[23] **viXra:1504.0248 [pdf]**
*submitted on 2015-04-30 17:49:37*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make three conjectures regarding a certain relation between the number 4320 and the squares of primes respectively four conjectures on squares of primes involving deconcatenation.

**Category:** Number Theory

[22] **viXra:1504.0239 [pdf]**
*submitted on 2015-04-29 15:08:09*

**Authors:** Kolosov Petya

**Comments:** 7 Pages.

In this paper described some new view and properties of the power function, the main aim of the work is to enter some new ideas. Also described expansion of power function, based on done research. Expansion has like Binominal theorem view, but algorithm not same.

**Category:** Number Theory

[21] **viXra:1504.0229 [pdf]**
*submitted on 2015-04-29 03:02:32*

**Authors:** .Wenlong Du

**Comments:** 6 Pages.

This paper studies the relationship between the prime divisor and Stirling's approximation. We get prime number theorem and its corrected value. We get bound for the error of the prime number theorem. Riemann hypothesis is established.

**Category:** Number Theory

[20] **viXra:1504.0217 [pdf]**
*submitted on 2015-04-27 23:31:53*

**Authors:** Pratish R. Rao, Prashanth R. Rao

**Comments:** 3 Pages.

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. In a previous paper we conceptualized a new way of describing n!, using sequential cuts to an imaginary circle and derived a well known result. In this paper we use the same intuitive approach but reverse the cutting strategy by starting with n-cuts to the circle. We observe that this method leads us to estimate the approximate sum of an infinite convergent series involving factorials as unity.

**Category:** Number Theory

[19] **viXra:1504.0195 [pdf]**
*replaced on 2015-05-11 10:58:53*

**Authors:** Hashem Sazegar

**Comments:** 6 Pages.

Given the fact that the Gilbreath's Conjecture has been a major topic of research in Aritmatic progression for well over a Century,and as bellow: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 1 2 2 4 2 4 2 4 6 2 6 4 2 4 6 6 2 1 0 2 2 2 2 2 2 4 4 2 2 2 2 0 4 1 2 0 0 0 0 0 2 0 2 0 0 0 2 4 1 2 0 0 0 0 2 2 2 2 0 0 2 2 1 2 0 0 0 2 0 0 0 2 0 2 0 1 2 0 0 2 2 0 0 2 2 2 2 1 2 0 2 0 2 0 2 0 0 0 1 2 2 2 2 2 2 2 0 0 1 0 0 0 0 0 0 2 0 1 0 0 0 0 0 2 2 1 0 0 0 0 2 0 1 0 0 0 2 2 1 0 0 2 0 1 0 2 2 1 2 0 1 2 1 The Gilbreath’s conjecture in a way as easy and comprehensive as possible. He proposed that these differences, when calculated repetitively and left as bsolute values, would always result in a row of numbers beginning with 1,In this paper we bring elementary proof for this conjecture.

**Category:** Number Theory

[18] **viXra:1504.0164 [pdf]**
*submitted on 2015-04-20 12:46:56*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In Addenda to my previous paper “On the special relation between the numbers of the form 505+1008k and the squares of primes” I defined the notions of c/m-integers and g/s-integers and showed some of their applications. In a previous paper I conjectured that, beside few definable exceptions, the Fermat pseudoprimes to base 2 with two prime factors are c/m-primes, but I haven’t defined the “definable exceptions”. However, in this paper I confirm one of my constant beliefs, namely that the relations between the two prime factors of a 2-Poulet number are definable without exceptions and I make a conjecture about a generic formula of these numbers, namely that the most of them are s-primes and the exceptions must satisfy a given Diophantine equation.

**Category:** Number Theory

[17] **viXra:1504.0159 [pdf]**
*submitted on 2015-04-20 08:40:39*

**Authors:** Marius Coman

**Comments:** 6 Pages.

The study of the power of primes was for me a constant probably since I first encounter “Fermat’s last theorem”. The desire to find numbers with special properties, as is, say, Hardy-Ramanujan number, was another constant. In this paper I present a class of numbers, i.e. the numbers of the form n = 505 + 1008*k, where k positive integer, which, despite the fact that they don’t seem to be, prima facie, “special”, seem to have a strong connection with the powers of primes: for a lot of values of k (I show in this paper that for nine from the first twelve and I conjecture that for an infinity of the values of k), there exist p and q primes such that p^2 – q^2 + 1 = n. The special nature of the numbers of the form 505 + 1008*k is also highlight by the fact that they are (all the first twelve of them, as much I checked) primes or g/s-integers or c/m-integers (I define in Addenda to this paper the two new notions mentioned).

**Category:** Number Theory

[16] **viXra:1504.0153 [pdf]**
*submitted on 2015-04-19 19:18:27*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I present the observation that the formula p^2 – q^2 + 1, where p and q are primes with the special property that the sums of their digits are equal, leads often to primes (of course, having only the digital root equal to 1 due to the property of p and q to have same digital sum implicitly same digital root) or to special kinds of semiprimes: some of them named by me, in few previous papers, c/m-primes, and some of them named by me, in this paper, g-primes respectively s-primes. Note that I chose the names “g/s-primes” instead “g/s-semiprimes” not to exist confusion with the names “g/s-composites”, which I intend to define and use in further papers.

**Category:** Number Theory

[15] **viXra:1504.0151 [pdf]**
*submitted on 2015-04-20 03:29:27*

**Authors:** JinHua Fei

**Comments:** 14 Pages.

This paper use the methods of References [1], we got a good upper bound of exceptional real zero of the Dirichlet L- function.

**Category:** Number Theory

[14] **viXra:1504.0149 [pdf]**
*submitted on 2015-04-19 13:56:45*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I present the following observation: concatenating to the right the number p^2 – 1, where p is a prime of the form 6*k – 1, with the digit 1, is often obtained a prime or a c-prime; also, concatenating to the right the number p^2 – 1, where p is a prime of the form 6*k + 1, with the digit 1, is often obtained a prime or a m-prime.

**Category:** Number Theory

[13] **viXra:1504.0143 [pdf]**
*submitted on 2015-04-18 17:04:29*

**Authors:** R.J. Tang

**Comments:** 2 Pages.

There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled ‘earth-centric’ idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably then it seems natural numbers ought to have an equal standing as any other numbers, irrational, complex, or even numbers yet to be invented.
Is there any physical underlying reason for natural numbers’ special status? Or are the natural numbers just a convenient way for people to count and were invented by macro intelligent beings like us?
Since all natural numbers are mere derivatives of the number ‘1’, so let’s look closely at what this number one really means. There are two broad meaning of the number one, corresponding to different mental construct to define ‘1’. First it registers a definitive state of some physical attribute, such as ‘presence’ or ‘non-presence’. We can find its application in information theory, statistical physics, counting and etc. The second interpretation of number one is that it denotes the ‘wholeness’ of an entity. Yet another definition arises from set theory. Still another arises from the order of which one element in a sequence related to the other elements. Remarkably the concept of natural numbers can come from many different constructs, just as remarkable that natural numbers come from many different domains in the physical world.
In physics, natural numbers virtually have no sacred places prior to the establishment of quantum mechanics. After all, we don’t need any natural numbers in our gravity functions or the Maxwell electro-magnetic wave functions. Some sharp observers would argue that the ‘R squared’ contains a natural number 2. However on close examination the number 2 is merely a mathematical notation for a number multiplying by itself, and it has no actual physical corresponding object or attribute. The fact that there is no natural number in the formulae represents the idea that time-space is fundamentally smooth. For instance there is no such law in physics that requires 7 bodies (non quantum mechanical) to form a system in equilibrium.
Had we obtained calculus capability before we can count our fingers, we probably would have been more familiar with the number e than 1-2-3. We might have used e/2.718 to represent the mundane singletons. There is no logical requirement that we couldn’t or shouldn’t do it. It is all due to the accident that people happened to need to count their fingers earlier than the invention of calculus. There is no physical evidence that the number ‘2’ is more significant than the any other numbers in the natural world.
However with the standard model of quantum mechanics, energy is quantized, that is, it can only take natural numbers. This idea profoundly altered the status of natural numbers in physics and is a direct contradictory of the notion of ‘no center in the universe’ principal. In this sense it is far more unorthodox than the two relativity theories combined because the latter in fact enhance the ‘no center in the universe’ law. Why does the quantum have to be integer times of a certain energy level, and not an irrational number like square root of 17, or the quantity e? Does it really mean there are aristocrats in the number world, where some are nobler than others? Were the ancient Greek mathematicians right after all, who worshiped the sacredness of natural numbers and even threw the irrational number discoverer into the sea?
From this standpoint we can almost say that quantum theory has some bad taste among all branches of natural science.
Before the quantum theory got its germination, actually people should have noticed the unusual role natural numbers play in rudimentary chemistry. For instance, why two hydrogen atoms and not five, are supposed to combine with one oxygen atom to form a water molecule? If scientists are sharp enough back then they ought to be able to be alarmed by the oddity underlying the strange status of natural numbers. It could almost be an indirect way to deduce the quantized nature of electrons.
Fundamentally if natural numbers indeed play a very unusual role in nature, then nature resembles a codebook not just from a coarse analogy standpoint. It is the ultimate codebook filled with rules for a limited number of building block codes. The DNA code is an excellent example.
If it’s a codebook, inevitably it takes us to surmise if information itself is the ultimate being in the universe. It is probably not electrons, strings, quarks or whatever ‘entities’ people have claimed. It is the information that is the only tangible and verifiable entity out there. Everything else is a mirage or manifestation of some underlying information, the codebook.
In this sense physics has somewhat gone awry by focusing on the wrong things, the ‘attributes’ such as momentum, position and etc. Instead, information is what contemporary physicists talk about and experiment with. Otherwise, the physicists would have no right to laugh at the medieval scholars who based their intellectual work on the measurement of the distance between a subject and God’s throne.
The nature has revealed her latest hand of cards to us. It looks like it’s the final hand but no one can be sure of course.

**Category:** Number Theory

[12] **viXra:1504.0141 [pdf]**
*submitted on 2015-04-18 17:21:12*

**Authors:** Marius Coman

**Comments:** 8 Pages.

In this paper I define the “Smarandache-Coman sequences” as “all the sequences of primes obtained from the Smarandache concatenated sequences using basic arithmetical operations between the terms of such a sequence, like for instance the sum or the difference between two consecutive terms plus or minus a fixed positive integer, the partial sums, any other possible basic operations between terms like a(n) + a(n+2) – a(n+1), or on a term like a(n) + S(a(n)), where S(a(n)) is the sum of the digits of the term a(n) etc.”, and I also present few such sequences.

**Category:** Number Theory

[11] **viXra:1504.0140 [pdf]**
*submitted on 2015-04-18 21:01:02*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that there exist an infinity of primes of the form 2*p^2 – p – 2, where p is a Sophie Germain prime, I show first few terms from this set and few larger ones.

**Category:** Number Theory

[10] **viXra:1504.0138 [pdf]**
*submitted on 2015-04-19 02:33:16*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there exist an infinity of squares of primes of the form 109 + 420*k, also an infinity of primes of this form and an infinity of semiprimes p*g of this form such that q – p = 60.

**Category:** Number Theory

[9] **viXra:1504.0121 [pdf]**
*submitted on 2015-04-15 11:13:03*

**Authors:** Th. Guyer

**Comments:** 8 Pages.

The nicest possible ABC Formula in Mathematic.

**Category:** Number Theory

[8] **viXra:1504.0080 [pdf]**
*submitted on 2015-04-09 23:02:36*

**Authors:** Marius Coman

**Comments:** 5 Pages.

In this paper I show that many Smarandache concatenated sequences, well known for the common feature that contain very few terms which are primes (I present here The concatenated square sequence, The concatenated cubic sequence, The sequence of triangular numbers, The symmetric numbers sequence, The antisymmetric numbers sequence, The mirror sequence, The “n concatenated n times” sequence) contain (or conduct to, through basic operations between terms) very many numbers which are cm-integers (c-primes, m-primes, c-composites, m-composites).

**Category:** Number Theory

[7] **viXra:1504.0077 [pdf]**
*replaced on 2015-05-27 11:50:50*

**Authors:** Marius Coman

**Comments:** 96 Pages. Published by Education Publishing, USA. Copyright 2015 by Marius Coman.

In three of my previous published books, namely “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function” and “Two hundred and thirteen conjectures on primes”, I showed my passion for conjectures on sequences of integers. In spite the fact that some mathematicians stubbornly understand mathematics as being just the science of solving and proving, my books of conjectures have been well received by many enthusiasts of elementary number theory, which gave me confidence to continue in this direction. Part One of this book brings together papers regarding conjectures on primes, twin primes, squares of primes, semiprimes, different types of pairs or triplets of primes, recurrent sequences, sequences of integers created through concatenation and other sequences of integers related to primes. Part Two of this book brings together several articles which present the notions of c-primes, m-primes, c-composites and m-composites (c/m-integers), also the notions of g-primes, s-primes, g-composites and s-composites (g/s-integers) and show some of the applications of these notions (because this is not a book structured unitary from the beginning but a book of collected papers, I defined the notions mentioned in various papers, but the best definition of them can be found in Addenda to the paper numbered tweny-nine), in the study of the squares of primes, Fermat pseudoprimes and generally in Diophantine analysis. Part Three of this book presents the notions of “Coman constants” and “Smarandache-Coman constants”, useful to highlight the periodicity of some infinite sequences of positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers), respectively in the analysis of Smarandache concatenated sequences. Part Four of this book presents the notion of Smarandache-Coman sequences, id est sequences of primes formed through different arithmetical operations on the terms of Smarandache concatenated sequences. Part Five of this book presents the notion of Smarandache-Coman function, a function based on the well known Smarandache function which seems to be particularly interesting: beside other characteristics, it seems to have as values all the prime numbers and, more than that, they seem to appear, leaving aside the non-prime values, in natural order. This book of collected papers seeks to expand the knowledge on some well known classes of numbers and also to define new classes of primes or classes of integers directly related to primes.

**Category:** Number Theory

[6] **viXra:1504.0069 [pdf]**
*replaced on 2015-05-28 05:47:56*

**Authors:** Marius Coman

**Comments:** 5 Pages.

In two previous papers I presented the notion of “Coman constant” and showed how could highlight the periodicity of some infinite sequences of integers. In this paper I present the notion of “Smarandache-Coman constant”, useful in Diophantine analysis of Smarandache concatenated sequences.

**Category:** Number Theory

[5] **viXra:1504.0068 [pdf]**
*replaced on 2015-05-28 04:59:43*

**Authors:** Marius Coman

**Comments:** 1 Page.

In a previous paper I defined the notion of “Coman constant”, based on the digital root of a number and useful to highlight the periodicity of some infinite sequences of non-null positive integers. In this paper I present two sequences that, in spite the fact that their terms can have only few values for digital root, don’t seem to have a periodicity, in other words don’t seem to be characterized by a Coman constant.

**Category:** Number Theory

[4] **viXra:1504.0064 [pdf]**
*replaced on 2015-05-28 04:16:00*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present a notion based on the digital root of a number, namely “Coman constant”, that highlights the periodicity of some infinite sequences of non-null positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers etc).

**Category:** Number Theory

[3] **viXra:1504.0060 [pdf]**
*submitted on 2015-04-08 09:31:39*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In a previous paper I presented a type of numbers which seem to be often m-primes or m-composites (the numbers of the form 1nn...nn1, where n is a digit or a group of digits, repetead by an odd number of times). In this paper I present a type of numbers which seem to be often c-primes or c-composites. These are the numbers of the form 1abc (formed through concatenation, not the product 1*a*b*c), where a, b, c are three primes such that b = a + 6 and c = b + 6.

**Category:** Number Theory

[2] **viXra:1504.0056 [pdf]**
*submitted on 2015-04-08 04:42:08*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In previous papers I presented already few types of numbers which conduct through concatenation often to cm-integers. In this paper I present a type of numbers which seem to be often m-primes or m-composites. These are the numbers of the form 1nn...nn1 (in all of my papers I understand through a number abc the number where a, b, c are digits and through the number a*b*c the product of a, b, c), where n is a digit or a group of digits, repetead by an odd number of times.

**Category:** Number Theory

[1] **viXra:1504.0002 [pdf]**
*submitted on 2015-04-01 02:48:47*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In a previous paper I presented 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. Because the study of Fermat pseudoprimes is a constant passion for me, I observed that in many cases they have a behaviour which is similar with that of the squares of primes. Therefore, I checked if the property mentioned above applies to these numbers too. Indeed, concatenating two squares of primes, are often obtained semiprimes which are either c-primes, m-primes or cm-primes. Using just the squares of the first 13 primes greater than or equal to 7 are obtained not less then: 6 semiprimes which are c-primes, 31 semiprimes which are m-primes and 15 semiprimes which are cm-primes.

**Category:** Number Theory