plane right to the critical strip to the unit disk. Application of a generalized Poisson-Jensen formula (due to Nevanlinna) shows that the investigated function has only a finite number of zeros in the interior of the unit disk. We show, that the Li coefficients $\lambda_n = \sum_\rho (1-(1-1/\rho)^n)$ are positive for $n> 10^{24}$, and discuss consequences.
Category: Number Theory

[1658] viXra:1712.0679 [pdf] submitted on 2017-12-31 06:29:29

A New Binomial Formula for the Sum of Two Powers

Authors: Julian Beauchamp
Comments: 1 Page.

In this paper, we reveal a new binomial formula that expresses the sum of, or difference between two powers, a^x \pm b^y, as a binomial expansion of a single power, z. Like the standard binomial formula it includes the normal binomial coefficients, factors and indices, but includes an additional non-standard factor. The new formula (with an upper index z) mimics a standard binomial formula (to the power z) without the value of the binomial expansion changing even when z itself changes. This has exciting implications for certain diophantine equations. This short paper simply highlights its existence.
Category: Number Theory

[1657] viXra:1712.0669 [pdf] submitted on 2017-12-30 17:43:18

Taken ABaCk by Conjecturing Out-of-Box

Authors: Arthur Shevenyonov
Comments: 17 Pages. ABC conjecture

ABC conjecture and beyond, with cross-operational linkage hinting at broader convergence
Category: Number Theory

[1656] viXra:1712.0662 [pdf] submitted on 2017-12-29 15:51:45

The Chameleon Effect, the Binomial Theorem and Beal's Conjecture

Authors: Julian Beauchamp
Comments: 9 Pages.

In psychology, the Chameleon Effect describes how an animal's behaviour can adapt to, or mimic, its environment through non-conscious mimicry. In the first part of this paper, we show how $a^x - b^y$ can be expressed as a binomial expansion (with an upper index, $z$) that, like a chameleon, mimics a standard binomial formula (to the power $z$) without its own value changing even when $z$ itself changes. In the second part we will show how this leads to a proof for the Beal Conjecture. We finish by outlining how this method can be applied to a more generalised form of the equation.
Category: Number Theory

[1655] viXra:1712.0660 [pdf] submitted on 2017-12-29 05:27:48

The Last Theorem of Fermat. Correct Proof

Authors: Victor Sorokine
Comments: 3 Pages.

The proof is based on studying digits in the endings of different numbers in Fermat's equation.
Category: Number Theory

[1654] viXra:1712.0656 [pdf] submitted on 2017-12-29 08:47:39

Classify Positive Integers to Prove Collatz Conjecture by Mathematical Induction (Revised Version)

Authors: Zhang Tianshu
Comments: 24 Pages.

Positive integers which are able to be operated to 1 by the leftwards operational rule and generating positive integers which start with 1 to operate by the rightwards operational rule are one-to-one correspondence and the same. So, we refer to the bunch of integers’ chains to apply the mathematical induction, next classify positive integers to get comparable results via operations, such that finally summarize out a proof at substep according to beforehand prepared two theorems as judgmental criteria.
Category: Number Theory

[1653] viXra:1712.0653 [pdf] submitted on 2017-12-29 12:07:03

The Last Theorem of Fermat. Correct Proof (Russian)

Authors: Victor Sorokine
Comments: 2 Pages. Russian version

We study the digits at the end of different numbers in the Fermat's Equality, and arrive to a contradiction.
Category: Number Theory

[1652] viXra:1712.0641 [pdf] submitted on 2017-12-28 10:19:41

Alternate Proof for Zeta(n>1) is Irrational

Authors: Timothy W. Jones
Comments: 2 Pages. You may need to read Visualizing Zeta(n>1) and Proving its Irrationality by the same author.

This is an alternative proof that zeta(n>1) is irrational. It uses nested intervals and Cantor's Nested Interval Theorem. It is a follow up for the article Visualizing Zeta(n>1) and Proving its Irrationality.
Category: Number Theory

[1651] viXra:1712.0588 [pdf] submitted on 2017-12-24 11:13:39

CV For A Self Taught Math & Science Problem Solver and or Fixxer

Authors: Ricardo Gil
Comments: 5 Pages. I will attempt any problem in any subject, just provide some background.

The objective of this paper is to show people that I am now for hire($). While many scientist and mathematicians are bound by the laws of nature and physics,I am able to look beyond the laws of nature and physics and come up with solutions for virtually every problem(See my papers).While my degrees are in education I have had a hobby of submitting unsolicited solutions to the CIA for the last 20 years for free. Whether they use the solutions of not is not relevant or if they deny it or confirm it. What is relevant is that if you have a problem and are willing to pay for a solution via paypal Im willing to solve it. Submit it to Ricardo.gil@sbcglobal.net. I ask for a fair price for a viable solution. "& Ye shall know the truth and any project is achievable in 18 mos or less."
Category: Number Theory

[1650] viXra:1712.0572 [pdf] submitted on 2017-12-22 12:57:33

An Introduction to Multi-Dimensional Identity

Authors: J. Mitchell
Comments: 35 Pages.

Multi-dimensional identity refers to the many labels which describe any ‘Thing’ as it exists, meaning both its describable states of existence and whatever processes generate, connect and count across those states. It develops through and out of a base2 pattern that becomes a multi-layered function which generates, relates and counts base10 numbers.
Category: Number Theory

[1649] viXra:1712.0565 [pdf] submitted on 2017-12-23 01:17:56

Triviality of Twin Prime Conjecture

Authors: Divyendu Priyadarshi
Comments: 1 Page. i am not a professional mathematician, so if there is some silly mistake or misconceptions , please point out.

In this small paper, I have argued very simply that "Twin Prime Conjecture" is quite obvious and there is nothing to prove. In fact, it reduces to the hypothesis that prime numbers are infinite in number if we accept the quite random pattern of occurrences of of prime numbers on number line.
Category: Number Theory

[1648] viXra:1712.0561 [pdf] submitted on 2017-12-23 04:16:14

A Short Disproof of the Riemann Hypothesis

Authors: Igor Hrnčić
Comments: 5 Pages.

This paper disproves the Riemann hypothesis by analyzing the integral representation of the Riemann zeta function that converges absolutely in the root-free region.
Category: Number Theory

[1647] viXra:1712.0554 [pdf] submitted on 2017-12-21 12:46:22

Conjecture that there is no a Square of an Odd Number to be as Well Lychrel Number

Authors: Marius Coman
Comments: 3 Pages.

In this paper I make the following two conjectures: (I) There exist an infinity of squares of odd numbers n^2 such that n^2 + R(n^2), where R(n^2) is the number obtained reversing the digits of n^2, is a palindromic number; (II) There is no a square of an odd number to be as well Lychrel number. Note that a Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers (process sometimes called the 196-algorithm, 196 being the smallest such number) – see the sequence A023108 in OEIS.
Category: Number Theory

[1646] viXra:1712.0543 [pdf] submitted on 2017-12-21 08:23:56

Conjecture that there is no a Poulet Number to be as Well Lychrel Number

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 P + R(P), where R(P) is the number obtained reversing the digits of P, is a palindromic number; (II) There is no a Poulet number to be as well Lychrel number. Note that a Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers (process sometimes called the 196-algorithm, 196 being the smallest such number) – see the sequence A023108 in OEIS.
Category: Number Theory

[1645] viXra:1712.0532 [pdf] submitted on 2017-12-20 16:05:37

Dark Matter or Antimatter a la Cold

Authors: Ricardo Gil
Comments: 1 Page. So let it be written so let it be done by CERN or any other MADD Scientist Club !!!

The objective of this paper is to suggest that a photon can be cooled to -273 Kelvin and the photon can be slowed down around 10 m/s**2 and then a positive charge can be added to the minimally charged electron of the photon to create a positron which would be dark matter or antimatter, No Que No Carnal???
Category: Number Theory

[1644] viXra:1712.0488 [pdf] submitted on 2017-12-17 05:24:36

Brief Solutions to Collatz Problem, Goldbach Conjecture and Twin Primes

Authors: Mesut Kavak
Comments: 5 Pages.

I published some solutions a time ago to Goldbach Conjecture, Collatz Problem and Twin Primes; but I noticed that there were some serious logic voids to explain the problems. After that I made some corrections in my another article; but still there were some mistakes. Even so, I can say it easily that here I brought exact solutions for them out by new methods back to the drawing board.
Category: Number Theory

[1643] viXra:1712.0483 [pdf] submitted on 2017-12-16 19:45:21

Electromagnetic Mass Reduction Repulsion Boots

Authors: Ricardo Gil
Comments: 2 Pages. This paper is about Human performance, mass reduction through repulsion and Mathematical Physics.

The purpose of this paper is to show how these electronic mass reduction repulsion boot(similar to magnetic field disruption TR3B (https://www.youtube.com/watch?v=au4hbUm4mMo) could be used on the Talos to reduce the weight of the suit.(https://www.youtube.com/watch?v=pFmFl5eE8vc).
Category: Number Theory

[1642] viXra:1712.0482 [pdf] submitted on 2017-12-17 02:19:30

Blue Rib Bridge or Hiway

Authors: Ricardo Gil
Comments: 1 Page. To make a Tesla car, put one or more car batteries in the front seat and run a jumper cable to the cigarette lighter.

The purpose of this paper is to point out the Blue Rib Bridge or Highway A.K.A the Pinn Oak or Hausmann Stargate.
Category: Number Theory

[1641] viXra:1712.0458 [pdf] submitted on 2017-12-14 09:17:53

Primes Obtained Concatenating 9p-12 with P^2 Where P Prime or Poulet Number

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following two conjectures: (1) There exist an infinity of primes obtained concatenating 9*p – 12 with p^2 where p is a prime (for example, such a prime is 208554289 obtained concatenating 9*233 – 12 = 2085 with 233^2 = 54289); (2) There exist an infinity of primes obtained concatenating 9*p – 12 with p^2 where p is a Poulet number (for example, such a prime is 155492989441 obtained concatenating 9*1729 – 12 = 15549 with 1729^2 = 2989441).
Category: Number Theory

[1640] viXra:1712.0457 [pdf] submitted on 2017-12-14 09:19:56

Primes Obtained Concatenating 2n+4 with 2n+4 Then with N Where N=3p and P Prime

Authors: Marius Coman
Comments: 1 Page.

In this paper I make the following conjecture: There exist an infinity of primes obtained concatenating 2*n + 4 with 2*n + 4 then with n where n = 3*p and p is a prime; for example, such primes are 19019093 obtained concatenating 190 = 2*(3*31) + 4 with 190 then with 93 = 3*31 or 12701270633 obtained concatenating 1270 = 2*(3*211) + 4 with 1270 then with 633 = 3*211. Note that for twenty-five from the first eighty primes p are obtained primes with this method.
Category: Number Theory

[1639] viXra:1712.0452 [pdf] submitted on 2017-12-15 03:07:11

Primes of the Form 2^a∙2^b∙2^c + D Where a, b, c, D of the Form 6k-1

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: For any a, b, c distinct numbers of the form 6*k – 1 there exist an infinity of numbers d of the form 6*h – 1 such that the number n = 2^a*2^b*2^c + d is prime. This is a formula that conducts often to primes and composites with very few prime factors; for instance, taking a = 5 and b = 11 are obtained seventeen primes for c and d both less than 100 (for c = 17, n is prime for six values of d up to 100: 17, 29, 35, 59, 71, 77)! Also note that for [a, b, c, d] = [59, 65, 71, 53] (all four less than or equal to 71) is obtained a prime with 59 digits!
Category: Number Theory

[1638] viXra:1712.0451 [pdf] submitted on 2017-12-15 03:09:04

Primes of the Form 2^a∙2^b∙2^c D Where a, b, c, D of the Form 6k+1

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: For any a, b, c distinct numbers of the form 6*k + 1 there exist an infinity of numbers d of the form 6*h + 1 such that the number n = 2^a*2^b*2^c - d is prime. This is a formula that conducts often to primes and composites with very few prime factors; for instance, taking a = 7 and b = 13 are obtained eighteen primes for c and d both less than 100 (for c = 19, n is prime for four values of d up to 100: 7, 19, 67, 91)! Also note that for [a, b, c, d] = [49, 55, 61, 61] (all four less than or equal to 61) is obtained a prime with 50 digits!
Category: Number Theory

[1637] viXra:1712.0441 [pdf] submitted on 2017-12-13 11:00:55

Natural Squarefree Numbers: Statistical Properties.

Authors: Helmut Preininger
Comments: 42 Pages.

n this paper we calculate for various sets X (some subsets of the natural numbers) the probability of an element a of X is also squarefree. Furthermore 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

[1636] viXra:1712.0434 [pdf] submitted on 2017-12-13 15:27:41

About the Conjecture of Syracuse

Authors: Antoine Balan
Comments: 2 pages, written in french

We show that the problem of Syracuse is a problem of complex analysis (Analytical Number Theory).
Category: Number Theory

[1635] viXra:1712.0421 [pdf] submitted on 2017-12-12 09:31:48

3400 Pin Oak Rd

Authors: Ricardo Gil
Comments: 2 Pages. Don't worry folks I have been there on foot and in a car. Remain Calm.Have fun with it.

The purpose of this paper is to explain a Stargate or Temporal anomaly on Pin Oak Road
Category: Number Theory

[1634] viXra:1712.0397 [pdf] submitted on 2017-12-10 06:30:48

Pie in Python_Piethon

Authors: Ricardo Gil
Comments: 2 Pages. If your computer can handle it the sky's the limit with regards to digits.

The objective of this paper is to provide everyone with a program in Piethon to be able to print 250,000 digits and if your computer allow to be able to print > 299792458 digits.
Category: Number Theory

[1633] viXra:1712.0396 [pdf] submitted on 2017-12-10 10:16:41

Frequency Topology of Encryption

Authors: Ricardo Gil
Comments: 3 Pages. Mosses equals 500 in the Torah.

The objective of this paper is simplify frequency topology of encryption and lininear Graphs that can be represent in dimension 2 or greater.
Category: Number Theory

[1632] viXra:1712.0384 [pdf] submitted on 2017-12-10 12:27:40

Discovering and Proving that Pi is Irrational, 2nd Edition

Authors: Timothy W. Jones
Comments: 11 Pages. This fixes a number of typos and adds two appendices to the 2010 article published by the MAA's Monthly.

Ivan Niven's proof of the irrationality of pi is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of pi. This simplifying concept yields a more motivated proof of the irrationality of pi and pi squared.
Category: Number Theory

[1631] viXra:1712.0366 [pdf] submitted on 2017-12-10 01:30:48

The Goldbach Conjecture

Authors: Barry Foster
Comments: 1 Page.

This treatment uses two simple facts and seems to confirm the Conjecture without providing an obvious method for discovering GP primes.
Category: Number Theory

[1630] viXra:1712.0359 [pdf] submitted on 2017-12-08 08:14:04

千古奇冤,素数有限

Authors: Liu Ran
Comments: 3 Pages.

素数个数有限,并且找出欧几里德证明的瑕疵,并举出证据
Category: Number Theory

[1629] viXra:1712.0353 [pdf] submitted on 2017-12-09 04:18:48

Goldbach Conjecture Proof

Authors: Bado idriss olivier
Comments: 6 Pages.

In this paper, we are going to give the proof of the Goldbach conjecture by introducing the lemma which implies Goldbach conjecture. first of all we are going to prove that the lemma implies Goldbach conjecture and in the following we are going to prove the validity of the lemma by using Chébotarev-Artin theorem's, Mertens formula and the Principle of inclusion - exclusion of Moivre
Category: Number Theory

[1628] viXra:1712.0352 [pdf] submitted on 2017-12-09 04:21:53

Legendre Conjecture

Authors: Bado idriss olivier
Comments: 5 Pages.

In this paper, we are going to give the proof of legendre conjecture by using the Chebotarev -Artin 's theorem ,Dirichlet arithmetic theorem and the principle inclusion-exclusion of Moivre
Category: Number Theory

[1627] viXra:1712.0342 [pdf] submitted on 2017-12-07 19:37:22

Disjunctive Sequence are Rare

Authors: F.L.B.Périat
Comments: 2 Pages.

Voici la démonstration que les nombres univers sont infiniment rare.
Category: Number Theory

[1626] viXra:1712.0202 [pdf] submitted on 2017-12-06 19:30:08

Conjecture de Brocard et Nouvelle Conjecture

Authors: Réjean Labrie
Comments: 1 Page.

542 Place Macquet
Category: Number Theory

[1625] viXra:1712.0098 [pdf] submitted on 2017-12-05 06:02:16

Proof of Goldbach's Conjecture and Twin Prime Conjecture

Authors: Choe Ryujin
Comments: 6 Pages.

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

[1624] viXra:1712.0073 [pdf] submitted on 2017-12-03 17:31:02

The Goldbach's Theorem

Authors: Leszek W. Guła
Comments: 1 Page.

The Goldbach's Theorem
Category: Number Theory

[1623] viXra:1711.0417 [pdf] submitted on 2017-11-25 10:19:09

素数分布无规律

Authors: Liu Ran
Comments: 2 Pages.

A new way to prove prime number distribution being no law.
Category: Number Theory

[1622] viXra:1711.0353 [pdf] submitted on 2017-11-19 03:41:41

Conjecture that Any Square of a Prime P^2 Can be Written as P+q+(nq-N±1) Where Q and nq-N±1 Primes

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: Any square of a prime p^2, where p > 3, can be written as p + q + (n*q – n + 1) or as p + q + (n*q - n – 1), where q and n*q – n + 1 respectively n*q - n – 1 are primes and n positive integer. Examples: 11^2 = 121 = 11 + 37 + (2*37 – 1), where 37 and 2*37 – 1 = 73 are primes; 13^2 = 169 = 13 + 53 + (2*53 – 3), where 53 and 2*53 – 3 = 103 are primes. An equivalent formulation of the conjecture is that for any prime p, p > 3, there exist n positive integer such that one of the numbers q = (p^2 – p + n – 1)/(n + 1) or q = p^2 – p + n + 1)/(n + 1) is prime satisfying also the condition that p^2 – p – q is prime.
Category: Number Theory

[1621] viXra:1711.0343 [pdf] submitted on 2017-11-18 03:29:59

A Method of Obtaining Large Primes Based on Carmichael Numbers

Authors: Marius Coman
Comments: 2 Pages.

Playing with Carmichael numbers, a set of numbers I’ve always been fond of (I’ve “discovered” Fermat’s “Little” Theorem and the first few Carmichael numbers before I know they had already been discovered), I noticed that the formula C + 81*2^(4*d), where C is a Carmichael number and d one of its prime factors, gives often primes or products of very few primes. For instance, for C = 1493812621027441 are obtained in this manner three primes: 2918779690625137, 6729216728661136606577017055290271857 and 644530914387083488233375393598279808770191171433362641802841314053534708129737067311868017 (a 90-digit prime!), respectively for d = 11, d = 29 and d = 73.
Category: Number Theory

[1620] viXra:1711.0330 [pdf] submitted on 2017-11-17 01:34:01

A Set of Poulet Numbers Defined by an Interesting Relation Between Their Prime Factors

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1 and dn be the least respectively the largest prime factors of the number P1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form P1 + |P1 – dn^2|*d1, where |P1 – dn^2| is the absolute value of P1 – dn^2. Example: for Poulet number P1 = 1729 = 7*13*19 is obtained through this operation Poulet number P2 = 11305 (1729 – 19^2 = 1368 and 1729 + 1368*7 = 11305). Note that from 11 from the first 30 Poulet numbers (P1) were obtained through this method Poulet numbers (P2).
Category: Number Theory

[1619] viXra:1711.0307 [pdf] submitted on 2017-11-14 06:41:14

Question 405 : pi and G

Authors: Edgar Valdebenito
Comments: 3 Pages.

This note presents some formulas involving pi and G (Catalan constant).
Category: Number Theory

[1618] viXra:1711.0303 [pdf] submitted on 2017-11-14 06:51:50

Question 416 : Pi , Integral Representations

Authors: Edgar Valdebenito
Comments: 3 Pages.

This note presents some elementary integrals for pi.
Category: Number Theory

[1617] viXra:1711.0296 [pdf] submitted on 2017-11-14 04:44:03

A Brief Proof of the Collatz Conjecture

Authors: Kurmet Sultan
Comments: 9 Pages.

Earlier in [http://vixra.org/abs/1708.0177] the author presented a proof of the Collatz conjecture, based on the regularities of numbers of the form 6n ± 1, formed as a result of calculating the Collatz function. In [http://vixra.org/abs/1708.0177] there were many tables, figures, definitions, examples and explanations, which created difficulties in the perception of the material. Taking this into account, in this paper we give a shortened version of the proof of the Collatz conjecture.
Category: Number Theory

[1616] viXra:1711.0291 [pdf] submitted on 2017-11-12 10:24:36

The Irrationality of Trigonometric and Hyperbolic Functions

Authors: Timothy W. Jones
Comments: 6 Pages.

This article simplifies Niven and others proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores unified approach to classic irrationality and transcendence proofs.
Category: Number Theory

[1615] viXra:1711.0283 [pdf] submitted on 2017-11-12 23:30:43

Riemann Hypothesis Rendered as not Provable

Authors: Colin James III
Comments: 1 Page. © Copyright 2017 by Colin James III All rights reserved.

A sentence to test is if known zeroes imply other zeroes. This effectively tests if a location of zeroes (trivial based on even numbers) and a location of zeroes (non trivial based on odd numbers) imply possibly another location of zeroes as a tautology, because the question is "Are there possibly other zeroes".
Category: Number Theory

[1614] viXra:1711.0276 [pdf] submitted on 2017-11-11 13:07:08

Monty Hall Problem

Authors: Dariusz Dudało
Comments: 1 Page.

Monty Hall problem
Category: Number Theory

[1613] viXra:1711.0267 [pdf] submitted on 2017-11-10 23:39:44

Conjecture that Any Square of a Prime Can be Obtained Through an Unusual Operation on the Numbers 360k+72

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture: The square of any odd prime can be obtained from the numbers of the form 360*k + 72 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number 360*k + 72; than for any square of a prime p^2 there exist k such that (d1 - 1)*(d2 - 1)*...*(dn - 1) + 1 = p^2. Example: for p^2 = 13^2 = 169 there exist k = 17 such that from 360*17 + 72 = 6192 = 2^4*3^2*43 is obtained 1^4*2^2*42 + 1 = 169. I also conjecture that any absolute Fermat pseudoprime (Carmichael number) can be obtained through the presented formula, which attests again the special relation that I have often highlighted between the nature of Carmichael numbers and the nature of squares of primes.
Category: Number Theory

[1612] viXra:1711.0262 [pdf] submitted on 2017-11-10 11:00:19

An Unusual Operation on a Set of Poulet Numbers Which Conducts to Another Set of Poulet Numbers

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following conjecture on Poulet numbers: There exist an infinity of Poulet numbers P2 obtained from Poulet numbers P1 in the following way: let d1, d2, ..., dn be the (not distinct) prime factors of the number P1 – 1, where P1 is a Poulet number; than there exist an infinity of Poulet numbers P2 of the form (d1 + 1)*(d2 + 1)*...*(dn + 1) + 1. Example: for Poulet number P1 = 645 is obtained through this operation Poulet number P2 = 1729 (644 = 2*2*7*23 and 3*3*8*24 + 1 = 1729). Note that from more than one Poulet number P1 can be obtained the same Poulet number P2 (from both 1729 and 6601 is obtained 46657).
Category: Number Theory

[1611] viXra:1711.0258 [pdf] submitted on 2017-11-09 13:06:12

The Squared Case of Pi^n is Irrational Gives Pi is Transcendental

Authors: Timothy W. Jones
Comments: 3 Pages.

This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.
Category: Number Theory

[1610] viXra:1711.0249 [pdf] submitted on 2017-11-08 09:49:05

On Expansion of Convergence Domain of Dirichlet Series Determining the Riemann Zeta Function

Authors: I. N. Tukaev
Comments: 3 Pages.

This paper proves that the Dirichlet series determining the Riemann zeta function converges within a domain of a real component of complex variable equal to one, with an imaginary component non-equal to zero.
Category: Number Theory

[1609] viXra:1711.0247 [pdf] submitted on 2017-11-07 09:41:06

Elementary Identities for Quocient of q -Series

Authors: Edigles Guedes
Comments: 14 Pages.

We demonstrate some elementary identities for quocient of q-series.
Category: Number Theory

[1608] viXra:1711.0239 [pdf] submitted on 2017-11-07 03:53:46

BBP - High - Precision Arithmetic

Authors: Edgar Valdebenito
Comments: 3 Pages.

This note presents two BBP-type formulas
Category: Number Theory

[1607] viXra:1711.0236 [pdf] submitted on 2017-11-06 18:00:00

Some Elementary Identities in Q-Series and the Generating Functions of the (M,k)-Capsids and (M, R1, R2)-Capsids

Authors: Edigles Guedes
Comments: 16 Pages.

We demonstrate some elementary identities for q-series involving the q-Pochhammer symbol, as well as an identity involving the generating functions of the (m,k)-capsids and (m, r1, r2)-capsids.
Category: Number Theory

[1606] viXra:1711.0203 [pdf] submitted on 2017-11-05 20:45:06

A Step -by- Step Proof of Beal’s Conjecture

Authors: Zhang Tianshu
Comments: 21 Pages.

In this article, we first classify A, B and C according to their respective odevity, and thereby get rid of two kinds which belong not to AX+BY=CZ. Then, affirm AX+BY=CZ in which case A, B and C have at least a common prime factor by several concrete equalities. After that, prove AX+BY≠CZ in which case A, B and C have not any common prime factor by mathematical induction with the aid of the symmetric law of odd numbers whereby even number 2W-1HZ as symmetric center after divide the inequality in four. Finally, reach a conclusion that the Beal’s conjecture holds water via the comparison between AX+BY=CZ and AX+BY≠CZ under the given requirements.
Category: Number Theory

[1605] viXra:1711.0202 [pdf] submitted on 2017-11-06 02:56:59

Sums of Arctangents and Sums of Products of Arctangents

Authors: Kunle Adegoke
Comments: 17 Pages.

We present new infinite arctangent sums and infinite sums of products of arctangents. Many previously known evaluations appear as special cases of the general results derived in this paper.
Category: Number Theory

[1604] viXra:1711.0140 [pdf] submitted on 2017-11-04 16:02:17

Interesting Formulas for the Fibonacci Sequence

Authors: José de Jesús Camacho Medina
Comments: 3 Pages.

This article disseminates a series of new and interesting mathematical formulas for the fibonacci sequence as product of the investigations of the author since 2015.
Category: Number Theory

[1603] viXra:1711.0134 [pdf] submitted on 2017-11-05 04:46:28

The Ulam Numbers up to One Trillion

Authors: Philip Gibbs, Judson McCranie
Comments: 8 Pages.

All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps.
Category: Number Theory

[1602] viXra:1711.0130 [pdf] submitted on 2017-11-03 10:35:22

The Irrationality and Transcendence of e Connected

Authors: Timothy W. Jones
Comments: 3 Pages. Suitable for first year, first term calculus students: just derivatives of polynomials.

Using just the derivative of the sum is the sum of the derivatives a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.
Category: Number Theory

[1601] viXra:1711.0128 [pdf] submitted on 2017-11-03 22:24:48

Theorem of Prime Pairs

Authors: Choe Ryujin
Comments: 4 Pages.

Theorem of prime pairs
Category: Number Theory

[1600] viXra:1711.0127 [pdf] submitted on 2017-11-03 23:29:58

Demonstration de la Conjecture de Polignac

Authors: Bado idriss olivier
Comments: 7 Pages.

In this paperwe give the proof Polignac Conjecture by using Chebotarev -Artin theorem ,Mertens formula and Poincaré sieve For doing that we prove that .Let's X be an arbitrarily large real number and n an even integer we prove that there are many primes p such that p+n is prime between sqrt(X) and X
Category: Number Theory

[1599] viXra:1711.0109 [pdf] submitted on 2017-11-02 11:47:31

The Qq'-Calculus

Authors: Antoine Balan
Comments: 5 Pages.

We present here the qq'-caculus which generalize the q-calculus but is however limited.
Category: Number Theory

[1598] viXra:1710.0353 [pdf] submitted on 2017-10-30 19:15:50

New and Interesting Mathematical Formulas

Authors: José de Jesús Camacho Medina
Comments: 6 Pages.

This article disseminates a series of new and interesting mathematical formulas, there are formulas of prime numbers, fibonacci sequence, square root and others as product of the investigations of the author since 2011.
Category: Number Theory

[1597] viXra:1710.0348 [pdf] submitted on 2017-10-31 03:21:01

Two Conjectures on Novak-Carmichael Numbers

Authors: Marius Coman
Comments: 2 Pages.

In this paper I make the following two conjectures on Novák-Carmichael numbers: (1) There exist an infinity of Novák-Carmichael numbers of the form (30n + p)*(30n + q) – p*q for any [p, q] distinct primes of the form 6k + 1; (2) There exist an infinity of Novák-Carmichael numbers of the form (30n + p)*(30n + q) - p*q for any [p, q] distinct primes of the form 6k – 1, where k > 1. See the sequence A124240 in OEIS for Novák-Carmichael numbers (numbers n such that a^n ≡ 1 (mod n) for every a coprime to n).
Category: Number Theory

[1596] viXra:1710.0347 [pdf] submitted on 2017-10-31 03:23:06

Three Conjectures on Novák-Carmichael Numbers

Authors: Marius Coman
Comments: 1 Page.

In this paper I make the following three conjectures on Novák-Carmichael numbers: (1) There exist an infinity of Novák-Carmichael numbers of the form (6k + 1)*(12k + 1)*(18k + 1) – 1; (2) There exist an infinity of Novák-Carmichael numbers of the form (6k - 1)*(12k - 1)*(18k - 1) + 1; (3) There exist an infinity of Novák-Carmichael numbers C such that C + 1 is a Poulet number. See the sequence A124240 in OEIS for Novák-Carmichael numbers (numbers n such that a^n ≡ 1 (mod n) for every a coprime to n).
Category: Number Theory

[1595] viXra:1710.0339 [pdf] submitted on 2017-10-31 19:02:00

(JAMCS - Original Research Article - 28.10.2017) The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths) (Journal of Advances in Mathematics and Computer Science)

Authors: Andrei Lucian Dragoi
Comments: 32 Pages.

This article proposes a synthesized classification of some Goldbach-like conjectures, including those which are "stronger" than the Binary Goldbach's Conjecture (BGC) and launches a new generalization of BGC briefly called "the Vertical Binary Goldbach's Conjecture" (VBGC), which is essentially a meta-conjecture, as VBGC states an infinite number of conjectures stronger than BGC, which all apply on "iterative" primes with recursive prime indexes (i-primeths). VBGC was discovered by the author of this paper in 2007 and perfected (by computational verifications) until 2017 by using the arrays of matrices of Goldbach index-partitions, which are a useful tool in studying BGC by focusing on prime indexes. VBGC distinguishes as a very important conjecture of primes, with potential importance in the optimization of the BGC experimental verification (including other possible theoretical and practical applications in mathematics and physics) and a very special self-similar property of the primes set. Keywords: Primes with prime indexes; i-primeths; the Binary Goldbach Conjecture; Goldbach-like conjectures; the Vertical Binary Goldbach Conjecture. 2010 mathematics subject classification: 11N05 (Distribution of primes, URL: http://www.ams.org/msc/msc2010.html?t=11N05&btn=Current) OFFICIAL LINKS OF THIS PUBLISHED (OPEN) PEER-REVIEWED ARTICLE: http://www.sciencedomain.org/issue/3151 http://www.journalrepository.org/media/journals/JAMCS_69/2017/Oct/Andrei2522017JAMCS36895.pdf http://www.sciencedomain.org/review-history/21625 http://www.sciencedomain.org/metrics/21625
Category: Number Theory

[1594] viXra:1710.0335 [pdf] submitted on 2017-10-29 21:39:34

A Brief Investigation Into Two Sets of Elliptic Curves

Authors: Lulu Karami
Comments: 17 Pages.

This submission is more or less an amateur exposition on a specific elliptic curve, discussing counting points over finite fields as well as constructing an associated $L$-function and pinning down the affiliated special value $L(E, 1)$ for the elliptic curve $E$ primarily discussed throughout this piece. The techniques and tools presented can be carried over to infinitely many elliptic curves partitioned into two sets depending on 'twists' of two specific curves; one of which happens to be the curve previously, and vaguely, mentioned.
Category: Number Theory

[1593] viXra:1710.0333 [pdf] submitted on 2017-10-30 05:47:22

A Solution of the Fermat’s Last Theorem

Authors: José Francisco García Juliá
Comments: 2 Pages.

It is obtained a solution of the Fermat’s last theorem.
Category: Number Theory

[1592] viXra:1710.0332 [pdf] submitted on 2017-10-30 07:53:11

Three Limits

Authors: Edgar Valdebenito
Comments: 5 Pages.

This note presents three limits for 1/pi
Category: Number Theory

[1591] viXra:1710.0331 [pdf] submitted on 2017-10-30 07:57:53

Question 408: A Trigonometric Formula

Authors: Edgar Valdebenito
Comments: 2 Pages.

This note presents a simple formula for pi
Category: Number Theory

[1590] viXra:1710.0263 [pdf] submitted on 2017-10-23 08:03:39

Question 404: Three Formulas and Some Fractals

Authors: Edgar Valdebenito
Comments: 9 Pages.

This note presents three formulas involving pi and some fractals.
Category: Number Theory

[1589] viXra:1710.0245 [pdf] submitted on 2017-10-22 16:36:56

Mathematical Deterministic Reductionism

Authors: Paris Samuel Miles-Brenden
Comments: 2 Pages. None.

None.
Category: Number Theory

[1588] viXra:1710.0242 [pdf] submitted on 2017-10-22 16:39:39

Mathematical Modular Closure

Authors: Paris Samuel Miles-Brenden
Comments: 4 Pages. None.

None.
Category: Number Theory

[1587] viXra:1710.0209 [pdf] submitted on 2017-10-18 15:14:30

Prime Numbers as a Function of a Geometric Progression

Authors: Leif R. Uppström, Daniel Uppström
Comments: 9 Pages.

In mathematical literature it is asked for a computable function or efficient algorithm to find all, or at least a large subset, of the prime numbers. This paper shows that all primes can be characerised by their reciprocal period length L and its figure value R. These parameters are given for each prime after inversion to an infinitely repeated period and are used to group all primes into disjoint sets that arise as a function of a geometric progression. This theory suggests new ways to enumerate and find large primes.
Category: Number Theory

[1586] viXra:1710.0205 [pdf] submitted on 2017-10-19 02:50:41

An Approximation to the Prime Counting Function Through the Sum of Consecutive Prime Numbers

Authors: Juan Moreno Borrallo
Comments: 6 Pages.

In this paper it is proved that the sum of consecutive prime numbers under the square root of a given natural number is asymptotically equivalent to the prime counting function. Also, it is proved another asymptotic relationship between the sum of the first prime numbers up to the integer part of the square root of a given natural number and the prime counting function.
Category: Number Theory

[1585] viXra:1710.0174 [pdf] submitted on 2017-10-17 01:35:25

Theorem of Prime Pair Distribution

Authors: Choe Ryujin
Comments: 4 Pages.

Theorem of prime pair distribution
Category: Number Theory

[1584] viXra:1710.0169 [pdf] submitted on 2017-10-17 09:44:37

Prime Set Representation

Authors: Steven Shawcross
Comments: 9 Pages. A version of this paper is copyrighted by Steven Shawcross, 2003.

The integer 2 satisfies the divisibility definition of a prime number: it is only divisible by itself and 1. The integer 1 also satisfies this definition, and yet, mathematicians generally do not consider 1 a prime. Rather 1 merits a class of its own, belonging neither to the prime nor composite class. In divisibility theory, 2 does occupy a special subclass within the class of prime numbers: it is the only even prime. This paper introduces a theory of numbers called the Prime Set Representation Theory. This theory utilizes the odd primes and does not rely on the primeness of 2. In Prime Set Representation Theory, the odd primes are building blocks of the theory; all integers, including 2, have representations in terms of them. The import of the theory is not to dislodge the integer 2 from its solitary, even-prime status. The theory's efficacy is a better understanding of the distribution of primes, twin primes, and primes of the form x^2 + 1. A natural extension of the theory yields valid and strikingly direct approximation formulas for these prime classifications. The same theory furnishes a new and improved approximation to the number of Goldbach pairs associated with general even number 2n (the improvement is relative to Sylvester's formula for Goldbach pairs, but the formula performs well vis-à-vis the Hardy-Littlewood formulas in the ranges tested).
Category: Number Theory

[1583] viXra:1710.0145 [pdf] submitted on 2017-10-12 05:04:05

Visualizing Zeta(n) and Proving Its Irrationality

Authors: Timothy W. Jones
Comments: 19 Pages.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(2n+1), n>1, is irrational.
Category: Number Theory

[1582] viXra:1710.0129 [pdf] submitted on 2017-10-11 11:54:50

Statistical Relationships Involving Benford's Law, the Lognormal Distribution, and the Summation Theorem

Authors: Robert C. Hall
Comments: 28 Pages.

Regarding Benford's law, many believe that the statistical data sources follow a Benford's law probability density function(1/xLn(10))when, in actuality, it follows a Lognormal probability density function. The only data that strictly follows a Benford's law probability density function is an exponential function i.e. a number (base) raised to a power x. The other sets of data conform to a Lognormal distribution and, as the standard deviation approaches infinity, approximates a true Benford distribution. Also, the so called Summation theorem whereby the sum of the values with respect to the first digits is a uniform distribution only applies to an exponential function. The data derived from the aforementioned Lognormal distribution is more likely to conform to a Benford like distribution as the data seems to indicate.
Category: Number Theory

[1581] viXra:1710.0113 [pdf] submitted on 2017-10-10 06:32:08

Kurmet's First Theorem and Simple Proof Fermat's Last Theorem

Authors: Kurmet Sultan
Comments: 2 Pages. This is the Russian version of the manuscript.

The paper describes the First theorem of Kurmet and a simple proof of the Last theorem of Fermat, which was obtained on the basis of Kurmet's First Theorem.
Category: Number Theory

[1580] viXra:1710.0112 [pdf] submitted on 2017-10-10 06:35:55

Kurmet's Second Theorem and Simple Proof Catalan’s Conjecture

Authors: Kurmet Sultan
Comments: 2 Pages. This is the Russian version of the manuscript.

In this paper we describe the Second Theorem of Kurmet and give a simple proof of Catalan’s conjecture on the basis of Kurmet's Second Theorem.
Category: Number Theory

[1579] viXra:1710.0109 [pdf] submitted on 2017-10-09 03:05:33

FLT Proof N=4

Authors: Maik Becker-Sievert
Comments: 1 Page.

Fermats Last Theorem n=4 One line proof
Category: Number Theory

[1578] viXra:1710.0099 [pdf] submitted on 2017-10-10 01:50:30

Proving the Erdös-Straus Conjecture from Infinite to Finite Equalities

Authors: Zhang Tianshu
Comments: 19 Pages.

We first classify all integers ≥2 into eight kinds, and that formulate each of seven kinds therein into a sum of three unit fractions. For remainder one kind, we classify it into three genera, and that formulate each of two genera therein into a sum of three unit fractions. For remainder one genus, we classify it into five sorts, and that formulate each of three sorts therein into a sum of three unit fractions. For remainder two sorts i.e. 4/(49+120c) and 4/(121+120c) with c≥0, we prove them by logical inference. But miss out 3587 concrete fractions to await computer programming to solve the problem that express each of them into a sum of three unit fractions.
Category: Number Theory

[1577] viXra:1710.0048 [pdf] submitted on 2017-10-04 20:55:25

Proof of Riemann Hypothesis

Authors: Choe Ryujin
Comments: 1 Page.

Proof of Riemann hypothesis
Category: Number Theory

[1576] viXra:1710.0042 [pdf] submitted on 2017-10-03 11:01:23

Proof of the Twin Prime Conjecture

Authors: Dieter Sengschmitt
Comments: 15 Pages.

I can proof that there are infinitely many twin primes. The twin prime counting function π2(n), which gives the number of twin primes less than or equal to n for any natural number n, is for lim⁡n→∞ π2(n)= 2 C2 [π(n)]^2/n where π(n) is the prime counting function and C2 is the so-called twin prime constant with C2=0,6601618…
Category: Number Theory

[1575] viXra:1710.0038 [pdf] submitted on 2017-10-03 16:37:46

An Alternate Proof of the Prime Number Theorem

Authors: Robert C. Hall
Comments: 2 Pages.

An attempt is made to derive the probability density function of the sum of prime numbers, which is x/Ln(x). This does appear to be quite accurate in predicting the sum of prime numbers less than 100,000( within 0.124%). Given this assertion, an attempt is made to derive the probability density function of the distribution of the prime numbers themselves.
Category: Number Theory

[1574] viXra:1710.0017 [pdf] submitted on 2017-10-02 02:24:57

François Mendzina Essomba pi Formulas (3)

Authors: Mendzina Essomba Francois
Comments: 2 Pages.

four new formulas for pi
Category: Number Theory

[1573] viXra:1710.0015 [pdf] submitted on 2017-10-02 03:09:57

A Cousin of One of Ramanujan's Identities

Authors: Lulu Karami
Comments: 4 Pages.

This submission gives a closed form identity similar to one given by Ramanujan. A formula for infinitely many similar identities is presented here as well.
Category: Number Theory

[1572] viXra:1709.0428 [pdf] submitted on 2017-09-28 17:54:57

(Published Version) Addendum to Paper Entitled "Do Prime Numbers Obey a Three Dimensional Double Helix?"

Authors: Peter Bissonnet
Comments: 13 Pages.

This paper again specifies the major points of the article “Do Prime Numbers Obey a Three-Dimensional Double Helix?” [1] which was received on February 16, 2006 by Hadronic Journal. New information has been added and elucidated upon, such as why the numbers 2 and 3 are not considered true prime numbers, and why s in the following formulas for 6s − 1 and for 6s + 1 is really a composite number equal to the sum of two other numbers, suggesting that s is always to be considered as an integer. Other new information is added as well, such as how an engineer in a matter of seconds decomposed a large prime product into its constituent primes using basic software and won a contract for his firm.
Category: Number Theory

[1571] viXra:1709.0417 [pdf] submitted on 2017-09-28 08:14:22

About the Twin Primes Conjecture

Authors: Ramón Ruiz
Comments: 26 Pages. This document is written in Spanish

Twin Primes Conjecture: “There are infinitely many primes p such that (p + 2) is also prime”. In this document I have used the prime numbers theorem enunciated by Carl Friedrich Gauss and the prime numbers theorem in arithmetic progressions. These two theorems applied to a combination of two arithmetic progressions of module 30 that contain prime numbers, allows us to develop a nonprobability general formula to calculate, approximately, the number of prime pairs, p and (p + 2), that are lesser than a number x. This research is based on a approach designed solely to demonstrate the Twin Prime Conjecture and the Binary Goldbach Conjecture.
Category: Number Theory

[1570] viXra:1709.0411 [pdf] submitted on 2017-09-27 14:14:11

On the Riemann Hypothesis and Other Brain Teasers

Authors: John Smith
Comments: 7 Pages.

Abstract In 1963, a game show called Lets Make A Deal began in the United States. On the show, the host - Monty Hall - would present contestants with the choice of 3 doors, behind only 1 of which was a car. A contestant would pick a door such as No. 1, and Monty, who knew what was behind the doors, would open another door, say No. 2, revealing a goat. Monty would then ask the contestant if they wanted to change their selection to door No. 3. It is widely accepted that the contestant should change doors on the basis that the chances of the car being behind door 3 are 2/3, whereas the chances of the car being behind door 1 are only 1/3. But by appeal to congruities that exist between this seemingly innocuous and simple problem and variety of deeper and less tractable problems, the Monty Hall Problem is revealed as the tip of a great intellectual iceberg.
Category: Number Theory

[1569] viXra:1709.0410 [pdf] submitted on 2017-09-27 14:15:31

On Fermat's Last Theorem

Authors: John Smith
Comments: 2 Pages.

In 1986 AndrewWiles published a ground-breaking proof of Fermat's Last Theorem. But in spite of the rarity and the significance of the achievement, the underlying reasoning is so convoluted that it would be be extremely difficult -if not impossible- for any but a tiny minority of specialists to understand it. Most must simply take the word of Wiles and his fellow experts that Fermat's Last Theorem has been proved. But the conjecture itself -that no 3 positive integers can satisfy the equation x^n + y^n = z^n for any positive-integer value of n greater than 2- is so simple that a school child could understand it, and Fermat himself claimed that he possessed a proof, one that -if it existed- must have been expressed in the language of 17th century mathematics, and the language of 21st century high school mathematics. Ye there can be no such proof: this note outlines a complimentary but alternative argument to that employed by Wiles that shows why no 17th century proof of the theorem is possible.
Category: Number Theory

[1568] viXra:1709.0408 [pdf] submitted on 2017-09-27 19:45:24

Fermat's Proof of Fermat's Last Theorem

Authors: Johnny E Magee
Comments: 18 Pages.

Focusing on the properties and constraints of the decompositions of Fermat’s equation and its elements --and employing only basic arithmetic and algebraic techniques that would have been known to Fermat-- we identify certain specific requirements necessary for c, of (an + bn) = cn, to be an integer, and establish that these requirements can only be met at n = 2. AMS Subject Classification (2010): Primary, 11D41 Key words and phrases: Fermat’s Last Theorem, Fermat’s Equation, Binomial Theorem
Category: Number Theory

[1567] viXra:1709.0375 [pdf] submitted on 2017-09-24 18:23:32

An Essay on the Zeroes of an L-Function

Authors: Matanari Shimoinuda
Comments: 28 Pages.

This article is the summary of the spectral interpretation of critical zeroes of an L-function by Alain Connes. I try to examine the subject from the view of the representation theory and add some comments.
Category: Number Theory

[1566] viXra:1709.0312 [pdf] submitted on 2017-09-22 01:36:01

The Distribution of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

In this paper, we analyze the behavior of prime numbers.
Category: Number Theory

[1565] viXra:1709.0295 [pdf] submitted on 2017-09-20 06:46:03

Fermat’s Zero Theorem

Authors: Faisal Amin Yassein Abdelmohssin
Comments: 4 Pages.

Fermat’s zero theorem is stated as follows: It is impossible to separate a square of a difference of two natural numbers into two squares of differences, or a cube power of a difference into two cube powers of differences, or a fourth power of a difference into two fourth powers, or in general, any power higher than the first, into two like powers of differences.
Category: Number Theory

[1564] viXra:1709.0288 [pdf] submitted on 2017-09-19 04:55:48

Special Rule for Certain Prime Numbers

Authors: Ranganath G. Kulkarni
Comments: 2 Pages.

A quadratic equation for prime numbers is assumed to be true that satisfy the following four rules. Some prime numbers violate these rules. Whereas some non prime numbers satisfy the four rules. They are not prime, therefore to make them violate the fourth rule we need to study how to choose the value of m and n so as make the quadratic equation as the primes generating formula.
Category: Number Theory

[1563] viXra:1709.0258 [pdf] submitted on 2017-09-17 13:33:28

Fermat's Theorem. Proof by 2 Operations

Authors: Victor Sorokine
Comments: 2 Pages.

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
Category: Number Theory

[1562] viXra:1709.0257 [pdf] submitted on 2017-09-17 13:35:16

Fermat's Theorem. Proof by 2 Operations French

Authors: Victor Sorokine
Comments: 2 Pages. French version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
L'égalité de Fermat est contradictoire entre les deuxièmes chiffres des facteurs du nombre A.
Category: Number Theory

[1561] viXra:1709.0256 [pdf] submitted on 2017-09-17 13:36:32

Fermat's Theorem. Proof by 2 Operations Russian

Authors: Victor Sorokine
Comments: 2 Pages. Russian version

The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number А.
Суть противоречия. Равенство Ферма противоречиво по вторым цифрам сомножителей числа А.
Category: Number Theory

[1560] viXra:1709.0227 [pdf] submitted on 2017-09-15 05:23:56

A Minor Theorem Related with the Fermat Conjecture

Authors: José Francisco García Juliá
Comments: 2 Pages.

It is obtained a minor theorem related with the Fermat conjecture.
Category: Number Theory

[1559] viXra:1709.0128 [pdf] submitted on 2017-09-11 05:10:30

Theorems on Pythagorean Triples and Prime Numbers

Authors: Faisal Amin Yassein Abdelmohssin
Comments: 3 Pages.

Relationships among natural numbers constituting a Pythagorean triple (PT) and between these natural numbers constituting the Pythagorean triples (PTs) and Prime Numbers (PNs) have been found. These relationships are formulated as theorems; first theorem is that the natural numbers constituting a Pythagorean triple (PT) satisfy a certain equation related to sum of their differences; second theorem is that differences of sum of the natural numbers constituting a Pythagorean triple (PT) are prime numbers.
Category: Number Theory

[1558] viXra:1709.0092 [pdf] submitted on 2017-09-08 12:19:26

Question 383 : Nonlinear Equation , Euler Numbers , Number Pi

Authors: Edgar Valdebenito
Comments: 4 Pages.

This note presents some formulas for pi.
Category: Number Theory

Replacements of recent Submissions

[720] viXra:1801.0140 [pdf] replaced on 2018-01-15 09:33:09

A Simple Proof that Zeta(2) is Irrational

Authors: Timothy W. Jones
Comments: 6 Pages. Gives a counter to the geometric counter example.

We prove that partial sums of $\zeta(2)-1=z_2$ 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 $z_2$ and the limit of the exclusions leaves only irrational numbers.
Category: Number Theory

[719] viXra:1801.0140 [pdf] replaced on 2018-01-14 03:44:54

A Simple Proof that Zeta(2) is Irrational

Authors: Timothy W. Jones
Comments: 6 Pages. An easier set theoretical proof has been added.

We prove that a partial sum of $\zeta(2)-1=z_2$ is not given by any single decimal in a number base given by a denominator of its 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 $z_2$ and the limit of the exclusions leaves only irrational numbers. This is a set theoretical proof. We also give a topological proof using nested intervals and Cantor's intersection theorem.
Category: Number Theory

[718] viXra:1801.0140 [pdf] replaced on 2018-01-13 09:51:26

A Simple Proof that Zeta(2) is Irrational

Authors: Timothy W. Jones
Comments: 6 Pages. Correction of upper bound.

We prove that a partial sum of $\zeta(2)-1=z_2$ is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that there are an infinite number of partial sums in one interval of the form $X_{k^2}=[.(x-1),.x]$ where $.x$ is a single, non-zero decimal in a number base of the denominators of the terms of $z_2$, here $k^2$. Using this property we show that $z_2$ is contained in an open interval inside $X_{k^2}$. As all possible rational values of $z_2$ are the endpoints of these $X_k$ intervals, $z_2$ must be irrational.
Category: Number Theory

[717] 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

[716] viXra:1801.0052 [pdf] replaced on 2018-01-14 21:28:41

A Trivially Simple Proof of Fermat's Last Theorem

Authors: Philip A. Bloom
Comments: Pages.

We formulate an algebraic identity that has positive integral Fer- mat triples equal to (x, y, z) of x^n + y^n = z^n. We assume that x^n + y^n = z^n has a positive integral Fermat triple for any given value of n greater or equal to three, and we derive a contradiction. Hence, for any given n greater or equal to three, there is a null set of positive integral (x, y, z).
Category: Number Theory

[715] 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

[714] viXra:1712.0384 [pdf] replaced on 2017-12-22 10:37:51

Discovering and Proving that Pi is Irrational, 2nd Edition

Authors: Timothy W. Jones
Comments: 13 Pages. Additional clarifications and appendix added.

Ivan Niven's proof of the irrationality of pi is often cited because it is brief and uses only calculus. However it is not well motivated. Using the concept that a quadratic function with the same symmetric properties as sine should when multiplied by sine and integrated obey upper and lower bounds for the integral, a contradiction is generated for rational candidate values of pi. This simplifying concept yields a more motivated proof of the irrationality of pi and pi squared.
Category: Number Theory

[713] viXra:1712.0359 [pdf] replaced on 2017-12-12 02:10:44

千古奇冤,素数有限

Authors: Liu Ran
Comments: 6 Pages.

素数个数有限,并且找出欧几里德证明的瑕疵,并举出证据
Category: Number Theory

[712] viXra:1712.0135 [pdf] replaced on 2017-12-30 12:15:53

Proof of Beal’s Conjecture and Related Examples

Authors: Kamal Barghout
Comments: 21 Pages. The material in this article is copyrighted. Please obtain authorization to use any part of the manuscipt from the author

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stands upon that the LHS of Beal’s conjecture represents the sum of two monomials of like terms. The monomial on the RHS of Beal’s conjecture can be built by combining the two monomials on the LHS. By representing any number in exponential form of single power as having a unique base-unit it is to be proved that any number in exponential form to be added to it to yield a sum in exponential form of single power must have the same base-unit by virtue of the two numbers having a “block-form” with a building block of their common base-unit. By conversion of the addition process of the two exponential numbers to multiplication, the GCF of the two terms on the LHS of Beal’s conjecture can be factored. Upon factorization of the GCF and making use of power rules, it must be combined with the sum of the two coefficients of the two terms to yield the monomial on the RHS of the conjecture, confirming the proposition that they must have a common and unique base-unit to successfully combine.
Category: Number Theory

[711] viXra:1712.0135 [pdf] replaced on 2017-12-18 06:46:20

Proof of Beal’s Conjecture and Related Examples

Authors: Kamal Barghout
Comments: 21 Pages. The material in this article is copyrighted. Please obtain authorization for use of any part of the artilce from the author.

In this article we prove Beal’s conjecture by deductive reasoning by means of elementary algebraic methods. The main assertion in the proof stands upon that the LHS of Beal’s equation represents the sum of two monomial functions with common variable. The monomial function on the RHS of Beal’s equation can be built from the sum of the two monomials on the LHS. The Greatest Common Factor (GCF) of the two terms on the LHS of the equation is a number in exponential form of single power whose base is the common variable of the two monomials. Upon factorization of the GCF, it must be combined with the sum of the two coefficients of the terms to yield the monomial on the RHS of the equation.
Category: Number Theory

[710] viXra:1711.0417 [pdf] replaced on 2017-11-25 21:37:21

素数分布无规律

Authors: Liu Ran
Comments: 2 Pages.

A new way to prove prime number distribution being no law.
Category: Number Theory

[709] viXra:1711.0296 [pdf] replaced on 2017-11-17 23:40:31

A Brief Proof of the Collatz Conjecture

Authors: Kurmet Sultan
Comments: 9 Pages. Russian version

In this paper we give a brief proof of the Collatz conjecture. It is shown that it is more efficient to start calculating the Collatz function C (n) from odd numbers 6m ± 1. It is further proved that if we calculate by the formula ((6n ± 1)·2^q -1) / 3 on the basis of a sequence of numbers 6n ± 1, increasing the exponent of two by 1 at each iteration, then to each number of the form 6n ± 1 there will correspond a set whose elements are numbers of the form 3t, 6m-1 and 6m + 1. Moreover, all sets are disjoint. Then it is shown that if we construct micro graphs of numbers by combining the numbers 6n ± 1 with their elements of the set 3t, 6m-1 and 6m + 1, then combine the micro graphs by combining equal numbers 6n ± 1 and 6m ± 1, then a tree-like fractal graph of numbers. A tree-like fractal graph of numbers, each vertex of which corresponds to numbers of the form 6m ± 1, is a proof of the Collatz conjecture, since any of its vertices is connected with a finite vertex connected with unity.
Category: Number Theory

[708] viXra:1711.0291 [pdf] replaced on 2017-11-30 10:59:47

The Irrationality of Trigonometric and Hyperbolic Functions

Authors: Timothy W. Jones
Comments: 7 Pages. This version adds an example using Leibniz tables.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.
Category: Number Theory

[707] viXra:1711.0291 [pdf] replaced on 2017-11-16 09:44:04

The Irrationality of Trigonometric and Hyperbolic Functions

Authors: Timothy W. Jones
Comments: 6 Pages. Clarifications of lemmas, slight re-organization.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.
Category: Number Theory

[706] viXra:1711.0291 [pdf] replaced on 2017-11-14 09:52:19

The Irrationality of Trigonometric and Hyperbolic Functions

Authors: Timothy W. Jones
Comments: 6 Pages. Minor clarifications and edits of previous version.

This article simplifies Niven's proofs that cos and cosh are irrational when evaluated at non-zero rational numbers. Only derivatives of polynomials are used. This is the third article in a series of articles that explores a unified approach to classic irrationality and transcendence proofs.
Category: Number Theory

[705] viXra:1711.0258 [pdf] replaced on 2017-11-10 12:40:06

The Squared Case of Pi^n is Irrational Gives Pi is Transcendental

Authors: Timothy W. Jones
Comments: 6 Pages. A more complete bibliography is included.

This is companion article to The Irrationality and Transcendence of e Connected. In it the irrationality of pi^n is proven using the same lemmas used for e^n. Also the transcendence of pi is given as a simple extension of this irrationality result.
Category: Number Theory

[704] viXra:1711.0134 [pdf] replaced on 2017-11-10 10:10:06

The Ulam Numbers up to One Trillion

Authors: Philip Gibbs, Judson McCranie
Comments: 9 Pages.

All Ulam numbers up to one trillion are computed using an efficient linear-time algorithm. We report on the distribution of the numbers including the positions of the largest gaps.
Category: Number Theory

[703] viXra:1711.0130 [pdf] replaced on 2017-11-09 06:45:07

The Irrationality and Transcendence of e Connected

Authors: Timothy W. Jones
Comments: 3 Pages. Slight corrections.

Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.
Category: Number Theory

[702] viXra:1711.0130 [pdf] replaced on 2017-11-04 11:58:03

The Irrationality and Transcendence of e Connected

Authors: Timothy W. Jones
Comments: 3 Pages. Suitable for first year, first term calculus students: just derivatives of polynomials.

Using just the derivative of the sum is the sum of the derivatives and simple undergraduate mathematics a proof is given showing e^n is irrational. The proof of e's transcendence is a simple generalization from this result.
Category: Number Theory

[701] viXra:1711.0109 [pdf] replaced on 2017-11-05 03:05:10

The QQ'-Calculus

Authors: Antoine Balan
Comments: 5 Pages.

We introduce a generalization of the q-calculus, which we call qq'-calculus. Some formulas are obtained; however the theory remains limited.
Category: Number Theory

[700] viXra:1711.0109 [pdf] replaced on 2017-11-02 16:01:26

The QQ'-Calculus

Authors: Antoine Balan
Comments: 5 Pages.

We introduce here a generalization of the q-calculus which we call the qq'-calculus. Some limited formulas are obtained like the Taylor's expansion formula.
Category: Number Theory

[699] viXra:1710.0145 [pdf] replaced on 2017-12-30 12:01:50

Visualizing Zeta(n>1) and Proving Its Irrationality

Authors: Timothy W. Jones
Comments: 17 Pages. Replaces use of Cantor's Diagonal Method with a set topological proof.

A number system is developed to visualize the terms and partials of zeta(n>1). This number system consists of radii that generate sectors. The sectors have areas corresponing to all rational numbers and can be added via a tail to head vector addition. Dots on the circles give an un-ambiguous cross reference to decimal systems in all bases. We show, in the proof section of this paper, first that all partials require decimal bases greater than the last denominator used in the partial, then that this can be used to make a sequence of nested intervals with rational endpoints. Using Cantor's Nested Interval theorem this gives the convergence point of zeta series and disallows rational values, thus proving the irrationality of zeta(n>1).
Category: Number Theory

[698] viXra:1710.0145 [pdf] replaced on 2017-12-21 08:13:33

Visualizing Zeta(n>1) and Proving Its Irrationality

Authors: Timothy W. Jones
Comments: 20 Pages. Typos corrected with some further clarifications.

A number system is developed to visualize the terms and partials of zeta(n>1). This number system consists of radii that generate sectors. The sectors have areas corresponing to all rational numbers and can be added via a tail to head vector addition. Dots on the circles give an un-ambiguous cross reference to decimal systems in all bases. We show, in the proof section of this paper, using a modification of Cantor's diagonal method, all zeta(n>1) require a infinite decimal in all bases. This establishes the result.
Category: Number Theory

[697] viXra:1710.0145 [pdf] replaced on 2017-11-02 10:23:05

Visualizing Zeta(n>=2) and Proving Its Irrationality

Authors: Timothy W. Jones
Comments: 20 Pages. Some corrections and explanations added.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(n), n>1, is irrational.
Category: Number Theory

[696] viXra:1710.0145 [pdf] replaced on 2017-10-21 05:31:18

Visualizing Zeta(n>=2) and Proving Its Irrationality

Authors: Timothy W. Jones
Comments: 19 Pages.

Using concentric circles that form sector areas of rational areas, an adaptation of Cantor's diagonal method shows that zeta(2n+1), n>1, is irrational.
Category: Number Theory

[695] viXra:1709.0312 [pdf] replaced on 2017-12-07 21:18:35

The Distribution of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 2 Pages.

In this paper, we find the axiomatic pattern of prime numbers.
Category: Number Theory

[694] viXra:1709.0312 [pdf] replaced on 2017-10-27 20:31:16

The Distribution of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.
Category: Number Theory

[693] viXra:1709.0312 [pdf] replaced on 2017-10-24 21:31:09

The Distribution of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.
Category: Number Theory

[692] viXra:1709.0312 [pdf] replaced on 2017-09-24 03:38:04

The Distribution of Primes

Authors: Ihsan Raja Muda Nasution
Comments: 1 Page.

In this paper, we find the axiomatic pattern of prime numbers.
Category: Number Theory