# Number Theory

## 1503 Submissions

 viXra:1503.0267 [pdf] submitted on 2015-03-31 11:18:19

### The Property of Poulet Numbers to Create Through Concatenation Semiprimes Which Are C-Primes or M-Primes

Authors: Marius Coman

In this paper I present a very interesting characteristic of Poulet numbers, namely the property that, concatenating two of such numbers, is often obtained a semiprime which is either c-prime or m-prime. Using just the first 13 Poulet numbers are obtained 9 semiprimes which are c-primes, 20 semiprimes which are m-primes and 9 semiprimes which are cm-primes (both c-primes and m-primes).
Category: Number Theory

 viXra:1503.0264 [pdf] submitted on 2015-03-31 06:17:57

### Two Formulas for Obtaining Primes and cm-Integers

Authors: Marius Coman

In this paper I present two very interesting and easy formulas that conduct often to primes or cm-integers (c-primes, m-primes, cm-primes, c-composites, m-composites, cm-composites).
Category: Number Theory

 viXra:1503.0253 [pdf] submitted on 2015-03-30 06:23:08

### An Analysis of Four Smarandache Concatenated Sequences Using the Notion of cm-Integers

Authors: Marius Coman

In this paper I show that Smarandache concatenated sequences presented here (i.e. The consecutive numbers sequence, The concatenated odd sequence, The concatenated even sequence, The concatenated prime sequence), sequences well known for the common feature that contain very few terms which are primes, per contra, contain very many terms which are c-primes, m-primes, c-reached primes and m-reached primes (notions presented in my previous papers, see “Conjecture that states that any Carmichael number is cm-composite” and “A property of repdigit numbers and the notion of cm-integer”).
Category: Number Theory

 viXra:1503.0242 [pdf] submitted on 2015-03-29 10:39:36

### A Property of Repdigit Numbers and the Notion of cm-Integer

Authors: Marius Coman

In this paper I want to name generically all the numbers which are either c-primes, m-primes, cm-primes, c-composites, m-composites or cm-composites with the name “cm-integers” and to present what seems to be a special quality of repdigit numbers (it’s about the odd ones) namely that are often cm-integers.
Category: Number Theory

 viXra:1503.0234 [pdf] submitted on 2015-03-29 03:33:20

### Three Conjectures on Probably Infinite Sequences of Primes Created Through Concatenation of Primes with the Powers of 2

Authors: Marius Coman

In this paper I present three conjectures, i.e.: (1) For any prime p greater than or equal to 7 there exist n, a power of 2, such that, concatenating to the left p with n the number resulted is a prime (2) For any odd prime p there exist n, a power of 2, such that, subtracting one from the number resulted concatenating to the right p with n, is obtained a prime (3) For any odd prime p there exist n, a power of 2, such that, adding one to the number resulted concatenating to the right p with n, is obtained a prime.
Category: Number Theory

 viXra:1503.0227 [pdf] submitted on 2015-03-28 15:35:18

### The Notions of C-Reached Prime and M-Reached Prime

Authors: Marius Coman

In spite the fact that I wrote seven papers on the notions (defined by myself) of c-primes, m-primes, c-composites and m-composites (see in my paper “Conjecture that states that any Carmichael number is a cm-composite” the definitions of all these notions), I haven’t thinking until now to find a connection, beside the one that defines, of course, such an odd composite n, namely that, after few iterative operations on n, is reached a prime p, between the number n and the prime p. This is what I try to do in this paper, and also to give a name to this prime p, namely, say, “reached prime”, and, in order to distinguish, because a number can be same time c-prime and m-prime, respectively c-composite and m-composite, “c-reached prime” or “m-reached prime”.
Category: Number Theory

 viXra:1503.0222 [pdf] submitted on 2015-03-28 13:05:45

### Journée de Pi, 3/14/15 pi Day

Authors: Simon Plouffe
Comments: 30 Pages. The conference is in french

Conférence pour la journée de Pi à Marseille le 3/14/15. Conference on Pi Day of 3/14/15 in Marseille
Category: Number Theory

 viXra:1503.0219 [pdf] submitted on 2015-03-27 19:39:02

### Equipotency of E# with FGH_w

Authors: Sbiis Saibian

The goal in this article is to demonstrate that E# is indeed on the order of ω. Formally this means that for every member of FGH_ω there is a function in E# with at least the same growth rate, and that f_w(n) the smallest member of FGH which eventually dominates over all functions within E#. It will be demonstrated that a certain family of functions of order-type "w" in E# dominates over corresponding members in FGH_w, thus showing that for every function in FGH_w there is a function in E# which grows at least as fast. Then it will be shown how f_w(n) diagonalizes over this family of functions and must eventually dominate every member of this family.
Category: Number Theory

 viXra:1503.0217 [pdf] submitted on 2015-03-28 01:40:00

### An Intuitive Conceptualization of n! and Its Application to Derive a Well Known Result

Authors: Prashanth R. Rao

n! is defined as the product 1.2.3………n and it popularly represents the number of ways of seating n people on n chairs. We conceptualize another way of describing n! using sequential cuts to an imaginary circle and derive the following well known result
Category: Number Theory

 viXra:1503.0216 [pdf] submitted on 2015-03-28 02:39:54

### Three Conjectures on Twin Primes Involving the Sum of Their Digits

Authors: Marius Coman

Observing the sum of the digits of a number of twin primes, I make in this paper the following three conjectures: (1) for any m the lesser term from a pair of twin primes having as the sum of its digits an odd number there exist an infinity of lesser terms n from pairs of twin primes having as the sum of its digits an even number such that m + n + 1 is prime, (2) for any m the lesser term from a pair of twin primes having as the sum of its digits an even number there exist an infinity of lesser terms n from pairs of twin primes having as the sum of its digits an odd number such that m + n + 1 is prime and (3) if a, b, c, d are four distinct terms of the sequence of lesser from a pair of twin primes and a + b + 1 = c + d + 1 = x, then x is a semiprime, product of twin primes.
Category: Number Theory

 viXra:1503.0214 [pdf] submitted on 2015-03-27 15:12:46

### An Interesting Recurrent Sequence Whose First 150 Terms Are Either Primes, Powers of Primes or Products of Two Prime Factors

Authors: Marius Coman

I started this paper in ideea to present the recurrence relation defined as follows: the first term, a(0), is 13, then the n-th term is defined as a(n) = a(n–1) + 6 if n is odd and as a(n) = a(n-1) + 24, if n is even. This recurrence formula produce an amount of primes and odd numbers having very few prime factors: the first 150 terms of the sequence produced by this formula are either primes, power of primes or products of two prime factors. But then I discovered easily formulas even more interesting, for instance a(0) = 13, a(n) = a(n–1) + 10 if n is odd and a(n) = a(n-1) + 80, if n is even (which produces 16 primes in first 20 terms!). Because what seems to matter in order to generate primes for such a recurrent defined formula a(0) = 13, a(n) = a(n–1) + x if n is odd and as a(n) = a(n-1) + y, if n is even, is that x + y to be a multiple of 30 (probably the choice of the first term doesn’t matter either but I like the number 13).
Category: Number Theory

 viXra:1503.0213 [pdf] submitted on 2015-03-27 11:14:09

### Two Conjectures on Squares of Primes, Involving Twin Primes and Pairs of Primes p, q, Where Q=p+4

Authors: Marius Coman

In this paper I make a conjecture which states that there exist an infinity of squares of primes that can be written as p + q + 13, where p and q are twin primes, also a conjecture that there exist an infinity of squares of primes that can be written as 3*q - p - 1, where p and q are primes and q = p + 4.
Category: Number Theory

 viXra:1503.0208 [pdf] submitted on 2015-03-27 07:37:51

### Seven Conjectures on the Triplets of Primes p, q, R Where Q=p+4 and R=p+6

Authors: Marius Coman

In this paper I make seven conjectures on the triplets of primes [p, q, r], where q = p + 4 and r = p + 6, conjectures involving primes, squares of primes, c-primes, m-primes, c-composites and m-composites (the last four notions are defined in previous papers, see for instance the paper “Conjecture that states that any Carmichael number is a cm-composite”.
Category: Number Theory

 viXra:1503.0207 [pdf] submitted on 2015-03-27 09:13:43

### Two Conjectures on Squares of Primes Involving the Sum of Consecutive Primes

Authors: Marius Coman

In this paper I make a conjecture which states that there exist an infinity of squares of primes of the form 6*k - 1 that can be written as a sum of two consecutive primes plus one and also a conjecture that states that the sequence of the partial sums of odd primes contains an infinity of terms which are squares of primes of the form 6*k + 1.
Category: Number Theory

 viXra:1503.0161 [pdf] submitted on 2015-03-21 18:13:55

### A Simple Algorithm to Express Any Odd Composite Number that is a Product of K-Primes not Necessarily Distinct as a Sum of Exactly K Unequal Terms

Authors: Prashanth R. Rao

Abstract: If N is an odd composite number that can be written as a product of k-primes not necessarily distinct, then we have devised a simple algorithm that would allow us to express N as the sum of exactly k terms all distinct derived using its prime factors.
Category: Number Theory

 viXra:1503.0134 [pdf] submitted on 2015-03-16 11:18:29

### Conjecture that States That, Beside Few Definable Exceptions, Poulet Numbers Are Either C-Primes, M-Primes, C-Composites or M-Composites

Authors: Marius Coman

In one of my previous paper, “Conjecture that states than any Carmichael number is a cm-composite”, I defined the notions of c-prime, m-prime, cm-prime, c-composite, m-composite and cm-composite. I conjecture that all Poulet numbers but a set of few definable exceptions belong to one of these six sets of numbers.
Category: Number Theory

 viXra:1503.0125 [pdf] submitted on 2015-03-16 04:35:07

### Formula for Generating C-Primes and M-Primes Based on Squares of Primes

Authors: Marius Coman

In this paper I present a formula, based on squares of primes, which seems to generate a large amount of c-primes and m-primes (I defined the notions of c-primes and m-primes in my previous paper “Conjecture that states that any Carmichael number is a cm-composite”).
Category: Number Theory

 viXra:1503.0123 [pdf] submitted on 2015-03-15 12:53:57

### Two Formulas Based on C-Chameleonic Numbers Which Conducts to C-Primes and the Notion of C-Chameleonic Number

Authors: Marius Coman

In one of my previous papers I defined chameleonic numbers as the positive composite squarefree integers C not divisible by 2, 3 or 5 having the property that the absolute value of the number P – d + 1 is always a prime or a power of a prime, where d is one of the prime factors of C and P is the product of all prime factors of C but d. In this paper I revise this definition, I introduce the notions of c-chameleonic numbers and m-chameleonic numbers and I show few interesting connections between c-primes and c-chameleonic numbers (I defined the notions of a c-prime in my paper “Conjecture that states that any Carmichael number is a cm-composite”).
Category: Number Theory

 viXra:1503.0119 [pdf] replaced on 2015-03-17 03:05:30

### Formula Based on Squares of Primes Which Conducts to Primes, C-Primes and M-Primes

Authors: Marius Coman

In my previous paper “Conjecture that states that any Carmichael number is a cm-composite” I defined the notions of c-prime, m-prime and cm-prime, odd positive integers that can be either primes either semiprimes having certain properties, and also the notions of c-composites, m-composites and cm-composites. In this paper I present a formula based on squares of primes which seems to lead often to primes, c-primes, m-primes and cm-primes.
Category: Number Theory

 viXra:1503.0117 [pdf] submitted on 2015-03-14 13:41:19

### Investigations on the Chebyshev Functions II: Discrete Formulas for First and Second Functions

Authors: Edigles Guedes

In present article, we create discrete formulas for first and second Chebyshev functions.
Category: Number Theory

 viXra:1503.0114 [pdf] replaced on 2015-03-17 03:00:34

### Conjecture that States that Any Carmichael Number is a cm-Composite

Authors: Marius Coman

In two of my previous papers I defined the notions of c-prime respectively m-prime. In this paper I will define the notion of cm-prime and the notions of c-composite, m-composite and cm-composite and I will conjecture that any Carmichael number is a cm-composite.
Category: Number Theory

 viXra:1503.0112 [pdf] submitted on 2015-03-14 09:38:18

### Operation Based on Multiples of Three and Concatenation for Obtaining Primes and M-Primes and the Definition of a M-Prime

Authors: Marius Coman

In this paper I show how, concatenating to the right the multiples of 3 with the digit 1, obtaining the number m, respectively with the number 11, obtaining the number n, by the simple operation n – m + 1, under the condition that both m and n are primes, is obtained often (I conjecture that always) a prime or a composite r = p(1)*p(2)*..., where p(1), p(2), ... are the prime factors of r, which have the following property: there exist p(k) and p(h), where p(k) is the product of some distinct prime factors of r and p(h) the product of the other distinct prime factors such that the number p(k) + p(h) – 1 is m-prime and I also define a m-prime.
Category: Number Theory

 viXra:1503.0110 [pdf] submitted on 2015-03-14 06:42:24

### Operation Based on Squares of Primes for Obtaining Twin Primes and Twin C-Primes and the Definition of a C-Prime

Authors: Marius Coman

In this paper I show how, concatenating to the right the squares of primes with the digit 1, are obtained primes or composites n = p(1)*p(2)*...*p(m), where p(1), p(2), ..., p(m) are the prime factors of n, which seems to have often (I conjecture that always) the following property: there exist p(k) and p(h), where p(k) is the product of some distinct prime factors of n and p(h) the product of the other distinct prime factors such that the numbers p(k) + p(h) ± 1 are twin primes or twin c-primes and I also define the notion of a c-prime.
Category: Number Theory

 viXra:1503.0094 [pdf] submitted on 2015-03-12 15:05:58

### A Formula that Produces from Any Prime P of the Form 11+30k Probably an Infinity of Semiprimes qr Such that R+q=30h

Authors: Marius Coman

In this paper I make a conjecture which states that from any prime p of the form 11 + 30*k can be obtained, through a certain formula, an infinity of semiprimes q*r such that r + q = 30*m, where m non-null positive integer.
Category: Number Theory

 viXra:1503.0093 [pdf] submitted on 2015-03-12 16:09:51

### Consecutive Natural Numbers Constant

Authors: Andrea Pignataro

The goal of this paper is to demonstrate that there exists a constant, a supposedly irrational and transcendental number, that relates all consecutive natural numbers n (taken from 1) when mutually divided as (n+1)/n and n/(n+1).
Category: Number Theory

 viXra:1503.0089 [pdf] replaced on 2015-03-12 14:34:56

### Formula Involving Primorials that Produces from Any Prime P Probably an Infinity of Semiprimes qr Such that R+q-1=np

Authors: Marius Coman

In this paper I make a conjecture involving primorials which states that from any odd prime p can be obtained, through a certain formula, an infinity of semiprimes q*r such that r + q - 1 = n*p, where n non-null positive integer.
Category: Number Theory

 viXra:1503.0083 [pdf] submitted on 2015-03-11 22:14:41

### A Proof of the Abc Conjecture (After Third Modification)

Authors: Zhang Tianshu

First, we get rid of three kinds from A+B=C according to their respective odevity and gcf (A, B, C) =1. After that, expound relations between C and raf (A, B, C) by the symmetric law of odd numbers. Finally, we have proven C ≤ Cε [raf (A, B, C)] 1+ ε in which case A+B=C and gcf (A, B, C) =1.
Category: Number Theory

 viXra:1503.0082 [pdf] submitted on 2015-03-12 03:12:47

### Condition of Ajay K PRASAD’S Proof of GOLDBACH’S Conjecture

By analyzing the recently published paper of Ajay K Prasad on Goldbach’s conjecture, I have obtained the exact solution of his paper.
Category: Number Theory

 viXra:1503.0058 [pdf] submitted on 2015-03-08 18:44:42

### On the Natural Logarithm Function and its Applications

Authors: Edigles Guedes

In present article, we create new integral representations for natural logarithm function, the Euler-Mascheroni constant, the natural logarithm of Riemann zeta function and the first derivative of Riemann zeta function.
Category: Number Theory

 viXra:1503.0028 [pdf] submitted on 2015-03-04 01:48:36

### Another Infinite Sequence Based on Mar Function that Abounds in Primes and Semiprimes

Authors: Marius Coman

In one of my previous paper, namely “The mar reduced form of a natural number”, I introduced the notion of mar function, which is, essentially, nothing else than the digital root of a number, and I also presented, in another paper, a sequence based on mar function that abounds in primes. In this paper I present another sequence, based on a relation between a number and the value of its mar reduced form (of course not the intrisic one), sequence that seem also to abound in primes and semiprimes.
Category: Number Theory

 viXra:1503.0026 [pdf] submitted on 2015-03-03 16:08:31

### The Notions of Mar Constants and Smarandache Mar Constants

Authors: Marius Coman

In one of my previous paper, “The mar reduced form of a natural number”, I introduced the notion of mar function, which is, essentially, nothing else than the digital root of a number, but defined as an aritmethical function, on the operations of addition, multiplication etc. in such way that it could be used in various applications (Diophantine equations, divizibility problems and others). In this paper I present two notions, useful in Diophantine analysis of Smarandache concatenated sequences or different classes of numbers (sequences of squares, cubes, triangular numbers, polygonal numbers, Devlali numbers, Demlo numbers etc).
Category: Number Theory

 viXra:1503.0025 [pdf] submitted on 2015-03-03 16:49:58

### An Infinite Sequence Based on Mar Function that Abounds in Primes

Authors: Marius Coman