[16] **viXra:1008.0089 [pdf]**
*submitted on 30 Aug 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This
is the Book proof. In this paper using Jiang function J_{2}(ω) we prove that the new prime
theorems (441)-(490) contain infinitely many prime solutions and no prime solutions.From (6)
we are able to find the smallest solution. π_{k}(N_{0},2) ≥ 1. This is the Book theorem.

**Category:** Number Theory

[15] **viXra:1008.0088 [pdf]**
*submitted on 31 Aug 2010*

**Authors:** Tong Xin Ping

**Comments:** 4 pages, In Chinese

We have inclusion-exclusion formula of π(N) and inclusion-exclusion formula of r_{2}(N). Make use of
inclusion-exclusion formula, we can obtain Hardy-Littlewood Conjecture (A).

**Category:** Number Theory

[14] **viXra:1008.0087 [pdf]**
*submitted on 30 Aug 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This
is the Book proof. In this paper using Jiang function J_{2}(ω) we prove that the new prime
theorems (541)-(590) contain infinitely many prime solutions and no prime solutions.From (6)
we are able to find the smallest solution. π_{k}(N_{0},2) ≥ 1. This is the Book theorem.

**Category:** Number Theory

[13] **viXra:1008.0086 [pdf]**
*submitted on 30 Aug 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 69 pages

Using Jiang function we are able to prove almost all prime problems in prime distribution. This
is the Book proof. In this paper using Jiang function J_{2}(ω) we prove that the new prime
theorems (491)-(540) contain infinitely many prime solutions and no prime solutions.From (6)
we are able to find the smallest solution. π_{k}(N_{0},2) ≥ 1. This is the Book theorem.

**Category:** Number Theory

[12] **viXra:1008.0082 [pdf]**
*submitted on 13 Mar 2010*

**Authors:** Sylvester Smith

**Comments:** 9 pages

Searching through the Archives of the Arizona State University,
I found interesting sequences of numbers and problems
related to them. I display some of them, and the readers
are welcome to contribute with solutions or ideas.

**Category:** Number Theory

[11] **viXra:1008.0080 [pdf]**
*submitted on 27 Aug 2010*

**Authors:** Chun-Xuan Jiang

**Comments:** 69 pages

Using Jiang function we prove that the new prime theorems (391)-(440) contain infinitely many
prime solutions and no prime solutions.

**Category:** Number Theory

[10] **viXra:1008.0069 [pdf]**
*submitted on 25 Aug 2010*

**Authors:** A.A.K. Majumdar

**Comments:** 217 pages

It was in mid-nineties of the last century when I received a letter from Professor Ion Patrascu of
the Fratii Buzesti College, Craiova, Romania, with lots of enclosures, introducing me with this
new branch of Mathematics. Though my basic undergraduate degree is in Mathematics, my
research field at that time was Operations Research and Mathematical Programming.

**Category:** Number Theory

[9] **viXra:1008.0064 [pdf]**
*submitted on 23 Aug 2010*

**Authors:** Tong Xin Ping

**Comments:** 3 pages, In Chinese

This upper bound estimation prevailed over upper bound estimation of Chen Jing Run

**Category:** Number Theory

[8] **viXra:1008.0062 [pdf]**
*submitted on 22 Aug 2010*

**Authors:** Robert G. Wilson V

**Comments:** 3 pages

"Smarandache consecutive sequences" is the nth member of the consecutive sequence, e. g. Sm(11)=1234567891011, and RSm(11)=1110987654321.
Following is the prime version of "Smarandache consecutive sequences"

**Category:** Number Theory

[7] **viXra:1008.0061 [pdf]**
*submitted on 22 Aug 2010*

**Authors:** Richard Pinch

**Comments:** 6 pages

Charles Ashbacher [1] has posed a number of questions relating
to the pseudo-Smarandache function Z(n). In this note we show that
the ratio of consecutive values Z(n + 1)/Z(n) and Z(n - 1)/Z(n) are unbounded;
that Z(2n)/Z(n) is unbounded; that n/Z(n) takes every integer
value infinitely often; and that the series Σ_{n} 1/Z(n)^{α} is convergent for any
α > 1.

**Category:** Number Theory

[6] **viXra:1008.0054 [pdf]**
*submitted on 20 Aug 2010*

**Authors:** Tong Xin Ping

**Comments:** 4 pages, In Chinese

According to five assumptions, get five proofs

**Category:** Number Theory

[5] **viXra:1008.0036 [pdf]**
*submitted on 12 Aug 2010*

**Authors:** J. S. Markovitch

**Comments:** 4 pages

The number of primes in the inclusive intervals defined by consecutive
Fibonacci numbers exhibits interesting behavior between the Fibonacci
numbers 55 and 196418. Specifically, starting with the interval [55,
89] through the interval [121393,196418] the ratio of the number of
primes in successive intervals is a value that alternates high, low, high,
low, etc.

**Category:** Number Theory

[4] **viXra:1008.0022 [pdf]**
*replaced on 29 Nov 2011*

**Authors:** Morgan D. Rosenberg

**Comments:** 11 pages

Presented herein is a proof of Fermat's Last Theorem, which is not only short
(relative to Wiles' 109 page proof), but is also performed using relatively
elementary mathematics. Particularly, the binomial theorem is utilized, which
was known in the time of Fermat (as opposed to the elliptic curves of Wiles'
proof, which belong to modern mathematics). Using the common integer expression
a^{n} + b^{n} = c^{n} for Fermat's Last Theorem, the
substitutions c = b+i and b = a+j are made,
where i and j are integers. Using a Taylor expansion (i.e., in the form of the
binomial theorem), Fermat's Last Theorem reduces to (see paper) and what remains
to be proven, from this equation, is that (see paper) only has rational solutions for
n=1 and n=2. This proof is presented herein, thus proving that
a^{n} + b^{n} = c^{n} only has
integer solutions for a, b and c for integer values of the exponent n=1 or n=2.

**Category:** Number Theory

[3] **viXra:1008.0021 [pdf]**
*submitted on 8 Aug 2010*

**Authors:** Tong Xin Ping

**Comments:** 2 pages, In Chinese

Don't confuse quantitative change and qualitative change.

**Category:** Number Theory

[2] **viXra:1008.0006 [pdf]**
*submitted on 4 Aug 2010*

**Authors:** Tong Xin Ping

**Comments:** 2 Pages. In Chinese

The method of the quantitative change can not solve the problem of the qualitative change.

**Category:** Number Theory

[1] **viXra:1008.0001 [pdf]**
*replaced on 20 Nov 2010*

**Authors:** Valery Demidovich

**Comments:** v2 is 29 Pages in Russian, v3 is 28 pages in English

The work maintenance: attempt to solve a problem about definition of set of simple numbers-twins is made.
In work absolutely new approach which is based on algorithm of a sieve of Eratosfena is applied.

**Category:** Number Theory