[50] **viXra:1004.0140 [pdf]**
*submitted on 10 Mar 2010*

**Authors:** Henry Ibstedt

**Comments:**
13 pages.

This article has been inspired by questions asked by Charles
Ashbacher in the Journal of Recreational Mathematics, vol. 29.2. It concerns the
Smarandache Deconstructive Sequence. This sequence is a special case of a more
general concatenation and sequencing procedure which is the subject of this
study. Answers are given to the above questions. The properties of this kind of
sequences are studied with particular emphasis on the divisibility of their terms by
primes.

**Category:** Number Theory

[49] **viXra:1004.0135 [pdf]**
*submitted on 30 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

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

**Category:** Number Theory

[48] **viXra:1004.0134 [pdf]**
*submitted on 30 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that there are infinitely many primes P such that each of
jP^{3} + 7 - j is a prime.

**Category:** Number Theory

[47] **viXra:1004.0133 [pdf]**
*submitted on 30 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that there are infinitely many primes P such that each of
jP^{3} + 5 - j is a prime.

**Category:** Number Theory

[46] **viXra:1004.0132 [pdf]**
*submitted on 30 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that there are infinitely many primes P such that 2P^{3} + 1 and
P^{3} + 2 are all prime.

**Category:** Number Theory

[45] **viXra:1004.0131 [pdf]**
*submitted on 30 Apr 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^{2} + k - j is a prime, if J_{2} (ω) = 0 then there are finitely many primes P such
that each of jP^{2} + k - j is a prime.

**Category:** Number Theory

[44] **viXra:1004.0126 [pdf]**
*replaced on 4 May 2010*

**Authors:** Philip Gibbs

**Comments:** 4 pages

A Smarandache friendly prime pair (SFPP) is a pair of prime numbers (p,q), p < q, such that
the product pq is equal to the sum of all primes from p to q inclusive. Previously four such
pairs were known: (2,5), (3,13), (5,31) and (7,53). Now a fifth one is found by a brute force
computer search. A heuristic approximation can be to estimate the expected number of SFPPs in
a given interval. The result suggests that the probability of further pairs existing is about 0.07.

**Category:** Number Theory

[43] **viXra:1004.0125 [pdf]**
*submitted on 10 Mar 2010*

**Authors:** Felice Russo

**Comments:**
3 pages.

In this paper a question posed in [1] and concerning the Smarandache
friendly prime pairs is analysed.

**Category:** Number Theory

[42] **viXra:1004.0123 [pdf]**
*submitted on 27 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove x^{2} + y^{4} (J. Friedlander and H. Iwaniec, The polynomial
x^{2} + y^{4} Captures its primes, Ann. Math., 148(1998) 945-1040)

**Category:** Number Theory

[41] **viXra:1004.0122 [pdf]**
*submitted on 27 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove x^{3} + 2y^{3} (D. R. Heath-Brown, prime represented by x^{3} + 2y^{3}, Acta Math., 186(2001)1-84).

**Category:** Number Theory

[40] **viXra:1004.0119 [pdf]**
*submitted on 24 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove and P_{1} = P^{9} ± m and P_{1} = (2P)^{9} ± n

**Category:** Number Theory

[39] **viXra:1004.0118 [pdf]**
*submitted on 24 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove and P_{1} = P^{P0} ± m and P_{1} = (2P)^{p0} ± n

**Category:** Number Theory

[38] **viXra:1004.0117 [pdf]**
*submitted on 24 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove and P_{1} = P^{5} ± m and P_{1} = (2P)^{5} ± n

**Category:** Number Theory

[37] **viXra:1004.0116 [pdf]**
*submitted on 24 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that 1P = P ± m and 1 P = 2P ± n have infinitely many

**Category:** Number Theory

[36] **viXra:1004.0115 [pdf]**
*replaced on 14 Jun 2010*

**Authors:** Jose Javier Garcia Moreta

**Comments:** 10 pages

We review the Wu-Sprung potential adding a correction involving a
fractional derivative of Riemann Zeta function, we study a global semiclassical analysis
in order to fit a Hamiltonian H=T+V fitting to the Riemann zeros and another new
Hamiltonian whose energy levels are precisely the prime numbers, through these paper
we use the notation log_{e} (x) = ln(x) = log(x) for the logarithm , also unles we specify
Σ_{γ} h(γ) means that we sum over ALL the imaginary parts of the nontrivial zero
on both the upper and lower complex plane.

**Category:** Number Theory

[35] **viXra:1004.0111 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture P: m^{2} +1 and m^{2} + 3 [4].

**Category:** Number Theory

[34] **viXra:1004.0110 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture N: x^{3} + y^{3} + z^{3} [4].

**Category:** Number Theory

[33] **viXra:1004.0109 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove Hardy-Littlewood conjecture M: x^{3} + y^{3} + k [4].

**Category:** Number Theory

[32] **viXra:1004.0108 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove Hardy-Littlewood conjecture K: x^{3} + k [4].

**Category:** Number Theory

[31] **viXra:1004.0107 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture F: am^{2} + bm+ c [4].

**Category:** Number Theory

[30] **viXra:1004.0106 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture B: P, P + k [4].

**Category:** Number Theory

[29] **viXra:1004.0105 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove binary Goldbach conjecture and N = P1 + ... + Pn [4]

**Category:** Number Theory

[28] **viXra:1004.0104 [pdf]**
*submitted on 20 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

Using Jiang function we prove that Jiang prime k -tuple theorem is true[1-3] and
Hardy-Littlewood prime k -tuple conjecture is false[4-8]. The tool of additive prime number
theory is basically the Hardy-Littlewood prime tuple conjecutre, but can not prove and count
any prime problems[6].

**Category:** Number Theory

[27] **viXra:1004.0088 [pdf]**
*submitted on 18 Apr 2010*

**Authors:** Tong Xin Ping

**Comments:** 6 Pages, In Chinese

The implicit function in the Hardy-Littewood conjecture

**Category:** Number Theory

[26] **viXra:1004.0087 [pdf]**
*submitted on 10 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:**
2 pages.

In this short note we study the existence and number of solutions in the
set of integers (Z) and in the set of natural numbers (N) of Diophantine equations of
second degree with two unknowns of the general form ax^{2} - by^{2} = c .

**Category:** Number Theory

[25] **viXra:1004.0071 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove that such that (see paper) has infinitely many prime
solutions.

**Category:** Number Theory

[24] **viXra:1004.0070 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove Hardy-Littlewood conjecture E : x^{2} + 1

**Category:** Number Theory

[23] **viXra:1004.0069 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove that such that P_{n} = 2 P_{1}P_{2} ... P_{n-1} has infinitely many prime
solutions.

**Category:** Number Theory

[22] **viXra:1004.0068 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove that there exist infinitely many primes P such that each of
(j)^{n} P + (k - j)^{n} is a prime.

**Category:** Number Theory

[21] **viXra:1004.0067 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

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

**Category:** Number Theory

[20] **viXra:1004.0066 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 3 pages

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

**Category:** Number Theory

[19] **viXra:1004.0060 [pdf]**
*submitted on 8 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove that n x a^{n} ± 1 has infinitely many prime solutions
and n x 2^{n} ± 1 have finite prime solutions.

**Category:** Number Theory

[18] **viXra:1004.0059 [pdf]**
*submitted on 8 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that 3 x a^{3} ± 1 has infinitely many prime solutions

**Category:** Number Theory

[17] **viXra:1004.0058 [pdf]**
*submitted on 8 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 pages

Using Jiang function we prove that 2 x a^{2} ± 1 has infinitely many prime solutions

**Category:** Number Theory

[16] **viXra:1004.0045 [pdf]**
*submitted on 6 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove the finite Mersenne primes and the finite repunits primes.

**Category:** Number Theory

[15] **viXra:1004.0044 [pdf]**
*submitted on 6 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove the finite fermat primes.

**Category:** Number Theory

[14] **viXra:1004.0043 [pdf]**
*submitted on 6 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 7 pages

Why we have five fingers. We suggest two principles: (1) the prime principle and (2)
the symmetric principle. We prove that 1, 3, 5, 7, 11, 23, 47, and 2, 4, 6, 10, 14, 22, 46,
94 are the most stable numbers, which are the basic building-blocks in clusters and
nanostructures. The prime principle is the mathematical foundations for clusters and
nanosciences. It is a theory of everything.

**Category:** Number Theory

[13] **viXra:1004.0042 [pdf]**
*submitted on 6 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 2 pages

Using Jiang function we prove prime theorem: P_{2} = aP_{1} + b, Polignac
theorem and Goldbach theorem.

**Category:** Number Theory

[12] **viXra:1004.0041 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 1 pages

As a generalization of the factorial and double factorial one defines the kfactorial
of n as the below product of all possible strictly positive factors (see paper)

**Category:** Number Theory

[11] **viXra:1004.0040 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

Back and Forth Factorials

**Category:** Number Theory

[10] **viXra:1004.0038 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 20 pages

Browsing through my fifth to twelfth grade years of preoccupation for creation I
discovered a notebook of Number Theory.
I liked to play with numbers as Tudor Arghezi (1880-1967) - our second national
Romanian poet {after the genial poet Mihai Eminescu (1850-1889)} - played with words.
I was so curious and amazed by the numbers' properties.
Interesting theorems, equations, and inequalities!
Such fascinating people who dedicated their research to numbers, just for the sake of
science!
I collected many results and tried to write a handbook of mathematicians and their
results.

**Category:** Number Theory

[9] **viXra:1004.0034 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that x^{6} + 1091 has no prime solutions.

**Category:** Number Theory

[8] **viXra:1004.0033 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 15 - j is a prime.

**Category:** Number Theory

[7] **viXra:1004.0032 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 9 - j is a prime.

**Category:** Number Theory

[6] **viXra:1004.0031 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + k - j is a prime.

**Category:** Number Theory

[5] **viXra:1004.0030 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 7 - j is a prime.

**Category:** Number Theory

[4] **viXra:1004.0029 [pdf]**
*submitted on 4 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 1 page

Using Jiang function we prove that there exist infinitely many primes P such that each jP + 5 - j is a prime.

**Category:** Number Theory

[3] **viXra:1004.0028 [pdf]**
*submitted on 5 Apr 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 13 pages

As it is well known, the Riemann hypothesis on the zeros of the ζ(s)
function has been assumed to be true in various basic developments of
the 20-th century mathematics, although it has never been proved to
be correct. The need for a resolution of this open historical problem
has been voiced by several distinguished mathematicians. By using preceding
works, in this paper we present comprehensive disproofs of the
Riemann hypothesis. Moreover, in 1994 the author discovered the arithmetic
function J_{n}(ω) that can replace Riemann's ζ(s) function in view of
its proved features: if J_{n}(ω) ≠ 0, then the function has infinitely many
prime solutions; and if J_{n}(ω) = 0, then the function has finitely many
prime solutions. By using the Jiang J_{2}(ω) function we prove the twin
prime theorem, Goldbach's theorem and the prime theorem of the form
x^{2} + 1. Due to the importance of resolving the historical open nature
of the Riemann hypothesis, comments by interested colleagues are here
solicited.

**Category:** Number Theory

[2] **viXra:1004.0027 [pdf]**
*replaced on 9 Jul 2011*

**Authors:** Chun-Xuan Jiang

**Comments:** 413 pages

In my works (see the bibliography at the end of the Preface) I often expressed
the view that the protracted lack of resolution of fundamental problems in science
signals the needs of basically new mathematics. This is the case, for example, for:
quantitative representations of biological structures; resolution of the vexing
problem of grand-unification; invariant treatment of irreversibility at the classical and
operator levels; identification of hadronic constituents definable in our spacetime;
achievement of a classical representation of antimatter; and other basic open
problems.

**Category:** Number Theory

[1] **viXra:1004.0020 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 2 pages

Back and Forth Summands

**Category:** Number Theory