[30] **viXra:1704.0355 [pdf]**
*submitted on 2017-04-26 17:54:24*

**Authors:** H. L. Mitchell

**Comments:** 14 Pages.

This paper presents a sieve for the twin primes using a double application of the Sieve of Eratosthenes to the k index for multiples of 6 centered around the primes ranging from 5 to sqrt N. Considering the set {(6k-1, 6k+1)}, we look at the exact pattern in which multiples of primes p occur in the set with respect to the values of k. By deleting the values of k such that 6k-1 is composite, we get the set of all pairs such that 6k-1 is prime, then from these we delete those in which 6k+1 is composite. That leaves us with the set of all twin primes less than N.
We derive a mathematical formula accordingly using the Euler product formula, the Prime
Number Theorem, Mertens’ Theorem and the Hardy-Littlewood Conjecture.
The main results are:
1) A sieve for the twin primes similar to the sieve of Eratosthenes for primes
2) A formula for the approximate number of twin primes less than N in terms of the number of
primes
3) The asymptotic formula for the number of twin primes less than N

**Category:** Number Theory

[29] **viXra:1704.0306 [pdf]**
*submitted on 2017-04-23 12:08:04*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following observation: Let d be a factor (not necessarily prime) of the Poulet number P such that d < sqr P and m the least number such that m*d*(d – 1) > (P – 1)/2. Let n be equal to P – m*d*(d – 1). Then often exist a set of Poulet numbers Q such that Q mod(m*d*(d – 1)) = n. For example, for P = 2047 = 23*89 and d = 23, where d < sqr 2047, the least m such that m*23*22 > (P – 1)/2 is equal to 3 (1518 > 1023, while, for 2, 1012 < 1023); so, n = 2047 – 3*23*22 = 2047 – 1518 = 529 and indeed there exist a set of Poulet numbers Q such that Q mod 1518 = 529; the formula 1518*x + 529 gives the Poulet numbers 2047, 6601, 15709, 30889 (...) for x = 1, 4, 10, 20 (...).

**Category:** Number Theory

[28] **viXra:1704.0296 [pdf]**
*submitted on 2017-04-22 23:35:39*

**Authors:** Marius Coman

**Comments:** 1 Page.

In a previous paper, “Poulet numbers in Smarandache prime partial digital sequence and a possible infinite set of primes” I conjectured that there exist an infinity of Poulet numbers which admit a deconcatenation in prime numbers. In this paper I conjecture that there exist an infinity of Poulet numbers which admit a deconcatenation in two prime numbers p and q where q = p + 30*k, where k integer.

**Category:** Number Theory

[27] **viXra:1704.0295 [pdf]**
*submitted on 2017-04-22 23:37:50*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I conjecture that there exist an infinity of primes obtained concatenating four consecutive numbers, the largest one from them being a Poulet number. For example, 1726172717281729 is such a prime, obtained concatenating the numbers 1726, 1727, 1728 and 1729, where 1729 is a Poulet number (see the sequence A030471 in OEIS for primes which are concatenation of four consecutive numbers).

**Category:** Number Theory

[26] **viXra:1704.0293 [pdf]**
*submitted on 2017-04-23 04:09:15*

**Authors:** Marius Coman

**Comments:** 2 Pages.

It is well known the story of the Hardy-Ramanujan number, 1729 (also a Poulet number), which is the smallest number expressible as the sum of two cubes in two different ways, but I have not met yet, not even in OEIS, the sequence of the Poulet numbers which can be written as x^3±y^3, sequence that I conjecture in this paper that is infinite. I also conjecture that there are infinite Poulet numbers which are centered cube numbers (equal to 2*n^3 + 3*n^2 + 3*n + 1), also which are centered hexagonal numbers (equal to 3*n^2 + 3*n + 1).

**Category:** Number Theory

[25] **viXra:1704.0274 [pdf]**
*replaced on 2017-04-22 06:37:27*

**Authors:** François Mendzina Essomba

**Comments:** 1 Page. extreme fomulas

I present in this small article two algorithms of calculation of pi, they are characterized by two extremities, one is the most convergent and the other the slowest of the imaginable formulas.

**Category:** Number Theory

[24] **viXra:1704.0260 [pdf]**
*submitted on 2017-04-20 07:53:36*

**Authors:** Edgar Valdebenito

**Comments:** 37 Pages.

This note presents a collection of fractals related with constant pi

**Category:** Number Theory

[23] **viXra:1704.0259 [pdf]**
*submitted on 2017-04-20 07:57:06*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

This note presents some series for pi constant.

**Category:** Number Theory

[22] **viXra:1704.0258 [pdf]**
*submitted on 2017-04-20 08:02:54*

**Authors:** Edgar Valdebenito

**Comments:** 7 Pages.

In this research , the autor has detailed about: Numerical evaluation of the Ramanujan-Göllnitz-Gordon continued fraction.

**Category:** Number Theory

[21] **viXra:1704.0240 [pdf]**
*submitted on 2017-04-19 11:32:59*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

This note presents some formulas related with Gelfond constant:exp(pi)

**Category:** Number Theory

[20] **viXra:1704.0225 [pdf]**
*submitted on 2017-04-17 17:28:46*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any n positive integer there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, a, b and a + b + n. Examples: for n = 0, the least such prime is 101 (1 + 0 + 0 = 1); for n = 1, the least such prime is 113 (1 + 1 + 1 = 3); for n = 2, the least such prime is 103 (1 + 0 + 2 = 3); for n = 3, the least such prime is 137 (1 + 3 + 3 = 7); for n = 4, the least such prime is 127 (1 + 2 + 4 = 7); for n = 5, the least such prime is 139 (1 + 3 + 5 = 9); for n = 6, the least such prime is 107 (1 + 0 + 6 = 7); for n = 7, the least such prime is 3313 (3 + 3 + 7 = 13).

**Category:** Number Theory

[19] **viXra:1704.0224 [pdf]**
*submitted on 2017-04-17 17:31:06*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I make the following conjecture: For any n even there exist an infinity of primes which can be deconcatenated in three numbers, i.e., from left to right, p, n and p + n, where p and p + n are primes. Examples: for n = 2, the least such prime is 11213 (11 + 2 = 13); for n = 4, the least such prime is 347 (3 + 4 = 7); for n = 6, the least such prime is 11617 (11 + 6 = 17); for n = 8, the least such prime is 5813 (5 + 8 = 13); for n = 10, the least such prime is 31013 (3 + 10 = 13); for n = 12, the least such prime is 51217 (5 + 12 = 17); for n = 14, the least such prime is 51419 (5 + 14 = 19); for n = 16, the least such prime is 431659 (43 + 16 = 59).

**Category:** Number Theory

[18] **viXra:1704.0210 [pdf]**
*submitted on 2017-04-16 19:36:57*

**Authors:** Zhang Tianshu

**Comments:** 13 Pages.

Due to exist forevermore uncorrelated limits of values of real number ε≥0, enable ABC conjecture to be able to be both proved and negated. In this article, we find a representative equality 1+2N(2N-2)=(2N-1)2 satisfying (2N-1)2>[Rad(1, 2N(2N-2), (2N-1)2)]1+ε, then both prove the ABC conjecture and negate the ABC conjecture according to two limits of values of ε.

**Category:** Number Theory

[17] **viXra:1704.0196 [pdf]**
*submitted on 2017-04-15 06:48:55*

**Authors:** François Mendzina Essomba, Gael Dieudonné Essomba Essomba

**Comments:** 7 Pages. algorithm, convergence and approximation

We give algorithms for the calculation of pi. These algorithms
can be easily developed in a linear manner and allows the calculation
of pi with an infinite degree of convergence. Of course, the calculation
of the second term passes through the first one, and it is necessary, as
this type of algorithms, for a larger memory for calculations contrary
to the formula BBP [1] whose execution corresponds to the order of
the desired number.
The advantage of our formulas in spite of the dificulty associated with extracting sin(x) lies in their degree of convergence, which is infinite, they prove the Borweins brothers hypothesis on the construction of algorithms At any speed as symbolized in our generic formula (8) of this paper.
These formulas for the most part are totally new :
We had found several other formulas of pi l

**Category:** Number Theory

[16] **viXra:1704.0146 [pdf]**
*replaced on 2017-04-25 10:45:44*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 6 Pages.

In this paper we define a new Mellin discrete convolution, which is related to Perron's formula. Also we introduce new explicit formulae for arithmetic function which generalize the explicit formulae of Weil.

**Category:** Number Theory

[15] **viXra:1704.0129 [pdf]**
*replaced on 2017-04-19 11:21:34*

**Authors:** Barry Foster

**Comments:** 2 Pages.

This attempt uses Bertrand’s postulate.

**Category:** Number Theory

[14] **viXra:1704.0121 [pdf]**
*submitted on 2017-04-10 07:46:25*

**Authors:** Shi-YuanDong

**Comments:** 2 Pages.

The prime partition of n!, On the Goldbach prime partition, and the algebraic sum of elements of prime.

**Category:** Number Theory

[13] **viXra:1704.0114 [pdf]**
*submitted on 2017-04-09 11:23:01*

**Authors:** Abdelghaffar Slimane

**Comments:** 3 Pages. Academic use only

We give a condition that an odd number in the neighborhood of a successive collatz numbers
set must verify to be a non-collatz number, and we use the result for odd numbers of the form 6k−1 at the boundary of a successive collatz numbers set.

**Category:** Number Theory

[12] **viXra:1704.0110 [pdf]**
*submitted on 2017-04-09 10:15:18*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This model can be also successfully used for forecasting purposes.

**Category:** Number Theory

[11] **viXra:1704.0102 [pdf]**
*submitted on 2017-04-09 00:26:53*

**Authors:** Marius Coman

**Comments:** 1 Page.

In this paper I make the following conjecture: there exist an infinity of Poulet numbers which can be written as a sum of two successive primes plus one (for the numbers that are the sum of two successive primes see the sequence A001043 in OEIS).

**Category:** Number Theory

[10] **viXra:1704.0101 [pdf]**
*replaced on 2017-04-14 23:03:01*

**Authors:** Chongxi Yu

**Comments:** 19 Pages.

Prime numbers are the basic numbers and are crucial important. There are many conjectures concerning primes are challenging mathematicians for hundreds of years. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. A kaleidoscope can produce an endless variety of colorful patterns and it looks like a magic, but when you open it, it contains only very simple, loose, colored objects such as beads or pebbles and bits of glass. Human is very easily cheated by 2 words, infinite and anything, because we never see infinite and anything, so we always make simple thing complex. Goldbach’s conjecture is about all very simple numbers, the pattern of prime numbers likes a “kaleidoscope” of numbers, we divided any even numbers into 5 groups and primes into 4 groups, Goldbach’s conjecture becomes much simpler. Here we give a clear proof for Goldbach's conjecture based on the fundamental theorem of arithmetic, the prime number theorem, and Euclid's proof that the set of prime numbers is endless.
Key words: Goldbach's conjecture, fundamental theorem of arithmetic, Euclid's proof of infinite primes, the prime number theorem

**Category:** Number Theory

[9] **viXra:1704.0098 [pdf]**
*submitted on 2017-04-08 09:49:00*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I state the following conjecture: let P be a Poulet number and n the integer for which the number (P – 1)/2^n is odd; then there exist an infinity of Poulet numbers for which the number q = (P – 1)/2^n – 2^n is prime.

**Category:** Number Theory

[8] **viXra:1704.0097 [pdf]**
*submitted on 2017-04-08 10:06:47*

**Authors:** Aaron Chau

**Comments:** 2 Pages.

或许在友善的下午茶叙上，笔者清心直说，既然代数无法正确筛选任一质数，这说明代数的
缺点是难免会把非质数来充当质数。不言而喻，数学毕竟不鼓励凭修饰把非质数来充当质数。
所以，虽然代数的工业用途广泛，但针对解决孪生质数猜想，算术才是一把能够开锁的钥匙。

**Category:** Number Theory

[7] **viXra:1704.0093 [pdf]**
*submitted on 2017-04-08 03:00:47*

**Authors:** Bing He

**Comments:** 16 Pages.

In this paper we introduce a finite field analogue for the Appell series F_{3} and
give some reduction formulae and certain generating functions for this function over finite fields.

**Category:** Number Theory

[6] **viXra:1704.0087 [pdf]**
*submitted on 2017-04-07 09:13:10*

**Authors:** Marius Coman

**Comments:** 2 Pages.

In this paper I conjecture that there are an infinity of primes which can be written as sqr ((p – q – 1)*p – q – 1), where p and q are successive primes, p > q.

**Category:** Number Theory

[5] **viXra:1704.0079 [pdf]**
*submitted on 2017-04-06 23:53:45*

**Authors:** C Sloane

**Comments:** 29 Pages. This paper is with several institutions and this submission is a time-stamp of authorship.

We discovered a beautiful symmetry to the equation x^n+y^n± z^n , first studied by Fermat, in a dependent variable t = x+y-z and the product (xyz) if we introduce a term we call the symmetric r = x^2+yz-xt-t^2. Once x^n+y^n± z^n is written in terms of powers of t, r and (xyz) we looked at the coefficient vs. exponent abstract space and found Lucas, Fibonacci and Convoluted Fibonacci sequences among other corollaries. We also found that 3 cases of a prime decomposition factor q of x^2+yz gave certain results for Fermat’s Last Theorem which could be eliminated if a forth case could also be solved. Intrigued by this, we then introduce partial congruence representations modulo a prime for this much harder forth case to find the ‘form’ of the solutions modulo q. The form of the solutions leads us to a cubic congruence method that solves the special and general cases. There are several pages and stages of the proofs where computer verification of the results is possible.

**Category:** Number Theory

[4] **viXra:1704.0058 [pdf]**
*submitted on 2017-04-05 08:41:51*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. This can also be used as a Universal Forecasting Model.

**Category:** Number Theory

[3] **viXra:1704.0056 [pdf]**
*submitted on 2017-04-05 10:40:16*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 10 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model. Also, this model can be used as a Universal Forecasting Model.

**Category:** Number Theory

[2] **viXra:1704.0029 [pdf]**
*submitted on 2017-04-03 11:20:53*

**Authors:** A. Zaganidis

**Comments:** 17 Pages. I have chosen the category Mathematics-Number Theory since most of the consequences of the present article are inside the number theory

In this work, we introduce the $n$-formal sequents and the formal numbers defined with the help of the second order logic. We give many concrete examples of formal numbers and $n$-formal sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-formal iff the sequent is composed by some closed hypotheses and a $n$-formal formula (an explicit sub-formula with one free variable which is only true with the unique natural number $n$), and no sub-sequent are composed by some closed sub-hypotheses and a $m$-formal explicit sub-formula with $m>1$. The definition is motivated by the intuition that the ``nature's hypotheses'' do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-formal sequent. Moreover, we postulate at second order of logic that the ``nature's hypotheses'' are not chosen randomly: the ``nature's hypotheses'' are the only hypotheses which give the largest formal number $N_Z=1497$. The Goldbach's conjecture, the Dubner's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) formal axiom. Finally, three open questions remain: Can we prove that a natural number is not formal? If a formal number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-formal? Does a sequent exist to make a difference between the definition of the $n$-formal sequents and the following variant of that definition: we look at the explicit sub-formulas of $\phi$ which induce the $m$-formal formulas instead of looking at the explicit sub-formulas of $\phi_{n-formal}$ which are $m$-formal formulas?

**Category:** Number Theory

[1] **viXra:1704.0012 [pdf]**
*submitted on 2017-04-02 07:11:13*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has presented a novel scheme of Universal Evolution Model.

**Category:** Number Theory