[7] **viXra:1205.0118 [pdf]**
*replaced on 2013-05-24 10:38:39*

**Authors:** Bertrand Wong

**Comments:** 25 Pages.

The author had published a paper on the solutions for the twin primes conjecture in an international mathematics journal in 2003 and had since then been working on the solutions for the Goldbach conjecture, which is another problem relating to the prime numbers. This paper, which comprises of 4 parts that are each self-contained, is a combination and modification of the author’s 2 papers published recently in another international mathematics journal. The expected mode of solving the Goldbach conjecture appears to be the utilisation of advanced calculus or analysis, e.g., by the summation, or, integration, of the reciprocals involving directly or indirectly the primes to see whether they converge or diverge, in order to get a “feel” of the pattern of the distribution of the primes. But, such a method of solving the problem has evidently not succeeded so far. Some other approach or approaches could be more appropriate. This paper brings up a number of such approaches.

**Category:** Number Theory

[6] **viXra:1205.0108 [pdf]**
*submitted on 2012-05-29 23:56:06*

**Authors:** Emmanuel Markakis, Christopher Provatidis, Nikiforos Markakis

**Comments:** 30 Pages.

This paper presents a deterministic process of finding all pairs (p,q) of odd numbers (composites and primes) of natural numbers ≥ 3 whose sum (p + q) is equal to a given even natural number 2n ≥ 6. Subsequently, based on the above procedure and also relying on the distribution of primes in the set of natural numbers, we propose a closed analytical formula, which estimates the number of primes which satisfy Goldbach’s conjecture for positive integers ≥ 6.

**Category:** Number Theory

[5] **viXra:1205.0099 [pdf]**
*submitted on 2012-05-25 06:49:29*

**Authors:** Maik Becker-Sievert

**Comments:** 1 Page.

(n+a)= 2n+(a-n)=2n-(n-a)
proofs direct
Goldbach Conjecture , 1 is prime
Levys = Lemoine's Conjecture with a,n are primes
Polignacs Conjecture

**Category:** Number Theory

[4] **viXra:1205.0077 [pdf]**
*submitted on 2012-05-19 16:01:41*

**Authors:** Germán Paz

**Comments:** 44 Pages. Although the results that are considered as main results have not been published, some of the theorems in this paper appear in Gen. Math. Notes. See: On the Interval [n,2n]: Primes, Composites and Perfect Powers, Gen. Math. Notes, 15(1) (2013), 1-15.

This paper is a LaTeX document which combines previously posted papers 'Infinitely Many Prime Numbers of the Form ap±b' (viXra:1202.0063 submitted on 2012-02-18 19:13:46; url: http://vixra.org/abs/1202.0063) and 'Solution to One of Landau's Problems' (viXra:1202.0061 submitted on 2012-02-18 21:49:14; url: http://vixra.org/abs/1202.0061) into one paper. The information contained in this paper is the same as the information contained in those two original papers. No new information or results are being added. ABSTRACT. In this paper it is proved that for every positive integer 'k' there are infinitely many prime numbers of the form n^2+k, which means that there are infinitely many prime numbers of the form n^2+1. In addition to this, in this document it is proved that if 'a' and 'b' are two positive integers which are coprime and also have different parity, then there are infinitely many prime numbers of the form ap+b, where 'p' is a prime number. Moreover, it is also proved that there are infinitely many prime numbers of the form ap-b. In other words, it is proved that the progressions ap+b and ap-b generate infinitely many prime numbers. In particular, all this implies that there are infinitely many prime numbers of the form 2p+1 (since the numbers 2 and 1 are coprime and have different parity), which means that there are infinitely many Sophie Germain Prime Numbers. This paper also proposes an important new conjecture about prime numbers called 'Conjecture C'. If this conjecture is true, then Legendre's Conjecture, Brocard's Conjecture and Andrica's Conjecture are all true, and also some other important results will be true.

**Category:** Number Theory

[3] **viXra:1205.0076 [pdf]**
*replaced on 2012-06-23 13:52:48*

**Authors:** Stephen Crowley

**Comments:** 9 Pages.

The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x)=⌊x−1⌋(x⌊x−1⌋+x−1) multiplied by s((s+1)/(s-1)). A finite-sum approximation to ζ(s) denoted by ζw(N;s) which has real roots at s=−1 and s=0 is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). A closed-form expression for the integral of ζw(N;s) over the interval s=-1..0 is given. The function χ(N;s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of ζw(N;s) and the reflection functions χ(N;s) are also provided. The values ζw(N;1−n) for integer values of n are found to be related to the Bernoulli numbers.

**Category:** Number Theory

[2] **viXra:1205.0042 [pdf]**
*submitted on 2012-05-07 20:38:15*

**Authors:** Chun-Xuan Jiang

**Comments:** 98 Pages.

Using Jiang function we prove the new prime theorems(1541)-(1590)

**Category:** Number Theory

[1] **viXra:1205.0019 [pdf]**
*submitted on 2012-05-04 05:53:24*

**Authors:** A.S.N. Misra

**Comments:** 8 Pages. I include an abstract with this upload as requested and also as the first paragraph of the PDF.

Here we argue that entropy is more fundamentally an analytical than an empirical concept, thus explaining its hitherto puzzling manifestation in the prime number distribution.
We suggest a precise formula for quantifying the presence of entropy in the continuum of positive integers and we use this breakthrough equation (connecting the world of pure mathematics with that of physics) to strongly suggest possible lines of proof of the Riemann hypothesis and the P versus NP problem and also the necessary primality of the number one.

**Category:** Number Theory