[13] **viXra:1202.0086 [pdf]**
*replaced on 2012-03-03 23:45:52*

**Authors:** Song Wen Miao

**Comments:** 3 Pages. CHINESE

ALL EYES ARE ON FERMAT LAST THEOREM WHO PROVED FORST

**Category:** Number Theory

[12] **viXra:1202.0085 [pdf]**
*submitted on 2012-02-28 04:04:30*

**Authors:** Predrag Terzich

**Comments:** 4 Pages.

We explore some properties of generalized Fermat primes of the form : F_n(2q)=(2q)^(2^n)+1 , where n > 1 and q is an odd prime number .

**Category:** Number Theory

[11] **viXra:1202.0079 [pdf]**
*replaced on 2012-05-18 13:09:21*

**Authors:** Stephen Crowley

**Comments:** 39 Pages.

The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) 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). 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 the reflection functions χ(N;s) are also provided. The values ζw(1-N;s) are found to be related to the Bernoulli numbers. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.

**Category:** Number Theory

[10] **viXra:1202.0067 [pdf]**
*submitted on 2012-02-19 15:28:34*

**Authors:** Carlos Giraldo Ospina

**Comments:** 2 Pages.

In this paper Legendre’s Conjecture and Brocard’s Conjecture are proved by determining the amount of prime numbers that are located between N^2 and (N+1)^2.

**Category:** Number Theory

[9] **viXra:1202.0066 [pdf]**
*replaced on 2012-02-24 18:01:30*

**Authors:** Stephen Crowley

**Comments:** 25 Pages.

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An analytic continuation formula for these hypergeometric functions exists and is used to derive some infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of hypergeometric functions at n − 1. The form might be considered to be a shift operator for the Riemann zeta function which leads to the curious values ζF(0) = I_0(2) − 1 and ζF(1) = Ei(1) − γ which involve a Bessel function of the first kind and an exponential integral respectively and differ from the values ζ(0) = −1/2 and ζ(1) = ∞ given by the usual method of continuation. Interpreting these “hypergeometrically continued” values of the zeta constants in terms of reciprocal common factor probability we have ζF(0)^-1 ~ 78.15% and ζF(1)^-1 ~ 75.88% which contrasts with the standard known values for sensible cases like ζ(2)^-1 ~ 60.79% and ζ(3)^-1 ~ 83.19%. The combinatorial definitions of the Stirling numbers of the second kind, and the 2-restricted Stirling numbers of the second kind are recalled because they appear in the differential equatlon satisfied by the hypergeometric representation of the polylogarithm. The notion of fractal strings is related to the (chaotic) Gauss map of the unit interval which arises in the study of continued fractions, and another chaotic map is also introduced called the “Harmonic sawtooth” whose Mellin transform is the (appropritately scaled) Riemann zeta function. These maps are within the family of what might be called “deterministic chaos”. Some number theoretic definitions are also recalled.

**Category:** Number Theory

[8] **viXra:1202.0063 [pdf]**
*submitted on 2012-02-18 19:13:46*

**Authors:** Germán Paz

**Comments:** 32 Pages, 15 pages of tables. On the tables that appear from pages 8 to 22, the numbers
that are located on the third column and have commas should rather have dots, since this work is
written in English. This detail does not change results at all.

In this paper 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) and infinitely many prime numbers of the form ap - b. In particular, all this proves that there
are infinitely many prime numbers of the form 2p + 1, which proves there are infinitely many Sophie
Germain Prime Numbers. This document also contains Lic. Carlos Giraldo Ospina's solution to the
Polignac's Conjecture and to the Twin Prime Conjecture, which is one of Landau's Problems. Previous
papers (written in Spanish language) were reviewed and approved by Lic. C. G. Ospina and versions of
those papers were posted by this person on his own website and on ABCdatos. You may search for the
papers' titles on the internet. You may also visit the websites that are mentioned in this paper.
This work was submitted to the Journal of Number Theory.

**Category:** Number Theory

[7] **viXra:1202.0061 [pdf]**
*submitted on 2012-02-18 21:49:14*

**Authors:** Germán Paz

**Comments:** 19 Pages. This paper was submitted to the Journal of Number Theory.

In this paper it is proved that for every positive integer 'k' there are infinitely many prime numbers of the form n^2+k. As a result, it is proved that there are infinitely many prime numbers of the form n^2+1. This document also proposes a new and important 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. Previous papers (written in Spanish language) were reviewed and approved by Carlos Giraldo Ospina (Lic. Matemáticas, USC, Cali, Colombia). This person posted versions of these papers at his own personal website and at ABCdatos. You may search for those papers on the internet. You may also visit the websites that are mentioned in this paper.

**Category:** Number Theory

[6] **viXra:1202.0056 [pdf]**
*submitted on 2012-02-17 10:43:30*

**Authors:** Chun-Xuan Jiang

**Comments:** 6 Pages.

Using cmplex hyperbolic functions and complex trionometric functions ,we reapear the Fermat marvelus proofs for Fermat last theorem

**Category:** Number Theory

[5] **viXra:1202.0055 [pdf]**
*replaced on 2012-03-23 11:28:14*

**Authors:** Choe Ryong Gil

**Comments:** 12 pages, 1 schema

The Riemann hypothesis is one of the most important unsolved problems in the modern mathematics. The Riemann hypothesis is closely related with the distribution of prime numbers. The Robin inequality is one of the famous criterions for the Riemann hypothesis. The Robin inequality is related with the sum of divisors function. In this paper we prove that the Robin inequality holds unconditionally. The main idea is to prove a certain inequality on the sum of divisors function, whch is equivalent to the Robin inequality.

**Category:** Number Theory

[4] **viXra:1202.0048 [pdf]**
*replaced on 2012-03-25 07:24:41*

**Authors:** Choe Ryong Gil

**Comments:** 7 pages

The Riemann hypothesis is well known. The Riemann hypothesis is related with many problems of the analytical number theory. And there have been found some propositions equivalent to one. In particular, the Robin inequality is one of the most famous criterions for the Riemann hypothesis. In this paper we would show one inequality on the sum of divisors function. This inequality is weaker than the Robin inequality, but equivalent to one.

**Category:** Number Theory

[3] **viXra:1202.0030 [pdf]**
*replaced on 2012-03-22 11:12:48*

**Authors:** Choe Ryong Gil

**Comments:** 25 pages, 2 tables

The Robin inequality is well known. This inequality is one of the most important and famous criterion for the Riemann hypothesis, but it is unsolved completely yet. In this paper we prove one inequality on the sum of divisors function. This inequality is closely related with the Robin inequality. The main idea is to introduce a new concept, which would be called a sigma-index of the natural number.

**Category:** Number Theory

[2] **viXra:1202.0029 [pdf]**
*replaced on 2012-02-15 02:03:24*

**Authors:** Predrag Terzich

**Comments:** 5 Pages.

We present deterministic primality test for Fermat numbers . Essentially this test is similar to the Lucas-Lehmer primality test for Mersenne numbers

**Category:** Number Theory

[1] **viXra:1202.0024 [pdf]**
*submitted on 2012-02-10 06:54:39*

**Authors:** Predrag Terzich

**Comments:** 4 Pages.

We present a theorem concerning the form of Mersenne
numbers . We also discuss a closed form expression which links prime numbers and natural logarithms .

**Category:** Number Theory