# Number Theory

## 1004 Submissions

 viXra:1004.0140 [pdf] submitted on 10 Mar 2010

### On a Concatenation Problem

Authors: Henry Ibstedt

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

 viXra:1004.0135 [pdf] submitted on 30 Apr 2010

### The New Prime Theorem (38)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that there are infinitely many primes P such that each of jP3 + k - j is a prime.
Category: Number Theory

 viXra:1004.0134 [pdf] submitted on 30 Apr 2010

### The New Prime Theorem (37)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that there are infinitely many primes P such that each of jP3 + 7 - j is a prime.
Category: Number Theory

 viXra:1004.0133 [pdf] submitted on 30 Apr 2010

### The New Prime Theorem (36)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that there are infinitely many primes P such that each of jP3 + 5 - j is a prime.
Category: Number Theory

 viXra:1004.0132 [pdf] submitted on 30 Apr 2010

### The New Prime Theorem (35)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that there are infinitely many primes P such that 2P3 + 1 and P3 + 2 are all prime.
Category: Number Theory

 viXra:1004.0131 [pdf] submitted on 30 Apr 2010

### The New Prime Theorem (34)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that if J2 (ω) ≠ 0 then there are infinitely many primes P such that each of jP2 + k - j is a prime, if J2 (ω) = 0 then there are finitely many primes P such that each of jP2 + k - j is a prime.
Category: Number Theory

 viXra:1004.0126 [pdf] replaced on 4 May 2010

### A Fifth Smarandache Friendly Prime Pair

Authors: Philip Gibbs

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

 viXra:1004.0125 [pdf] submitted on 10 Mar 2010

### On a Problem Concerning the Smarandache Friendly Prime Pairs

Authors: Felice Russo

In this paper a question posed in  and concerning the Smarandache friendly prime pairs is analysed.
Category: Number Theory

 viXra:1004.0123 [pdf] submitted on 27 Apr 2010

### The New Prime Theorem (33)

Authors: Chun-Xuan Jiang

Using Jiang function we prove x2 + y4 (J. Friedlander and H. Iwaniec, The polynomial x2 + y4 Captures its primes, Ann. Math., 148(1998) 945-1040)
Category: Number Theory

 viXra:1004.0122 [pdf] submitted on 27 Apr 2010

### The New Prime Theorem (32)

Authors: Chun-Xuan Jiang

Using Jiang function we prove x3 + 2y3 (D. R. Heath-Brown, prime represented by x3 + 2y3, Acta Math., 186(2001)1-84).
Category: Number Theory

 viXra:1004.0119 [pdf] submitted on 24 Apr 2010

### The New Prime Theorem (31)

Authors: Chun-Xuan Jiang

Using Jiang function we prove and P1 = P9 ± m and P1 = (2P)9 ± n
Category: Number Theory

 viXra:1004.0118 [pdf] submitted on 24 Apr 2010

### The New Prime Theorem (30)

Authors: Chun-Xuan Jiang

Using Jiang function we prove and P1 = PP0 ± m and P1 = (2P)p0 ± n
Category: Number Theory

 viXra:1004.0117 [pdf] submitted on 24 Apr 2010

### The New Prime Theorem (29)

Authors: Chun-Xuan Jiang

Using Jiang function we prove and P1 = P5 ± m and P1 = (2P)5 ± n
Category: Number Theory

 viXra:1004.0116 [pdf] submitted on 24 Apr 2010

### The New Prime Theorem (28)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0115 [pdf] replaced on 14 Jun 2010

### Corrections to the wu-Sprung Potential for the Riemann Zeros and a New Hamiltonian Whose Energies Are the Prime Numbers

Authors: Jose Javier Garcia Moreta

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 loge (x) = ln(x) = log(x) for the logarithm , also unles we specify Σγ 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

 viXra:1004.0111 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (27)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture P: m2 +1 and m2 + 3 .
Category: Number Theory

 viXra:1004.0110 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (26)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture N: x3 + y3 + z3 .
Category: Number Theory

 viXra:1004.0109 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (25)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture M: x3 + y3 + k .
Category: Number Theory

 viXra:1004.0108 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (24)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture K: x3 + k .
Category: Number Theory

 viXra:1004.0107 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (23)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture F: am2 + bm+ c .
Category: Number Theory

 viXra:1004.0106 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (22)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture B: P, P + k .
Category: Number Theory

 viXra:1004.0105 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (21)

Authors: Chun-Xuan Jiang

Using Jiang function we prove binary Goldbach conjecture and N = P1 + ... + Pn 
Category: Number Theory

 viXra:1004.0104 [pdf] submitted on 20 Apr 2010

### The New Prime Theorem (20)

Authors: Chun-Xuan Jiang

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.
Category: Number Theory

 viXra:1004.0088 [pdf] submitted on 18 Apr 2010

### The Implicit Function in the Hardy-Littewood Conjecture

Authors: Tong Xin Ping
Comments: 6 Pages, In Chinese

The implicit function in the Hardy-Littewood conjecture
Category: Number Theory

 viXra:1004.0087 [pdf] submitted on 10 Mar 2010

### Existence and Number of Solutions of Diophantine Quadratic Equations with Two Unknowns in Z and N

Authors: Florentin Smarandache

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 ax2 - by2 = c .
Category: Number Theory

 viXra:1004.0071 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (19)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0070 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (18)

Authors: Chun-Xuan Jiang

Using Jiang function we prove Hardy-Littlewood conjecture E : x2 + 1
Category: Number Theory

 viXra:1004.0069 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (17)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that such that Pn = 2 P1P2 ... Pn-1 has infinitely many prime solutions.
Category: Number Theory

 viXra:1004.0068 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (16)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0067 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (15)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0066 [pdf] submitted on 10 Apr 2010

### The New Prime Theorem (14)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0060 [pdf] submitted on 8 Apr 2010

### The New Prime Theorem (13)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that n x an ± 1 has infinitely many prime solutions and n x 2n ± 1 have finite prime solutions.
Category: Number Theory

 viXra:1004.0059 [pdf] submitted on 8 Apr 2010

### The New Prime Theorem (12)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that 3 x a3 ± 1 has infinitely many prime solutions
Category: Number Theory

 viXra:1004.0058 [pdf] submitted on 8 Apr 2010

### The New Prime Theorem (11)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that 2 x a2 ± 1 has infinitely many prime solutions
Category: Number Theory

 viXra:1004.0045 [pdf] submitted on 6 Apr 2010

### The New Prime Theorem (10) there Are Finite Mersenne Primes and there Are Finite Repunits Primes

Authors: Chun-Xuan Jiang

Using Jiang function we prove the finite Mersenne primes and the finite repunits primes.
Category: Number Theory

 viXra:1004.0044 [pdf] submitted on 6 Apr 2010

### The New Prime Theorem (9) there Are Finite Fermat Primes

Authors: Chun-Xuan Jiang

Using Jiang function we prove the finite fermat primes.
Category: Number Theory

 viXra:1004.0043 [pdf] submitted on 6 Apr 2010

### The Prime Principle In Clusters And Nanostructures

Authors: Chun-Xuan Jiang

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

 viXra:1004.0042 [pdf] submitted on 6 Apr 2010

### Prime Theorem: P2 = AP1 + b, Polignac Theorem and Goldbach Theorem

Authors: Chun-Xuan Jiang

Using Jiang function we prove prime theorem: P2 = aP1 + b, Polignac theorem and Goldbach theorem.
Category: Number Theory

 viXra:1004.0041 [pdf] submitted on 8 Mar 2010

### K-Factorial

Authors: Florentin Smarandache

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

 viXra:1004.0040 [pdf] submitted on 8 Mar 2010

### Back and Forth Factorials

Authors: Florentin Smarandache

Back and Forth Factorials
Category: Number Theory

 viXra:1004.0038 [pdf] submitted on 8 Mar 2010

### Souvenirs from the Empire of Numbers

Authors: Florentin Smarandache

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

 viXra:1004.0034 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (8)

Authors: Chun-Xuan Jiang

Using Jiang function we prove that x6 + 1091 has no prime solutions.
Category: Number Theory

 viXra:1004.0033 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (7)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0032 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (6)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0031 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (5)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0030 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (4)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0029 [pdf] submitted on 4 Apr 2010

### The New Prime Theorem (3)

Authors: Chun-Xuan Jiang

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

 viXra:1004.0028 [pdf] submitted on 5 Apr 2010

### Disproofs of Riemann's Hypothesis

Authors: Chun-Xuan Jiang

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 Jn(ω) that can replace Riemann's ζ(s) function in view of its proved features: if Jn(ω) ≠ 0, then the function has infinitely many prime solutions; and if Jn(ω) = 0, then the function has finitely many prime solutions. By using the Jiang J2(ω) function we prove the twin prime theorem, Goldbach's theorem and the prime theorem of the form x2 + 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

 viXra:1004.0027 [pdf] replaced on 9 Jul 2011

### Foundations of Santilli's Isonumber Theory

Authors: Chun-Xuan Jiang