[36] **viXra:1005.0107 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Yi Yuan, Zhang Wenpeng

**Comments:** 3 pages

see paper for abstract

**Category:** Number Theory

[35] **viXra:1005.0106 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Felice Russo

**Comments:** 13 pages

In this paper the main properties of Smarandache Square Complementary
function has been analyzed. Several problems still unsolved are reported too.

**Category:** Number Theory

[34] **viXra:1005.0105 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Sabin Tabirca, Tatiana Tabirca

**Comments:** 7 pages

In this article we present two new results concerning the Smarandache Ceil
function. The first result proposes an equation for the number of fixed-point number of
the Smarandache ceil function. Based on this result we prove that the average of the
Smarandache ceil function is Θ(n) .

**Category:** Number Theory

[33] **viXra:1005.0102 [pdf]**
*replaced on 19 Jun 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 33 pages

Using Jiang function we prove that the new prime theorems (45)-(70) contain infinitely many
prime solutions and no prime solutions.

**Category:** Number Theory

[32] **viXra:1005.0096 [pdf]**
*submitted on 24 May 2010*

**Authors:** Tong Xin Ping

**Comments:** 3 Pages, In Chinese

We can find all solutions of Goldbach conjecture (A) ling in the closed
interval [pr+1, N-pr-1], and we can obtain expression of the number of solutions of
Goldbach conjecture (A).

**Category:** Number Theory

[31] **viXra:1005.0092 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

In this short paper we propose four conjectures in synthetic geometry that generalize
Erdos-Mordell Theorem, and three conjectures in number theory that generalize Fermat
Numbers.

**Category:** Number Theory

[30] **viXra:1005.0088 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function J_{2}(ω) we prove that jP^{n} + 9 - j contain infinitely many prime solutions.

**Category:** Number Theory

[29] **viXra:1005.0087 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that jP^{8} + k - j contain infinitely many prime solutions.

**Category:** Number Theory

[28] **viXra:1005.0086 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that jP^{7} + k - j contain infinitely many prime solutions.

**Category:** Number Theory

[27] **viXra:1005.0085 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that jP^{6} + k - j contain infinitely many prime solutions.

**Category:** Number Theory

[26] **viXra:1005.0084 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that jP^{5} + k - j contain infinitely many prime solutions.

**Category:** Number Theory

[25] **viXra:1005.0083 [pdf]**
*submitted on 21 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that if J_{2}(ω) ≠ 0 then there are infinitely many primes P
such that each of jP^{4} + k - j is a prime, J_{2}(ω) = 0 then there are finite primes P such
that each of jP^{4} + k - j is a prime.

**Category:** Number Theory

[24] **viXra:1005.0067 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Felice Russo

**Comments:**
5 pages.

The Smarandache P and S persistence of a prime

**Category:** Number Theory

[23] **viXra:1005.0064 [pdf]**
*submitted on 15 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 16 Pages

We establish the Santilli's isomathematics based on the generalization of the modern mathematics.
(see paper for rest of abstract with equations)

**Category:** Number Theory

[22] **viXra:1005.0058 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 3 pages

We prove that for any partition of a set which contains an infinite arithmetic
(respectively geometric) progression into two subsets, at least one of these subsets
contains an infinite number of triplets such that each triplet is an arithmetic (respectively
geometric) progression.

**Category:** Number Theory

[21] **viXra:1005.0054 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Mladen V. Vassilev-Missana, Krassimir T. Atanassov

**Comments:** 67 pages, Book in Romanian, French and English. Proposed and solved problems for students' mathematical
competitions in number theory, algebra, geometry, trigonometry, calculus.

During the five years since publishing [2], we have obtained many
new results related to the Smarandache problems. We are happy to
have the opportunity to present them in this book for the enjoyment
of a wider audience of readers.
The problems in Chapter two have also been solved and published
separately by the authors, but it makes sense to collate them
here so that they can be better seen in perspective as a whole, particularly
in relation to the problems elucidated in Chapter one.
Many of the problems, and more especially the techniques employed
in their solution, have wider applicability than just the Smarandache
problems, and so they should be of more general interest to
other mathematicians, particularly both professional and amateur
number theorists.

**Category:** Number Theory

[20] **viXra:1005.0049 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 112 pages

The development of mathematics continues in a rapid rhythm, some unsolved problems
are elucidated and simultaneously new open problems to be solved appear.

**Category:** Number Theory

[19] **viXra:1005.0047 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

In this paper we propose a method of solving a Nonlinear Diophantine Equation by
converting it into a System of Diophantine Linear Equations.

**Category:** Number Theory

[18] **viXra:1005.0042 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any there are infinitely many primes P such that each of jP^{P0} + j+1 is a prime.

**Category:** Number Theory

[17] **viXra:1005.0041 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any there are infinitely many primes P such that each of P^{P0} + 4^{n} is a prime.

**Category:** Number Theory

[16] **viXra:1005.0040 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any there are infinitely many primes kPsuch that each of P^{P0} + (2j)^{2} is a prime.

**Category:** Number Theory

[15] **viXra:1005.0039 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any there are infinitely many primes kPsuch that each of P^{P0} + j(j+1) is a prime.

**Category:** Number Theory

[14] **viXra:1005.0038 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
jP^{5} + j +1 is a prime.

**Category:** Number Theory

[13] **viXra:1005.0037 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
P^{5} + 4^{n} is a prime.

**Category:** Number Theory

[12] **viXra:1005.0036 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
P^{5} + (2j)^{2} is a prime.

**Category:** Number Theory

[11] **viXra:1005.0035 [pdf]**
*submitted on 11 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
P^{5} + j( j +1) is a prime.

**Category:** Number Theory

[10] **viXra:1005.0032 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
jP^{3} + j + 1 is a prime.

**Category:** Number Theory

[9] **viXra:1005.0031 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
P^{3} + 4^{n} is a prime.

**Category:** Number Theory

[8] **viXra:1005.0030 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each of
P^{3} + (2 j)^{2} is a prime.

**Category:** Number Theory

[7] **viXra:1005.0029 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove that P, P^{15} + j(j+1)(j=1,...,7) contain no prime solutions.

**Category:** Number Theory

[6] **viXra:1005.0028 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove that P, P^{9} + j(j+1)(j=1,...,7) contain no prime solutions.

**Category:** Number Theory

[5] **viXra:1005.0027 [pdf]**
*submitted on 9 May 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 Pages

Using Jiang function we prove for any k there are infinitely many primes P such that each
of P^{3} + j( j + 1) is a prime.

**Category:** Number Theory

[4] **viXra:1005.0025 [pdf]**
*submitted on 10 May 2010*

**Authors:** Steffen Bode

**Comments:** 6 Pages.

I establish the existence of a unique binary pattern inherent to the 3n+1
step, and then use this binary pattern to prove the 3n+1 problem for all
positive integers.

**Category:** Number Theory

[3] **viXra:1005.0023 [pdf]**
*submitted on 11 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 20 pages

In this paper a small survey is presented on eighteen new functions and four new
sequences, such as: Inferior/Superior f-Part, Fractional f-Part, Complementary
function with respect with another function, S-Multiplicative, Primitive
Function, Double Factorial Function, S-Prime and S-Coprime Functions, Smallest
Power Function.

**Category:** Number Theory

[2] **viXra:1005.0017 [pdf]**
*submitted on 5 May 2010*

**Authors:** Mihály Bencze, Florentin Smarandache

**Comments:** 2 pages

About an Identity and its Applications

**Category:** Number Theory

[1] **viXra:1005.0008 [pdf]**
*submitted on 2 May 2010*

**Authors:** Tong Xin Ping

**Comments:** 3 Pages, In Chinese

Chen Jing Run proved that "On the representation of a large even integer as the sum of a
prime and the product of at most two primes" and lower bound estimations of the number of
solutions. Jiang Chun Xuan, Tong Xin Ping proved that "An even integer as the sum of a
prime and the product of two primes" and compute formula of the number of solutions. This
paper compares the accuracy of the three formulas

**Category:** Number Theory