[55] **viXra:1604.0391 [pdf]**
*submitted on 2016-04-30 19:01:30*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

1) The function of the prime number, the wave pattern of the zeta function are equivalent with the friendship of the sin wave of the having many kinds, cos wave by Fourier transform.
2) There are a sin wave, a cos wave and the point of intersection (0 points) with the axis on an axis. The point of intersection that left the axis (straight line) does not exist.
3) The top of the pulsation wave pattern has deep relation to prime number, mass, quark, 0 points, ...
4) Circle that is a trace of the circular motion is a quantum-mechanical autocoupling operator, an L meat operator.
5) Circle to assume a prime number a radius is a trace of the tops of the material wave of the pulsation principle, and Japanese yen and the point of intersection with the axis are zero points.
(6) As for the product indication equation of the oiler, a radius is a circle of 1 integral multiple.
(7) As for the quantum mechanics, a radius is a circle of the integral multiples of "h".
(8) As for the mass of the quantum-mechanical mass, a radius is a circle of the integral multiples of "m".
9) Circle of the prime number, all the circular center have a radius on 1/2 line. (Lehman expectation)
10) The top (prime number, mass) of the pulsation wave pattern becomes the straight line by レッジェ trace graph.
11) As for the レッジェ trace graph, square of the mass becomes the straight line.
12) 1/2h is the important fixed number to often come up in a quantum-mechanical equation.
13) 1/2 of the Lehman expectation is the straight line that is the mystery that 0 points form a line of the infinite unit.
14) A sine wave by the Fourier transform, a cosine wave and the point of intersection of the 1/2 straight line are 0 points.
15) The eddy of the solution (material wave) of the Schrodinger equation is equivalent with circular motion.

**Category:** Number Theory

[54] **viXra:1604.0386 [pdf]**
*submitted on 2016-04-30 12:49:08*

**Authors:** Terubumi Honjou

**Comments:** 3 Pages.

1) The function of the prime number, the wave pattern of the zeta function are equivalent with the friendship of the sin wave of the having many kinds, cos wave by Fourier transform.
2) There are a sin wave, a cos wave and the point of intersection (0 points) with the axis on an axis. The point of intersection that left the axis (straight line) does not exist.
3) The top of the pulsation wave pattern has deep relation to prime number, mass, quark, 0 points, ...
4) Circle that is a trace of the circular motion is a quantum-mechanical autocoupling operator, an L meat operator.
5) Circle to assume a prime number a radius is a trace of the tops of the material wave of the pulsation principle, and Japanese yen and the point of intersection with the axis are zero points.
(6) As for the product indication equation of the oiler, a radius is a circle of 1 integral multiple.
(7) As for the quantum mechanics, a radius is a circle of the integral multiples of "h".
(8) As for the mass of the quantum-mechanical mass, a radius is a circle of the integral multiples of "m".
9) Circle of the prime number, all the circular center have a radius on 1/2 line. (Lehman expectation)
10) The top (prime number, mass) of the pulsation wave pattern becomes the straight line by レッジェ trace graph.
11) As for the レッジェ trace graph, square of the mass becomes the straight line.
12) 1/2h is the important fixed number to often come up in a quantum-mechanical equation.
13) 1/2 of the Lehman expectation is the straight line that is the mystery that 0 points form a line of the infinite unit.
14) A sine wave by the Fourier transform, a cosine wave and the point of intersection of the 1/2 straight line are 0 points.
15) The eddy of the solution (material wave) of the Schrodinger equation is equivalent with circular motion.

**Category:** Number Theory

[53] **viXra:1604.0357 [pdf]**
*submitted on 2016-04-26 20:35:55*

**Authors:** Terubumi Honjou

**Comments:** 9 Pages.

0 points and the distribution map of the prime number of the zeta function are expressed by a complex number coordinate.
The figure of elementary particle pulsation principle energy wave pattern is expressed by a complex number coordinate.
The figure of fusion synchronized a straight line and the horizon of the figure of elementary particle pulsation principle energy wave pattern where 0 points formed a line and fused with neither.
Four dimensions of lower domains express space on the horizon, and the prime number in the top of the material wave pulsates by a turn of the four-dimensional space as a top of the waves.
There are all the non-self-evident zero points of the zeta function on the horizon (three-dimensional space) of the figure of elementary particle pulsation principle energy wave pattern and is real part 1/2.
It fuses in a complex number coordinate and a figure of elementary particle pulsation principle energy wave pattern (complex number coordinate) that 0 points of a prime number and the zeta function present.

**Category:** Number Theory

[52] **viXra:1604.0345 [pdf]**
*submitted on 2016-04-26 03:04:31*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

A humble attempt is made at proving the twin prime conjecture. An argument involving a form derived from a set of characteristic equations and a parity argument is used in the proof.

**Category:** Number Theory

[51] **viXra:1604.0344 [pdf]**
*submitted on 2016-04-25 23:39:15*

**Authors:** Anthony J. Browne

**Comments:** 2 Pages.

Use of the harmonic numbers to create congruencies is discussed. Interesting relations to known congruencies are shown.

**Category:** Number Theory

[50] **viXra:1604.0342 [pdf]**
*submitted on 2016-04-25 08:41:45*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we diagnose the critical line.

**Category:** Number Theory

[49] **viXra:1604.0337 [pdf]**
*submitted on 2016-04-24 19:45:00*

**Authors:** Terubumi Honjou

**Comments:** 6 Pages.

A prime number was the top of the material wave in the theory physics and, in "a challenge to Lehman expectation which I announced in YOUTUBE for 2,012 years ," expected it so that a material wave and the point of intersection of the figure of pulsation energy wave pattern were non-self-evident zero points of the Lehman expectation.
I tried a prime number and the conversion of the equation indicating the connection with π (Circle) that Euler discovered recently.
As a result of the right side sprinkling π to a denominator, molecules of the left side of a go board of the product formula of π 2, and having converted it into the equation of the area of Japanese yen, a radius got an equation of Circle of the prime number. This suggests that expectation of 2012 saying that it is a prime number on the top of the material wave of the figure of pulsation energy wave pattern was right.

**Category:** Number Theory

[48] **viXra:1604.0327 [pdf]**
*replaced on 2017-04-10 02:53:47*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we analyze the behavior of prime numbers.

**Category:** Number Theory

[47] **viXra:1604.0324 [pdf]**
*replaced on 2016-05-20 01:33:52*

**Authors:** Anthony J. Browne

**Comments:** 5 Pages.

Approximations of square roots are discussed. A very close approximation to their decimal expansion is derived in the form of a simple fraction. Their relationship with the AKS test is also discussed.

**Category:** Number Theory

[46] **viXra:1604.0321 [pdf]**
*replaced on 2016-05-20 16:31:25*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Summing characteristic equations to find forms of theoretical functions in number theory will be discussed. Forms of many number theoretic functions will be derived. Although many may not be efficient in a computing sense for large numbers, the aim in this paper will simply be to explore what these forms are and show relationships between expressions.

**Category:** Number Theory

[45] **viXra:1604.0316 [pdf]**
*submitted on 2016-04-23 06:05:21*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The Tijdeman–Zagier conjecture, also known as Beal's conjecture, is a conjecture in number theory: –
If A^x+B^y=C^z,
Where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor. Equivalently,
There are no solutions to the above equation in positive integers A, B, C, x, y, z with A, B, and C being pairwise coprime and all of x, y, z being greater than 2.

**Category:** Number Theory

[44] **viXra:1604.0315 [pdf]**
*submitted on 2016-04-23 06:09:00*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The purpose of this paper is to show how a Pseudo Random pattern appears in Pi.The reason there are no repeating numbers in Pi is because there is a Pseudo Random process in Pi. The Pseudo Random Process causes no repeating numbers in Pi. As in the prime numbers A+B +/- 1=C there is a Pseudo Random Process in Pi. In Pi, characteristics of the Pseudo Random Process can be seen by taking the digits in Pi and doing a progression which starting with 3 take its square root. Then take the next two digits, add them up and take the square root. After progressing the patterns appear. At the 6th &7th series,11th & 12th series and 16th&17th series.

**Category:** Number Theory

[43] **viXra:1604.0295 [pdf]**
*replaced on 2016-05-01 04:58:57*

**Authors:** Jan Pavo Barukčić, Ilija Barukčić

**Comments:** 10 Pages. (C) Jan Pavo Barukčić, Münster and Ilija Barukčić, Jever, Germany, 2016.

Unfortunately, however, the relation between a finite and an infinite is not always so straightfor-ward. The infinite and the finite mutually related as sheer others are inseparable. A related point is that while the infinite is determined in its own self by the other of itself, the finite, the finite itself is determined by its own infinite. Each of both is thus far the unity of its own other and itself. The inseparability of the infinite and the finite does not mean that a transition of the finite into the infinite and vice versa is not possible. In the finite, as this negation of the infinite, we have the sat-isfaction that determinateness, alteration, limitation et cetera are not vanished, are not sublated. The finite is a finite only in its relation to its own infinite, and the infinite is only infinite in its rela-tion to its own finite. As will become apparent, the infinite as the empty beyond the finite is bur-dened by the fact that determinateness, alteration, limitation et cetera are vanished. The relation between the finite and the infinite finds its mathematical formulation in the division of one by zero. As we will see, it is +1/+0=+oo.

**Category:** Number Theory

[42] **viXra:1604.0259 [pdf]**
*submitted on 2016-04-17 14:22:16*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

In the simplest terms here is a counterexample to Fermat's Last Theorem and s solution to Beal's Conjecture. Dr. Andrew Wiles proved Fermat's Last Theorem but I think my solution below is an example for n=3 if allowed. It also satisfies Beal's conjecture and is a counterexample to Fermat’s Last Theorem.

**Category:** Number Theory

[41] **viXra:1604.0258 [pdf]**
*submitted on 2016-04-17 14:25:44*

**Authors:** Ricardo Gil

**Comments:** 11 Pages.

The purpose of this paper is to show that prime numbers are structured in a Pseudo Random manner. Like the Fibonacci or the Lucas sequence, the prime number sequence is a sequence in which 2 primes when added together (+ or -1) makes the next prime. The sum of the two primes, A+B(+or-1)=C dictates the next prime number in the sequence. Goldbach's conjecture is that every even integer is the sum of two primes, A+B(+or-1)=C is two primes +or-1 make up another prime and dictates the gap between the primes. Progressing along the prime number line is similar to the Fibonacci sequence and the Lucas sequence. In a sense the A+B(+or-1)=C is a sequence but for prime numbers. In the Pseudo Random Prime Number Sequence or A+B(+or -1)=c, 5+3-1=7 and 7+5-1=11. The "A" side progresses or dictates the progression and in the progression or sequence if 5 were used in 5+5+1=11 instead of 7+5-11 it would be out of order in the progression sequence.

**Category:** Number Theory

[40] **viXra:1604.0257 [pdf]**
*submitted on 2016-04-17 14:29:48*

**Authors:** Ricardo Gil

**Comments:** 8 Pages.

The purpose of this paper is to show how Riemann Zeros and Prime Numbers synchronize at N+6 and why there are no Riemann Zeros smaller than 14.

**Category:** Number Theory

[39] **viXra:1604.0255 [pdf]**
*submitted on 2016-04-17 16:02:09*

**Authors:** Ricardo Gil

**Comments:** 2 Pages.

The purpose of this paper is to provide algorithm that is 4 lines of code and that finds P & Q when N is given. It will work for RSA-1024 & RSA-2018 if the computer can float large numbers in PyCharm or Python.

**Category:** Number Theory

[38] **viXra:1604.0243 [pdf]**
*submitted on 2016-04-15 23:18:19*

**Authors:** Marius Coman

**Comments:** 150 Pages. Published by Education Publishing, USA. Copyright 2016 by Marius Coman.

The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (S sequences) but also on “Smarandache-Coman sequences of primes” (SC sequences), defined by the author as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation”: the SC sequences presented in this book are related, of course, to concatenation, but in three different ways: the S sequence is obtained by the method of concatenation but the operation applied on its terms is some other arithmetical operation; the S sequence is not obtained by the method of concatenation but the operation applied on its terms is concatenation, or both S sequence and SC sequence are using the method of concatenation. Part Two of this book, “Sequences of primes obtained by the method of concatenation”, brings together 51 articles which aim, using the mentioned method, to highlight sequences of numbers that are rich in primes or are liable to lead to large primes. The method of concatenation is applied to different classes of numbers, e.g. squares of primes, Poulet numbers, triangular numbers, reversible primes, twin primes, repdigits, factorials, primorials, in order to obtain sequences, possible infinite, of primes. Part Two of this book also contains a paper which lists a number of 33 sequences of primes obtained by the method of concatenation, sequences presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”.

**Category:** Number Theory

[37] **viXra:1604.0241 [pdf]**
*replaced on 2016-05-04 06:40:28*

**Authors:** F. Portela

**Comments:** 10 Pages.

We revisit a 25 years old approach of the twin primes conjecture, and after a simple adjustment, push it forward by means of simple sieves to an important conclusion.

**Category:** Number Theory

[36] **viXra:1604.0227 [pdf]**
*submitted on 2016-04-13 19:31:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) There exist an infinity of primes obtained concatenating two consecutive primorial numbers and adding 1 to the resulted number; example: concatenating the tenth and eleventh primorials then adding 1 is obtained the prime 6469693230200560490131; (II) There exist an infinity of primes obtained concatenating two consecutive primorial numbers and subtracting 1 from the resulted number; example: concatenating the ninth and tenth primorials then subtracting 1 is obtained the prime 2230928706469693229.

**Category:** Number Theory

[35] **viXra:1604.0226 [pdf]**
*submitted on 2016-04-13 19:32:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: let [p, q] be a pair of sexy primes (q = p + 6); then: (I) there exist an infinity of primes obtained concatenating 30*p with 30*q and adding 1 to the resulted number; example: for [p, q] = [23, 29], the number 690871 is prime; (II) there exist an infinity of primes obtained concatenating 30*p with 30*q and subtracting 1 from the resulted number; example: for [p, q] = [23, 29], the number 690869 is prime; (III) there exist an infinity of pairs of twin primes obtained concatenating 30*p with 30*q and adding/subtracting 1 from the resulted number; example: for [p, q] = [101, 107], the numbers 30303209 and 30303211 are primes.

**Category:** Number Theory

[34] **viXra:1604.0219 [pdf]**
*submitted on 2016-04-13 11:31:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: (I) There exist an infinity of primes q obtained deconcatenating to the right with 1 the Poulet numbers of the form 30*k + 1 then subtracting 1 (example: from P = 997465414921 is obtained q = 99746541491); (II) There exist an infinity of primes q obtained deconcatenating to the right with 1 the Poulet numbers of the form 30*k + 1 then adding 1 (example: from P = 996881835961 is obtained q = 99688183597); (III) There exist an infinity of primes q obtained deconcatenating to the right with 01 the Poulet numbers of the form 300*k + 1 then subtracting 1 (example: from P = 999666754801 is obtained q = 9996667547).

**Category:** Number Theory

[33] **viXra:1604.0218 [pdf]**
*submitted on 2016-04-13 11:33:58*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) For any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the factorial numbers and adding 1 to the resulted number; (II) for any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the factorial numbers and subtracting 1 from the resulted number. It is worth noting the pair of twin primes having 49 digits each obtained for k = 9: (5502622159812088949850305428800254892961651752959,
5502622159812088949850305428800254892961651752961).

**Category:** Number Theory

[32] **viXra:1604.0217 [pdf]**
*submitted on 2016-04-13 11:35:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) For any k positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the fibonorial numbers and adding 1 to the resulted number; (II) for any k non-null positive integer there exist a sequence having an infinity of prime terms obtained deconcatenating to the right with a group with k digits of 0 the fibonorial numbers and subtracting 1 from the resulted number. It is known that fibonorial numbers are defined as the products of nonzero Fibonacci numbers.

**Category:** Number Theory

[31] **viXra:1604.0216 [pdf]**
*submitted on 2016-04-13 08:42:42*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

In this note we show some solutions of the equation 4xz=4+y*y , and a relation with the constant pi

**Category:** Number Theory

[30] **viXra:1604.0200 [pdf]**
*submitted on 2016-04-12 07:24:12*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we diagnose the critical line.

**Category:** Number Theory

[29] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-15 16:12:25*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Inspection of the conjecture shows that it is a phenomenological model. Thus, a solution could be found through regression analysis. Numerical verification is possible through nominative determinism/visibility theory. By adding adaptive learning (AL) to the model, the model admits an important time variation in beliefs, which would be ruled out under rational expectations. Entropy can be given from a detailed molecular analysis of the system. In summary, perception consists of the selection, organization, and interpretation of stimuli. These factors affect the conduct of work. We include two inequalities on the log-volume change associated to appropriately chosen deformations.

**Category:** Number Theory

[28] **viXra:1604.0181 [pdf]**
*submitted on 2016-04-12 02:23:01*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: for any positive integer n > 1 there exist a sequence having an infinity of prime terms p, obtained concatenating to the right with 1 the terms of the sequence of concatenated n-th powers. For n = 2 the primes p are obtained concatenating with 1 to the right the terms of the Smarandache concatenated squares sequence; for n = 3 the primes p are obtained concatenating with 1 to the right the terms of the Smarandache concatenated cubic sequence.

**Category:** Number Theory

[27] **viXra:1604.0180 [pdf]**
*submitted on 2016-04-12 02:24:28*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I state the following two conjectures: (I) If p is a prime which admits deconcatenation in two primes p1 and p2, both of the form 6*k – 1, then there exist an infinity of primes q obtained concatenating q1 with q2, where q1 = 30*n – p1, q2 = 30*n – p2 and n positive integer; (II) If p is a prime which admits deconcatenation in two primes p1 and p2, both of the form 6*k + 1, then there exist an infinity of primes q obtained concatenating q1 with q2, where q1 = 30*n + p1, q2 = 30*n + p2 and n positive integer.

**Category:** Number Theory

[26] **viXra:1604.0179 [pdf]**
*submitted on 2016-04-12 02:25:49*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following four conjectures. Let q be the number obtained concatenating to the right with 1 the numbers p – 1, where p primes of the form 30*k + 11; then: (I) there exist an infinity of primes q; (II) there exist an infinity of semiprimes q = q1*q2, such that q2 + q1 - 1 is prime. Let q be the number obtained concatenating to the right with 1 the numbers p + 1, where p primes of the form 30*k + 11; then: (III) there exist an infinity of primes q; (IV) there exist an infinity of semiprimes q = q1*q2, such that q2 – q1 + 1 is prime.

**Category:** Number Theory

[25] **viXra:1604.0171 [pdf]**
*submitted on 2016-04-10 18:40:43*

**Authors:** Zhang Tianshu

**Comments:** 14 Pages.

In this article, the author gave a specific example to negate the ABC conjecture once and for all.

**Category:** Number Theory

[24] **viXra:1604.0169 [pdf]**
*submitted on 2016-04-10 13:18:17*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: For any positive integer n, n > 1, there exist a sequence having an infinity of prime terms obtained concatenating n consecutive numbers and then the resulting number, to the right, with 1. Examples: for n = 2, the sequence obtained this way contains the primes 10111, 15161, 18191, 21221 (...); for n = 9, the sequence obtained this way contains the primes 1234567891, 910111213141516171, 2021222324252627281, 2930313233343536371 (...).

**Category:** Number Theory

[23] **viXra:1604.0163 [pdf]**
*submitted on 2016-04-10 10:36:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: there exist an infinity of numbers q = (30*k + 7)*(60*k + 13) which admit a deconcatenation in two primes p1 and p2. Examples: for k = 2, q = 67*133 = 8911 which can be deconcatenated in p1 = 89 and p2 = 11; for k = 5, q = 157*313 = 49141 which can be deconcatenated in p1 = 491 and p2 = 41.

**Category:** Number Theory

[22] **viXra:1604.0162 [pdf]**
*submitted on 2016-04-10 10:38:56*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: For any digit from 1 to 9 there exist a sequence with an infinity of prime terms obtained concatenating to the right with 1 the partial sums of the repdigits. Examples: for repunit numbers 1, 11, 111 (...), concatenating the sum S(3) = 1 + 11 + 111 = 123 to the right with 1 is obtained 1231, prime; for repdigit numbers 3, 33, 333, 3333 (...), concatenating the sum S(4) = 3 + 33 + 333 + 3333 = 3702 to the right with 1 is obtained 37021, prime.

**Category:** Number Theory

[21] **viXra:1604.0161 [pdf]**
*submitted on 2016-04-10 06:06:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following two conjectures: (I) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “multiples of 3” sequence (defined as the sequence obtained through the concatenation of multiples of 3, in reverse order); such prime is, for example, 13330272421181512963; (II) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “odd multiples of 3” sequence (defined as the sequence obtained through the concatenation of odd multiples of 3, in reverse order); such prime is, for example, 145393327211593.

**Category:** Number Theory

[20] **viXra:1604.0160 [pdf]**
*submitted on 2016-04-10 06:07:45*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture three conjectures: (I) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “powers of 3” sequence (defined as the sequence obtained through the concatenation of powers of 3, in reverse order); such prime is, for example, 1243812793; (II) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “odd powers of 3” sequence (defined as the sequence obtained through the concatenation of odd powers of 3, in reverse order); such prime is, for example, 1243273; (III) there exist an infinity of primes p obtained concatenating to the left with 1 the terms of back concatenated “even powers of 3” sequence (defined as the sequence obtained through the concatenation of even powers of 3, in reverse order); such prime is, for example, 14782969531441590496561729819.

**Category:** Number Theory

[19] **viXra:1604.0158 [pdf]**
*submitted on 2016-04-10 02:09:11*

**Authors:** Marius Coman

**Comments:** 2 Pages.

I was studying the sequences of primes obtained applying concatenation to some well known classes of numbers, when I discovered that the second Poulet number, 561 (also the first Carmichael number, also a very interesting number – I wrote a paper dedicated to some of its properties), is also a triangular number. Continuing to look, I found, up to the triangular number T(817), if we note T(n) = n*(n + 1)/2 = 1 + 2 +...+ n, fifteen Poulet numbers. In this paper I state the conjecture that there exist an infinity of Poulet numbers which are also triangular numbers.

**Category:** Number Theory

[18] **viXra:1604.0147 [pdf]**
*submitted on 2016-04-09 01:41:21*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: There exist an infinity of primes p obtained concatenating to the right with 1 the triangular numbers.

**Category:** Number Theory

[17] **viXra:1604.0146 [pdf]**
*submitted on 2016-04-09 01:43:13*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following three conjectures: (I) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache reverse sequence; (II) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache back concatenated odd sequence; (III) There exist an infinity of primes p obtained concatenating to the left with 1 the terms of the Smarandache back concatenated square sequence.

**Category:** Number Theory

[16] **viXra:1604.0138 [pdf]**
*submitted on 2016-04-08 09:21:17*

**Authors:** Méhdi Pascal

**Comments:** 35 Pages. document en langue français

Ce papier contient deux petits résultats, le premier est sur une toute petite liaison qui lie les nombres parfaits avec les nombres de Carmichael. Le second résultat, est un simple exemple de traduction d’une méthode en fonction. A la fin de ce papier, je donne une introduction à la prochaine lettre qui montre que l’infinité des nombres premiers sous forme de n²+1 est lié à l’infinité des nombres premiers dans les deux progressions arithmétiques 4n+1 & n, par une simple identité asymptotique.

**Category:** Number Theory

[15] **viXra:1604.0132 [pdf]**
*submitted on 2016-04-08 03:51:10*

**Authors:** Pankaj Mani

**Comments:** 14 Pages.

In this paper, I try to look at Riemann Hypothesis from the Game Theoretical Perspective. As David Hilbert had visualized that advanced math is actually a game of symbols satisfying certain fixed rules. Indeed, here number theoretical system plays the Non-Cooperative game and more precisely the Game of Perfect Information.
Applying the technical Game Theoretic concepts, I have tried to show that Riemann Hypothesis is definitely true !
In case of any typos, please avoid them or else feel free to write to me. I shall correct them.
Author : Pankaj Mani,FRM,CQF
New Delhi, India
Email: manipankaj9@gmail.com

**Category:** Number Theory

[14] **viXra:1604.0110 [pdf]**
*submitted on 2016-04-06 01:14:55*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: Let p be a prime of the form 30*k + 17; then there exist an infinity of primes q obtained concatenating p – 1 with 3; example: 677, 797, 827, 857, 887, 947 are primes (successive primes of the form 30*k + 17) and the numbers 6763, 7963, 8263, 8563, 8863, 9463 are also primes. As an incidental observation, many of the semiprimes x*y obtained in the way defined have one of the following two properties: (i) y – x + 1 is a prime of the form 13 + 30*k; (ii) y – x + 1 is a prime of the form 19 + 30*k.

**Category:** Number Theory

[13] **viXra:1604.0105 [pdf]**
*submitted on 2016-04-05 08:25:12*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any term S(n) of the Smarandache consecutive numbers sequence (1, 12, 123, 1234, 12345, 123456, 1234567...) there exist an infinity of primes p such that the number q obtained concatenating S(n) both to the left and to the right with p is prime.

**Category:** Number Theory

[12] **viXra:1604.0103 [pdf]**
*submitted on 2016-04-05 04:39:52*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q with n then with q + 6 is prime; (II) let n be defined as in Conjecture 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q + 6 with n then with q is prime; (III) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k + 1; there exist an infinity of primes q of the form 6*h - 1 such that the number r obtained concatenating q with n then with q + 6 is prime; (IV) let n be defined as in Conjecture 3; there exist an infinity of primes q of the form 6*h - 1 such that the number r obtained concatenating q + 6 with n then with q is prime.

**Category:** Number Theory

[11] **viXra:1604.0101 [pdf]**
*submitted on 2016-04-04 16:43:39*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1 (e.g. n = 12345 for p = 5); there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q + 2 with n then with q is prime (e.g. for n = 12345 there exist q = 19 such that r = 211234519 is prime); (II) let n be a number obtained concatenating the positive integers from 1 to p, where p prime of the form 6*k – 1; there exist an infinity of primes q of the form 6*h + 1 such that the number r obtained concatenating q - 4 with n then with q is prime (e.g. for n = 12345 there exist q = 37 such that r = 331234537 is prime). I use the operator “]c[“ with the meaning “concatenated to”.

**Category:** Number Theory

[10] **viXra:1604.0032 [pdf]**
*submitted on 2016-04-04 12:03:24*

**Authors:** Marius Coman

**Comments:** 1 Page.

The triplets of primes [p, p + 2, p + 6] and [p, p + 4, p + 6] have already been studied: Hardy and Wright conjectured that there exist an infinity of such triplets. In this paper I make the following two conjectures on the triplets [p, p + 2, p + 6] and [p, p + 4, p + 6], but only p is required to be prime: (I) there exist an infinity of primes q obtained concatenating a prime p with p + 2 then with p + 6; example: for p = 11, the number q = 111317 is prime; (II) there exist an infinity of primes q obtained concatenating a prime p with p + 4 then with p + 6; example: for p = 241, the number q = 241245247 is prime.

**Category:** Number Theory

[9] **viXra:1604.0031 [pdf]**
*submitted on 2016-04-04 12:05:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures on the triplets [p, p + 2, p + 8] and [p, p + 6, p + 8]: (I) there exist an infinity of triplets of primes of the form [p, p + 2, p + 8]; (II) there exist an infinity of triplets of primes of the form [p, p + 6, p + 8]; (III) there exist an infinity of primes q obtained concatenating a prime p with p + 2 then with p + 8 (only p is necessary prime); (IV) there exist an infinity of primes q obtained concatenating a prime p with p + 6 then with p + 8 (only p is necessary prime).

**Category:** Number Theory

[8] **viXra:1604.0030 [pdf]**
*submitted on 2016-04-04 12:07:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures on the triplets [p, p + 4, p + 10] and [p, p + 6, p + 10]: (I) there exist an infinity of triplets of primes of the form [p, p + 4, p + 10]; (II) there exist an infinity of triplets of primes of the form [p, p + 6, p + 10]; (III) there exist an infinity of primes q obtained concatenating a prime p with p + 4 then with p + 10 (only p is necessary prime); (IV) there exist an infinity of primes q obtained concatenating a prime p with p + 6 then with p + 10 (only p is necessary prime).

**Category:** Number Theory

[7] **viXra:1604.0028 [pdf]**
*submitted on 2016-04-04 07:54:35*

**Authors:** Kolosov Petro

**Comments:** 10 Pages. -

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in {\mathbb{N}}$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems.

**Category:** Number Theory

[6] **viXra:1604.0023 [pdf]**
*submitted on 2016-04-03 14:07:45*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following conjecture: for any arithmetic progression a + b*k, where at least one of a and b is different than 1, that also satisfies the conditions imposed by the Dirichlet’s Theorem (a and b are positive coprime integers) is true that the sequence obtained by the consecutive concatenation of the terms a + b*k has an infinity of prime terms. Example: for [a, b] = [7, 11], the sequence obtained by consecutive concatenation of 7, 18, 29, 40, 51, 62, 73 (...) has the prime terms 718294051, 7182940516273 (...). If this conjecture were true, the fact that the Smarandache consecutive numbers sequence 1, 12, 123, 1234, 12345 (...) could have not any prime term (thus far there is no prime number known in this sequence, though there have been checked the first about 40 thousand terms) would be even more amazing.

**Category:** Number Theory

[5] **viXra:1604.0016 [pdf]**
*submitted on 2016-04-03 05:35:40*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: for any k multiple of 3, the sequence obtained by the consecutive concatenation of the numbers n*k + 1, where n positive integer, has an infinity of prime terms. Examples: for k = 3, the sequence 1, 14, 147, 14710 (...) has the prime terms 14710131619, 14710131619222528313437 (...); for k = 6, the sequence 1, 7, 13, 19 (...) has the prime terms 17, 17131925313743495561 (...).

**Category:** Number Theory

[4] **viXra:1604.0014 [pdf]**
*submitted on 2016-04-03 01:53:31*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: there exist an infinity of pairs of primes (p, q), where q – p = k, for any even number k, such that the number obtained concatenating p with k then with q is prime. Note that is not necessary, as is stipulated in the Polignac’s Conjecture, for the primes p and q to be consecutive (though, for the particular cases k = 2 and k = 4, of course that p and q are consecutive, which means that the conjecture above can be regarded as well as a stronger statement than the Twin primes Conjecture).

**Category:** Number Theory

[3] **viXra:1604.0011 [pdf]**
*submitted on 2016-04-01 10:57:18*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) there exist an infinity of triplets of consecutive primes [p, q, r] such that the number obtained concatenating p with x then with q then with y then with r, where x is the gap between p and q and y the gap between q and r, is prime. In other words, if we use the operator “]c[“ with the meaning “concatenating to”, p]c[x]c[q]c[y]c[r is prime for an infinity of triplets [p, q, r]. Example: for [p, q, r, x, y] = [11, 13, 17, 2, 4] the number 11213417 is prime; (II) for any pair of consecutive primes [p, q], p ≥ 7, there exist an infinity of primes r such that the number n = p]c[x]c[q]c[y]c[r is prime, where x is the gap between p and q and y the gap between q and r. Example: for [p, q] = [13, 17] there exist r = 61 such that n = 134174461 is prime (x = 4 and y = 44).

**Category:** Number Theory

[2] **viXra:1604.0003 [pdf]**
*submitted on 2016-04-01 05:42:08*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: for any k non-null positive integer there exist an infinity of pairs of sexy primes (p, q = p + 6) such that the number r formed concatenating p, repeatedly k times, with the digit 6 then with q is prime. Examples: for k = 1 there exist (p, q) = (11, 17) such that the number r = 11617 is prime; for k = 2 there exist the pair (p, q) = (31, 37) such that the number r = 316637 is prime.

**Category:** Number Theory

[1] **viXra:1604.0002 [pdf]**
*submitted on 2016-04-01 02:05:48*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: for any k positive integer there exist an infinity of primes p such that the number q, obtained concatenating (p – k) with p then, repeatedly k times, with the digit 1, is prime. Examples: for k = 1, there exist p = 19 such that q = 18191 is prime; for k = 2, there exist p = 5 such that q = 3511 is prime; for k = 3, there exist p = 7 such that q = 47111 is prime; for k = 4, there exist p = 37 such that q = 33371111 is prime; for k = 5, there exist p = 11 such that q = 61111111 is prime; for k = 6, there exist p = 17 such that q = 1117111111 is prime.

**Category:** Number Theory