**Previous months:**

2007 - 0703(3) - 0706(2)

2008 - 0807(1) - 0809(1) - 0810(1) - 0812(2)

2009 - 0901(2) - 0904(2) - 0907(2) - 0908(4) - 0909(1) - 0910(2) - 0911(1) - 0912(3)

2010 - 1001(3) - 1002(1) - 1003(55) - 1004(50) - 1005(36) - 1006(7) - 1007(11) - 1008(16) - 1009(21) - 1010(8) - 1011(7) - 1012(13)

2011 - 1101(14) - 1102(7) - 1103(13) - 1104(3) - 1105(1) - 1106(2) - 1107(1) - 1108(2) - 1109(3) - 1110(5) - 1111(4) - 1112(4)

2012 - 1201(2) - 1202(10) - 1203(6) - 1204(8) - 1205(7) - 1206(6) - 1207(5) - 1208(5) - 1209(11) - 1210(14) - 1211(10) - 1212(4)

2013 - 1301(5) - 1302(10) - 1303(16) - 1304(15) - 1305(12) - 1306(13) - 1307(25) - 1308(11) - 1309(9) - 1310(13) - 1311(15) - 1312(21)

2014 - 1401(20) - 1402(10) - 1403(27) - 1404(10) - 1405(17) - 1406(20) - 1407(34) - 1408(52) - 1409(47) - 1410(17) - 1411(16) - 1412(18)

2015 - 1501(14) - 1502(14) - 1503(35) - 1504(23) - 1505(19) - 1506(14) - 1507(16) - 1508(15) - 1509(15) - 1510(13) - 1511(9) - 1512(26)

2016 - 1601(14) - 1602(18) - 1603(78) - 1604(56)

Any replacements are listed further down

[1225] **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

[1224] **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

[1223] **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

[1222] **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

[1221] **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

[1220] **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

[1219] **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

[1218] **viXra:1604.0327 [pdf]**
*submitted on 2016-04-24 08:28:51*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

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

**Category:** Number Theory

[1217] **viXra:1604.0324 [pdf]**
*submitted on 2016-04-23 18:09:26*

**Authors:** Anthony J. Browne

**Comments:** 3 Pages.

A different approach involving roots and leading to the basis of the AKS test are introduced and discussed.

**Category:** Number Theory

[1216] **viXra:1604.0321 [pdf]**
*submitted on 2016-04-23 11:20:06*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of Characteristic equations are discussed and several number theoretic functions are derived.

**Category:** Number Theory

[1215] **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

[1214] **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

[1213] **viXra:1604.0295 [pdf]**
*submitted on 2016-04-20 13:24:30*

**Authors:** Ilija Barukčić

**Comments:** 7 Pages. Copyright © 2016 by Ilija Barukčić, Jever, Germany.

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

[1212] **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

[1211] **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

[1210] **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

[1209] **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

[1208] **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

[1207] **viXra:1604.0241 [pdf]**
*submitted on 2016-04-15 10:06:17*

**Authors:** F. Portela

**Comments:** 8 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 a possibly important conclusion.

**Category:** Number Theory

[1206] **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

[1205] **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

[1204] **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

[1203] **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

[1202] **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

[1201] **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

[1200] **viXra:1604.0201 [pdf]**
*submitted on 2016-04-12 07:14:59*

**Authors:** Jian Ye

**Comments:** 3 Pages.

Goldbach’s conjecture: symmetrical primes exists in natural numbers. the generalized Goldbach’s conjecture: symmetry of prime number in the former and tolerance coprime to arithmetic progression still exists.

**Category:** Number Theory

[1199] **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

[1198] **viXra:1604.0189 [pdf]**
*submitted on 2016-04-11 20:36:15*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Numerical verification is possible through nominative determinism (visibility theory). Adding learning (adaptive learning) to 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

[1197] **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

[1196] **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

[1195] **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

[1194] **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

[1193] **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

[1192] **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

[1191] **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

[1190] **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

[1189] **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

[1188] **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

[1187] **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

[1186] **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

[1185] **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

[1184] **viXra:1604.0132 [pdf]**
*submitted on 2016-04-08 03:51:10*

**Authors:** Pankaj Mani, Frm, Cqf

**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

[1183] **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

[1182] **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

[1181] **viXra:1604.0104 [pdf]**
*submitted on 2016-04-05 09:40:07*

**Authors:** Allen D. Allen

**Comments:** 5 Pages.

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving that there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a condition in which > changes to < so their transitive properties cannot be used. FLT implies that for an integral exponent n, the inequality changes over the interval with the minimum extent 1 ≤ n ≤ 3. For any exponent in the positive real numbers, a solution to Fermat’s equation occurs and inequality is replaced by equality at the instant when four distinct exponential curves collapse into two intersecting curves.

**Category:** Number Theory

[1180] **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

[1179] **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

[1178] **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

[1177] **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

[1176] **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

[1175] **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

[1174] **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

[1173] **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

[1172] **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

[1171] **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

[1170] **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

[1169] **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

[1168] **viXra:1603.0426 [pdf]**
*submitted on 2016-03-31 22:44:34*

**Authors:** Simon Plouffe

**Comments:** 14 Pages. news identities for primes, binomial sums and euler, bernoulli numbers

A survey is made based on finite sums of the polygamma function with rational arguments which are
D_(k,j)^n=∑_((m,n)=1)▒〖χ_j (n)ψ(k,m/n) 〗
Where, χ_j (n) is the j’th Dirichlet character and ψ(k,m/n) is the polygamma function of order k.
We use this representation to rewrite identities using a new notation for linear combinations of mathematical constants. Identities are given for prime numbers using irrational constants. For negative argument n we use the generalization of Espinosa and Moll[6], well implemented into Maple CAS.

**Category:** Number Theory

[1167] **viXra:1603.0425 [pdf]**
*submitted on 2016-03-31 14:09:17*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) there exist positive integers k such that the number (30*k)\\(30*k + p) is prime for an infinity of primes p. I used the operator “\\” with the meaning “concatenated to”; (II) there exist primes p such that the number (30*k)\\(30*k + p) is prime for an infinity of values of k; (III) there exist an infinity of primes of the form (30*k)\\(30*k + 1), where k positive integer.

**Category:** Number Theory

[1166] **viXra:1603.0424 [pdf]**
*submitted on 2016-03-31 14:10:42*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I define a Smarandache reconcatenated sequence Sr(n) as “the sequence obtained from the terms of a Smarandache concatenated sequence S(n), terms for which was applied the operation of consecutive concatenation” and I present six such sequences. Example: for Smarandache consecutive numbers sequence (1, 12, 123, 1234, 12345...), the Smarandache reconcatenated consecutive numbers sequence has the terms: 1, 112, 112123, 1121231234, 112123123412345...). According to the same pattern, we can define back reconcatenated sequences (the terms of the Smarandache back reconcatenated consecutive numbers sequence, noted Sbr(n), are 1, 121, 123121, 1234123121...).

**Category:** Number Theory

[1165] **viXra:1603.0414 [pdf]**
*submitted on 2016-03-30 18:19:23*

**Authors:** Brian Scannell

**Comments:** 28 Pages.

We look here at the geometry of zeta(3). By piling cubes a 3D shape is defined which has a volume of zeta(3). This shape is a double integral form for zeta(3). Considering the centroid of this shape leads to an experimental estimate for zeta(3). Cutting the shape parallel to the x axis reproduces the dilogarithmic relationship to zeta(3). Cutting the shape in the z axis reproduces the logarithmic version of Riemann’s formula for zeta(3). Geometrical considerations also reproduce formula for the polylog of a half Lin(1/2) for n=2 and 3.
These are illustrations of number geometry.

**Category:** Number Theory

[1164] **viXra:1603.0403 [pdf]**
*submitted on 2016-03-30 07:26:23*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I present the following four Smarandache type sequences: (I) The sequence of numbers obtained concatenating the positive integers of the form 6*k – 1; (II) The sequence of numbers obtained concatenating the primes of the form 6*k – 1; (III) The sequence of numbers obtained concatenating the positive integers of the form 6*k + 1; (IV) The sequence of numbers obtained concatenating the primes of the form 6*k + 1.

**Category:** Number Theory

[1163] **viXra:1603.0402 [pdf]**
*submitted on 2016-03-30 07:29:50*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: There exist an infinity of primes S(n+1) + S(n) - 1, where S(n) is a term in Smarandache concatenated odd sequence (which is defined as the sequence obtained through the concatenation of the first n odd primes).

**Category:** Number Theory

[1162] **viXra:1603.0394 [pdf]**
*submitted on 2016-03-29 05:08:39*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following two conjectures: (I) there exist an infinity of quintets of primes (p, p + 10, p + 30, p + 40, p + 60), where p is a prime of the form 6*k + 1; (II) there exist an infinity of primes of the form p\\(p + 10)\\(p + 30)\\(p + 40)\\(p + 60), where p is a number of the form 6*k + 1. I used the operator “\\” with the meaning “concatenated to”.

**Category:** Number Theory

[1161] **viXra:1603.0393 [pdf]**
*submitted on 2016-03-29 06:17:41*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following two conjectures: (I) there exist an infinity of quintets of primes (p, p + 20, p + 30, p + 50, p + 80), where p is a prime of the form 6*k - 1; (II) there exist an infinity of primes of the form p\\(p + 20)\\(p + 30)\\(p + 50)\\(p + 80), where p is a number of the form 6*k - 1. I used the operator “\\” with the meaning “concatenated to”.

**Category:** Number Theory

[1160] **viXra:1603.0391 [pdf]**
*submitted on 2016-03-29 03:17:35*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: There exist an infinity of primes S(n) + S(n + 1) - 1, where S(n) is a term in Smarandache-Wellin sequence (which is defined as the sequence obtained through the concatenation of the first n primes).

**Category:** Number Theory

[1159] **viXra:1603.0380 [pdf]**
*submitted on 2016-03-27 16:43:22*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist always a prime of the form p^2 + 18*m between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the primes 139 = 121 + 1*18) and 157 = 121 + 2*18; for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 289 + 1*18); (II) there exist always a prime of the form q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2], beside the pair [3, 5]; examples: for [p^2, q^2] = [11^2, 13^2] = [121, 169] there exist the prime 151 = 169 – 1*18); for [p^2, q^2] = [17^2, 19^2] = [289, 361] there exist the prime 307 = 361 – 3*18); (III) there exist an infinity of r primes of the form p^2 + 18*m or q^2 – 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating p^2 to the right with r is prime; example: 121139 is prime; (IV) there exist an infinity of r primes of the form p^2 + 18*m or q^2 - 18*n between the squares p^2 and q^2 of a pair of twin primes [p, q = p + 2] such that the number obtained concatenating q^2 to the left with r is prime; example: 139169 is prime. Of course, the conjectures (III) and (IV) imply that there exist an infinity of pairs of twin primes.

**Category:** Number Theory

[1158] **viXra:1603.0379 [pdf]**
*submitted on 2016-03-27 16:45:25*

**Authors:** Marius Coman

**Comments:** 11 Pages.

In this paper I list a number of 33 sequences of primes obtained by the method of concatenation; some of these sequences are 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

[1157] **viXra:1603.0370 [pdf]**
*submitted on 2016-03-26 23:53:47*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture on an infinity of subsequences of primes in Smarandache prime-partial-digital sequence, defined as the sequence of prime numbers which admit a deconcatenation into a set of primes: for any prime p which admits a deconcatenation in k primes larger than 3 is true that there exist a number of k sequences of primes P1, P2,...,Pk, each one having an infinity of prime terms which also admit a deconcatenation in prime numbers, obtained replacing a prime q in p with primes having the same digital root as q (example: for the prime 547 there exist an infinite sequence of primes obtained replacing 5 with primes having the digital root equal to 5 (2347, 13147, 14947, ...) and also an infinite sequence of primes obtained replacing 47 with primes having the digital root equal to 2 (5101, 5227, 5281,...).

**Category:** Number Theory

[1156] **viXra:1603.0369 [pdf]**
*submitted on 2016-03-26 23:55:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: Let’s consider the primes p obtained from composite numbers in the following way: concatenating the prime factors of a composite number n (example: for 31941 = 3*3*3*7*13*13, the concatenation of its prime factors is 33371313) is obtained either a prime (in which case this prime is p), either a composite; if is obtained a composite, is reiterated the operation until is obtained a prime (in which case this prime is p). I conjecture that there exist such prime p for every composite number.

**Category:** Number Theory

[1155] **viXra:1603.0367 [pdf]**
*submitted on 2016-03-27 04:47:03*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures: (I) For any prime p, p > 5, there exist a pair of primes (q1, q2), both having the group of their last digits equal to p, and a positive integer n, such that p = (n + 1)*q1 – n*q2 (examples: for p = 11, there exist the primes q1 = 211 and q2 = 311 and also the number n = 2 such that 11 = 3*211 – 2*311; for p = 29, there exist the primes q1 = 829 and q2 = 929 and also the number n = 8 such that 29 = 9*829 – 8*929); (II) For any q1 prime, q1 > 5, and any n non-null positive integer, there exist an infinity of primes q2, having the group of their last digits equal to q1, such that p = (n + 1)*q2 – n*q1 is prime; (III) For any q1 prime, q1 > 5, and any q2 prime having the group of its last digits equal to q1, there exist an infinity of positive integers n such that p = (n + 1)*q2 – n*q1 is prime.

**Category:** Number Theory

[1154] **viXra:1603.0362 [pdf]**
*submitted on 2016-03-25 21:27:44*

**Authors:** Octavian Cira, Florentin Smarandache

**Comments:** 400 Pages.

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile
periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, Romania).
This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira - with his algorithmic thinking and knowledge of Mathcad.

**Category:** Number Theory

[1153] **viXra:1603.0356 [pdf]**
*submitted on 2016-03-25 12:28:28*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper, we show that for all even integers “2n”, there exists infinite
positive integers “d” greater than one, such that their product “2nd” is a sum of two
primes. Any two odd primes add to give even integers. However this general method
does not allow us to understand the property or relationship among even numbers
numbers derived in this manner. On the other hand, our results suggests existence
of even integers of the specific form “2nd” that can be written as a sum of two
primes.

**Category:** Number Theory

[1152] **viXra:1603.0355 [pdf]**
*submitted on 2016-03-25 09:59:25*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) for any k having one of the values 1, 2, 4, 5, 7 or 8, there exist an infinity of primes obtained concatenating two primes that both have the digital root equal to k; (II) for any n positive integer, not divisible by 3, n ≥ 4, there exist primes obtained concatenating two primes that both have the digital sum equal to n; (III) there exist an infinity of values of n, positive integer, for which exist an infinity of primes obtained concatenating two primes that both have the digital sum equal to n.

**Category:** Number Theory

[1151] **viXra:1603.0353 [pdf]**
*submitted on 2016-03-25 02:50:23*

**Authors:** Marius Coman

**Comments:** 5 Pages.

In this paper I make the following eight conjectures: (Ia) for any p prime, p > 3, there exist an infinity of primes q such that the number n obtained concatenating p – 1 to the right with q^2 is prime; (Ib) there exist an infinity of terms in any of the sequences above (for any p) such that r = (p – 1)*q^2 + 1 is prime; (IIa) for any q prime, q > 3, there exist an infinity of primes p such that the number n obtained concatenating q^2 to the left with p – 1 is prime; (IIb) there exist an infinity of terms in any of the sequences above (for any q) such that r = (p – 1)*q^2 + 1 is prime; (IIIa) for any Poulet number P, not divisible by 3, there exist an infinity of primes q such that the number n obtained concatenating P – 1 to the right with q^2 is prime; (IIIb) there exist an infinity of terms in any of the sequences above (for any P) such that r = (P – 1)*q^2 + 1 is prime; (IVa) for any Poulet number Q, not divisible by 3 or 5, there exist an infinity of primes p such that the number n obtained concatenating Q^2 to the left with p - 1 is prime; (IVb) there exist an infinity of terms in any of the sequences above (for any Q) such that r = (p – 1)*Q^2 + 1 is prime.

**Category:** Number Theory

[1150] **viXra:1603.0347 [pdf]**
*submitted on 2016-03-24 07:52:47*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes p of the form (6*k - 1)]c[(6*k + 1)]c[(6*k - 1), where “]c[“ means “concatenated to" (example: for k = 4, the number p = 232523 is prime); (II) there exist an infinity of primes q of the form (6*k + 1)]c[(6*k - 1)]c[(6*k + 1) (example: for k = 2, the number p = 131113 is prime); (III) there exist an infinity of pairs of primes (p, q) = ((6*k - 1)]c[(6*k + 1)]c[(6*k - 1), ((6*k + 1)]c[(6*k – 1)]c[(6*k + 1)); example: for k = 5, (p, q) = (293129, 32931); note that, for such a pair (p, q), q – p = 19802; 1998002; 199980002 and so on; (IV) there exist, for any h positive integer, an infinity of primes q = p + m, where p is prime and m is the number obtained concatenating 1 with a number of h digits of 9 then with 8 then with the same number of h digits of 0 then with 2.

**Category:** Number Theory

[1149] **viXra:1603.0342 [pdf]**
*submitted on 2016-03-24 05:01:47*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following three conjectures: (I) there exist an infinity of primes p obtained concatenating even numbers n with 0 then with n + 2, then again with 0, then with n + 5 (example: for n = 44, the number p = 44046049 is prime). It is notable that are found chains with 4 primes p obtained for 4 consecutive even numbers n (example: 17201740177, 17401760177, 17601780181, 17801800183, obtained for 172, 174, 176, 178); (II) there exist an infinity of pairs of primes (p, q) obtained applying on two consecutive even numbers (m, n) the method of concatenation showed in the conjecture above (note that q – p = 20202; 2002002; 200020002 and so on); (III) there exist, for any k positive integer, an infinity of primes q = p + n, where p is prime and n is the number obtained concatenating 2 with a number of k digits of 0 then with 2 then again with the same number of k digits of 0 then again with 2.

**Category:** Number Theory

[1148] **viXra:1603.0341 [pdf]**
*submitted on 2016-03-23 15:58:15*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

The Kurepa conjecture states that the gcd (!n,n!) is equal to 2 for all n=2,3,….Although this conjecture has not been proven, in this paper we study the implication of a true Kurepa conjecture. We use (!n)/2 and (n!)/2 as components of two distinct arithmetic progressions and propose that both should have infinitely many primes as known from Dirichlet’s theorem of Arithmetic progressions and these are named as Dirichlet-Kurepa primes Type 1 and Type 2 in their honor.

**Category:** Number Theory

[1147] **viXra:1603.0336 [pdf]**
*submitted on 2016-03-23 09:12:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: there exist an infinity of primes q = 2*n – 1, where n is the sum of a reversible prime p of the form 6*k + 1 concatenated to the left with 1 and its reversal, also concatenated to the left with 1 (example: for p = 13, n = 113 + 131 = 244 and q = 244*2 – 1 = 487, prime).

**Category:** Number Theory

[1146] **viXra:1603.0328 [pdf]**
*submitted on 2016-03-22 20:13:29*

**Authors:** Constantin Dumitrescu, Vasile Seleacu

**Comments:** 137 Pages.

The Smarandache Function is defined as the smallest integer S(n) such that S(n)! is divisible by n.
The authors study properties of this function.

**Category:** Number Theory

[1145] **viXra:1603.0327 [pdf]**
*submitted on 2016-03-23 01:41:55*

**Authors:** Marius Coman

**Comments:** 13 Pages.

In this paper I list a number of 20 Smarandache concatenated sequences (for other lists and analyses on these sequences see “Smarandache Sequences” on Wolfram MathWorld and “The math encyclopedia of Smarandache type notions”, Educational Publishing, 2013) and I highlight the sets of primes distinguished among the terms of these sequences, but also I list 25 “sets of primes which can be obtained from the terms of Smarandache sequences using any arithmetical operation” (I named such primes Smarandache-Coman sequences of primes, see my previous papers “Fourteen Smarandache-Coman sequences of primes” and “Seven Smarandache-Coman sequences of primes”).

**Category:** Number Theory

[1144] **viXra:1603.0322 [pdf]**
*submitted on 2016-03-22 07:20:59*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: for any p prime, p > 5, there exist an infinity of k positive integers such that the number q obtained concatenating to the right p with p + 30*k is prime (examples: for p = 13, the least k for which q is prime is 2 because 1373 is prime; for p = 104729, the least k for which q is prime is 3 because 104729104819 is prime). It is notable the small values of k for which primes q are obtained, even in the case of primes p having 20 digits, so this formula could be a way to easily find, starting from a prime p, a prime q having twice as many digits!

**Category:** Number Theory

[1143] **viXra:1603.0321 [pdf]**
*submitted on 2016-03-21 17:06:19*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper we show that any two successive left factorials can only represent the first and second terms and not any other pair of successive terms in any arithmetic progression of positive integers.

**Category:** Number Theory

[1142] **viXra:1603.0320 [pdf]**
*submitted on 2016-03-21 18:42:18*

**Authors:** Islem Ghaffor

**Comments:** 2 Pages.

In this paper we give 2 progressions Vn and Tn have a relation with Collatz conjecture.

**Category:** Number Theory

[1141] **viXra:1603.0314 [pdf]**
*submitted on 2016-03-21 15:06:30*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following conjecture: for any prime p of the form 6*k + 1 there exist an infinity of primes n obtained concatenating p to the left with 3 and to the right with a square of prime q^2 (examples: for p = 13, the numbers n = 313289, 313961, 3131369 – obtained for q = 17, 31, 37 – are primes).

**Category:** Number Theory

[1140] **viXra:1603.0307 [pdf]**
*submitted on 2016-03-21 11:58:26*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures on primes: (I) there exist an infinity of primes q obtained concatenating to the left a prime p with the number (p – 1)/2 (example: for p = 23, q is the number obtained concatenating 23 to the left with (p – 1)/2 = 11, i.e. q = 1123, prime); (II) there exist an infinity of primes q obtained concatenating to the left a prime p with the number (p + 1)/2 (example: for p = 41, q is the number obtained concatenating 41 to the left with (p + 1)/2 = 21, i.e. q = 2141, prime); (III) there exist an infinity of pairs of primes (q1, q2) where q1 is obtained concatenating to the left a prime p with the number (p – 1)/2 and q2 is obtained concatenating to the left the same prime p with the number (p + 1)/2.

**Category:** Number Theory

[1139] **viXra:1603.0303 [pdf]**
*submitted on 2016-03-21 10:33:02*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures on squares of primes: (I) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number (p^2 + 1)/2 (example: for p = 17, p^2 = 289 and q is the number obtained concatenating 289 to the left with (p^2 + 1)/2 = 145, i.e. q = 145289, prime); (II) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number p + 12 (example: for p = 7, p^2 = 49 and q = 1949, prime); (III) there exist an infinity of primes q obtained concatenating to the left a square of a prime p^2 with the number p^ + 12 (example: for p = 11, p^2 = 121 and q = 133121, prime).

**Category:** Number Theory

[1138] **viXra:1603.0302 [pdf]**
*submitted on 2016-03-21 07:08:43*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes p such that 3*p – 10 is also prime; (II) there exist an infinity of triplets of primes (p, 2*p – 1, 3*p – 10); (III) there exist an infinity of primes q obtained concatenating a prime p to the right with 2*p – 1 and to the left with 3 (example: for p = 11, q = 31121, prime; (IV) there exist, for any n positive integer, n > 1, an infinity of primes q obtained concatenating a prime p to the right with n*p – n + 1 and to the left with 3 (examples: for n = 5 and p = 19, q = 31991, prime; for n = 8 and p = 13, q = 31397, prime).

**Category:** Number Theory

[1137] **viXra:1603.0299 [pdf]**
*submitted on 2016-03-20 20:48:07*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we transform the zeros in the critical line.

**Category:** Number Theory

[1136] **viXra:1603.0290 [pdf]**
*submitted on 2016-03-21 03:18:13*

**Authors:** Marius Coman

**Comments:** 4 Pages.

Though the well known Fermat’s conjecture on the diophantine equation x^n + y^n = z^n is named “Fermat’s big theorem”, in fact probably much more important for number theory is what is called “Fermat’s little theorem” which was the most important step up to that time in order to discover a primality criterion. This exceptional criterion of primality still has its exceptions: Fermat pseudoprimes, numbers which “behave” like primes though they are no primes; but they are still a class of numbers at least as interesting as the class of primes. Among Fermat pseudoprimes two classes of numbers are particularly distinguished: Poulet numbers (relative Fermat pseudoprimes) and Carmichael numbers (absolute Fermat pseudoprimes). The initial aim of this paper was only to see which Poulet numbers can be obtained concatenating primes (or, in other words, whichever admit a deconcatenation in prime numbers) but, inspired by a characteristic of a subset of Poulet numbers, I also made the following conjecture: there exist an infinity of primes p obtained concatenating to the right a prime q having the sum of the digits s(q) equal to a multiple of 5 with 3.

**Category:** Number Theory

[1135] **viXra:1603.0288 [pdf]**
*submitted on 2016-03-20 14:56:25*

**Authors:** Charles Ashbacher

**Comments:** 62 Pages.

The Smarandache Function is the smallest integer S(n) such that S(n)! is divisibil by n.

**Category:** Number Theory

[1134] **viXra:1603.0286 [pdf]**
*submitted on 2016-03-20 14:59:03*

**Authors:** Marius Coman

**Comments:** 113 Pages.

This book brings together fifty-two papers regarding primes and Fermat pseudoprimes, submitted by the author to the scientific database Research Gate. Part One of this book, “Sequences of primes and conjectures on them”, contains papers on sequences of primes, squares of primes, semiprimes, pairs, triplets and quadruplets of primes and conjectures on them. This part also contains papers on possible methods to obtain large primes, some of them based on concatenation, some of them on other arithmetical operations. It is also introduced a new notion, “Smarandache-Coman sequences of primes”, defined as “all sequences of primes obtained from the Smarandache sequences using any arithmetical operation”. Part Two of this book, “Sequences of Fermat pseudoprimes and conjecture on them”, contains sequences of Poulet and Carmichael numbers. Among these papers there is a list of thirty-six polynomials and formulas that generate sequences of Fermat pseudoprimes. Part Three of this book, “Prime producing quadratic polynomials”, contains three papers which list few already known such polynomials, that generate more than 20, 30 or even 40 primes in a row, and few such polynomials discoverd by the author himself (in a review of records in the field of prime generating polynomials, written by Dress and Landreau, two mathematicians well known for their contributions in this field, the author is mentioned with 18 prime producing quadratic polynomials). One of these three papers proposes 17 generic formulas that may generate prime producing quadratic polynomials.

**Category:** Number Theory

[1133] **viXra:1603.0268 [pdf]**
*submitted on 2016-03-20 03:45:41*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following four conjectures on the Smarandache prime-partial-digital sequence defined as the sequence of prime numbers which admit a deconcatenation into a set of primes: (I) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k + 1, such that n = m*h – h + 1 , where h positive integer; (II) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k - 1, such that n = m*h + h - 1 , where h positive integer; (III) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k + 1, such that n + m - 1 is prime or power of prime; (IV) there exist an infinity of primes p obtained concatenating two primes m and n, both of the form 6*k - 1, such that n - m + 1 is prime or power of prime. Note that almost all from the first 65 primes obtained concatenating two primes of the form 6k + 1 (exceptions: 3779, 4373, 6173, 6719, 6779), and all the first 65 primes obtained concatenating two primes of the form 6k - 1, belong to one of the four sequences considered by the conjectures above.

**Category:** Number Theory

[1132] **viXra:1603.0266 [pdf]**
*submitted on 2016-03-18 23:32:28*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following two conjectures: (I) there exist an infinity of primes q obtained concatenating a prime p with 9 then with p itself (example: p = 104593 is prime and q = 1045939104593 is also prime); (II) there exist an infinity of primes q obtained concatenating a prime p of the form 6*k – 1 with 9 then with p itself and subtracting 2 (example: p = 104471 is prime and q = 1044719104471 – 2 = 1044719104469 is also prime).

**Category:** Number Theory

[1131] **viXra:1603.0264 [pdf]**
*submitted on 2016-03-19 02:53:33*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: there exist, for any m prime of the form 6*k + 1, an infinity of primes n obtained concatenating a prime p with a prime q where q – p + 1 = m (example: for m = 457, prime, we have q - p + 1 = 457 for [p, q] = [11, 467], both primes, and the number n = 11467 is prime).

**Category:** Number Theory

[1130] **viXra:1603.0263 [pdf]**
*submitted on 2016-03-18 14:33:20*

**Authors:** Marius Coman

**Comments:** 4 Pages.

In this paper I make the following two conjectures: (I) there exist an infinity of primes obtained concatenating, once or repeatedly, an odd multiple n of 3 with 111, then raising the number obtained to the power 2, adding to it n and subtracting 1 (Examples: 3111^2 + 3 – 1 = 9678323, prime; 27111111^2 + 27 – 1 = 735012339654347, prime); (I) there exist an infinity of semiprimes obtained concatenating, once or repeatedly, an odd multiple n of 3 with 111, then raising the number obtained to the power 2, adding to it n and subtracting 1.

**Category:** Number Theory

[1129] **viXra:1603.0262 [pdf]**
*submitted on 2016-03-18 16:23:50*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: there exist an infinity of primes obtained concatenating the square of a prime p with p then with 1 and then subtracting 2 from the resulting number (example: 127^2 = 16129 and the number 161291271 – 2 = 161291269 is prime).

**Category:** Number Theory

[1128] **viXra:1603.0256 [pdf]**
*submitted on 2016-03-18 00:45:36*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following four conjectures: (I) there exist an infinity of primes of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k – 1; (II) there exist an infinity of semiprimes m*n of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k – 1, semiprimes having the property that n – m + 1 is prime; (III) there exist an infinity of primes of the form 2*(p*q*r) - 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k + 1; (IV) there exist an infinity of semiprimes m*n of the form 2*(p*q*r) + 1, where p, q = p + 6, r = q + 6 are odd numbers of the form 6*k + 1, semiprimes having the property that n – m + 1 is prime.

**Category:** Number Theory

[1127] **viXra:1603.0254 [pdf]**
*submitted on 2016-03-18 03:26:29*

**Authors:** Kunle Adegoke

**Comments:** 22 Pages.

Using a clear and straightforward approach, we prove new ternary (base 3) digit extraction BBP-type formulas for polylogarithm constants. Some known results are also rediscovered in a more direct and elegant manner. A previously unproved degree~4 ternary formula is also proved. Finally, a couple of ternary zero relations are established, which prove two known but hitherto unproved formulas.

**Category:** Number Theory

[1126] **viXra:1603.0248 [pdf]**
*submitted on 2016-03-16 11:45:12*

**Authors:** Prem Kumar

**Comments:** 3 Pages.

In this paper I discuss an algorithm which will solve a very famous puzzle involving a monkey, few men and some coconuts. The puzzle involves a group of n men who have an unknown amount of coconuts among them. At night, while the others are asleep, one of the men divides the coconuts in n parts and hides his share. While dividing, he discovers that there is one extra coconut, which he gives away to a monkey. Exactly the same thing happens with the rest of the men, one by one. They all hide their share, are left with one extra coconut that cannot be divided, which they give to the monkey. The next morning they again divide the coconuts together equally among themselves, with no extra coconut remaining this time. The puzzle is to find out the initial number of coconuts.

**Category:** Number Theory

[1125] **viXra:1603.0238 [pdf]**
*submitted on 2016-03-16 10:05:23*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following two conjectures on the Smarandache’s proper divisor products sequence where a term P(n) of the sequence is defined as the product of the proper divisors of n: (1) there exist an infinity of numbers n divisible by 3 such that the number obtained concatenating the value of P(n) to the right with 1 is prime; (2) there exist an infinity of numbers n divisible by 3 such that the number obtained concatenating the value of P(n) to the right with 1 is semiprime p*q with the property that q – p + 1 is prime.

**Category:** Number Theory

[1124] **viXra:1603.0237 [pdf]**
*submitted on 2016-03-16 10:07:03*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following three conjectures: (I) there exist an infinity of primes obtained concatenating the number P – 1 to the right with 1, where P is a Poulet number; (II) there exist an infinity of primes obtained concatenating the number P – 1 to the right with 11, where P is a Poulet number; (III) there exist an infinity of primes obtained concatenating the number P + 1 to the right with 11, where P is a Poulet number.

**Category:** Number Theory

[1123] **viXra:1603.0236 [pdf]**
*submitted on 2016-03-16 10:09:42*

**Authors:** Marius Coman

**Comments:** 3 Pages.

In this paper I make the following four conjectures: (I) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating p – 1 with the value of d; (II) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating twice p – 1 with the value of d; (III) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating p + 1 with the value of d; (II) there exist, for any prime p having the value of the last digit d equal to 1, respectively to 3, 7 or 9, an infinity of primes obtained concatenating twice p + 1 with the value of d.

**Category:** Number Theory

[1122] **viXra:1603.0233 [pdf]**
*submitted on 2016-03-15 17:51:42*

**Authors:** Allen D Allen

**Comments:** 4 Pages.

The only way to prove Fermat’s Last Theorem with logical rigor is to first prove Fermat’s Extended Last Theorem (FELT): If n is an integer greater than 2, then there cannot exist positive rational fractions r, s, and t, neither integral nor non-integral, such that r^n + s^n = t^n.

**Category:** Number Theory

[1121] **viXra:1603.0227 [pdf]**
*submitted on 2016-03-16 01:25:03*

**Authors:** Kunle Adegoke

**Comments:** 6 Pages.

Hitherto only a base 5 BBP-type formula is known for $\sqrt 5\log\phi$, where \mbox{$\phi=(\sqrt 5+1)/2$}, the golden ratio, ( i.e. Formula 83 of the April 2013 edition of Bailey's Compendium of \mbox{BBP-type} formulas). In this paper we derive a new binary BBP-type formula for this constant. The formula is obtained as a particular case of a BBP-type formula for a family of logarithms.

**Category:** Number Theory

[543] **viXra:1604.0321 [pdf]**
*replaced on 2016-04-28 15:15:04*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of Characteristic equations are discussed and several number theoretic functions are derived.

**Category:** Number Theory

[542] **viXra:1604.0321 [pdf]**
*replaced on 2016-04-23 23:06:35*

**Authors:** Anthony J. Browne

**Comments:** 11 Pages.

Sums of characteristic equations are discussed. Several number theoretic functions are derived and different techniques are introduced and discussed.

**Category:** Number Theory

[541] **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

[540] **viXra:1604.0241 [pdf]**
*replaced on 2016-04-19 15:27:35*

**Authors:** F. Portela

**Comments:** 9 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

[539] **viXra:1604.0241 [pdf]**
*replaced on 2016-04-17 17:32:39*

**Authors:** F. Portela

**Comments:** 8 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 a possibly important conclusion.

**Category:** Number Theory

[538] **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

[537] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-14 20:41:28*

**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

[536] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-14 14:04:15*

**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

[535] **viXra:1604.0189 [pdf]**
*replaced on 2016-04-12 10:36:44*

**Authors:** Nicholas R. Wright

**Comments:** 7 Pages.

We prove the integrality and modularity of the Birch and Swinnerton-Dyer conjecture with ERG Theory. Numerical verification is possible through nominative determinism (visibility theory). Adding learning (adaptive learning) to 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

[534] **viXra:1603.0248 [pdf]**
*replaced on 2016-04-14 03:21:54*

**Authors:** Prem kumar

**Comments:** 3 Pages.

In this paper I discuss an algorithm which will solve a very famous puzzle involving a monkey, few men and some coconuts. The puzzle involves a group of n men who have an unknown number of coconuts among them. At night, while the others are asleep, one of the men divides the coconuts in n parts and hides his share. While dividing, he discovers that there is one extra coconut, which he gives away to a monkey. Exactly the same thing happens with the rest of the men, one by one. They all hide their share, are left with one extra coconut that cannot be divided, which they give to the monkey. The next morning they again divide the coconuts together equally among themselves, with no extra coconut remaining this time. The puzzle is to find out the initial number of coconuts.

**Category:** Number Theory