Number Theory

Some Fourier Series - Identities

Authors: Edgar Valdebenito
Comments: 3 Pages.

We give some Fourier Series - Identities.
Category: Number Theory

Numbers: Part 3, " Ramanujan's Integral ".

Authors: Edgar Valdebenito
Comments: 4 Pages.

We recall a Ramanujan's integral: int(f(x),x=0..1)=(pi*pi)/15
Category: Number Theory

On Prime Numbers

Authors: Yuji Masuda
Comments: 1 Page.

This study focuses on primes.
Category: Number Theory

Riemann Hypothesis Elementary Proof

Authors: Shekhar Suman
Comments: 4 Pages.

ANALYTIC CONTINUATION AND SIMPLE APPLICATION OF ROLLE'S THEOREM
Category: Number Theory

Riemann Hypothesis

Authors: Shekhar suman
Comments: 11 Pages. Please send replies at shekharsuman068@gmail.com

Analytic continuation and monotonicity gives the zeroes
Category: Number Theory

The Riemann Hypothesis Proof

Authors: Shekhar Suman
Comments: 9 Pages. Please read once

We take the integral representation of the Riemann Zeta Function over entire complex plane, except for a pole at 1. Later we draw an equivalent to the Riemann Hypothesis by studying its monotonicity properties.
Category: Number Theory

The Riemann Hypothesis

Authors: Shekhar Suman
Comments: 7 Pages.

Analytical continuation gives a functional equation having nice properties. Further we give an equivalence of riemann hypotheis through its monotonicity in specific intervals
Category: Number Theory

A Final Proof of The abc Conjecture

Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages. Comments welcome. Submitted to the Ramanujan Journal.

In this paper, we consider the abc conjecture. As the conjecture c<rad^2(abc) is less open, we give firstly the proof of a modified conjecture that is c<2rad^2(abc). The factor 2 is important for the proof of the new conjecture that represents the key of the proof of the main conjecture. Secondly, the proof of the abc conjecture is given for \epsilon \geq 1, then for \epsilon \in ]0,1[. We choose the constant K(\epsion) as K(\epsilon)=2e^{\frac{1}{\epsilon^2} } for $\epsilon \geq 1 and K(\epsilon)=e^{\frac{1}{\epsilon^2}} for \epsilon \in ]0,1[. Some numerical examples are presented.
Category: Number Theory

God and Mathematical Beauty IV

Authors: Johannes Abdus Salam
Comments: 1 Page.

I discovered an evidence of the existence of God as the mathematically beautiful equality of the Euler product.
Category: Number Theory

On Certain Pi_{q}-Identities of W. Gosper

Authors: Bing He
Comments: 16 Pages. All comments are welcome

In this paper we employ some knowledge of modular equations with degree 5 to confirm several of Gosper's Pi_{q}-identities. As a consequence, a q-identity involving Pi_{q} and Lambert series, which was conjectured by Gosper, is proved. As an application, we confirm an interesting q-trigonometric identity of Gosper.
Category: Number Theory

The Josephus Numbers

Authors: Kouider Mohammed Ridha
Comments: 3 Pages.

According to Josephuse history we present a new numbers called The josephuse numbers. Hence we give explicit formulas to compute the Josephus-numbers J(n)where n is positive integer . Furthermore we present a new fast algorithm to calculate J(n). We also offer prosperities , and we generalized it for all positive real number non-existent, Finally we give .the proof of properties.
Category: Number Theory

Rational Distance

Authors: Radomir Majkic
Comments: 3 Pages.

There are countable many rational distance squares, one square for each rational trigonometric Pythagorean pair (s; c) : s^2+c^2=1 and a rational number r:
Category: Number Theory

The Prime Gaps Between Successive Primes to Ensure that there is Atleast One Prime Between Their Squares Assuming the Truth of the Riemann Hypothesis

Authors: Prashanth R. Rao
Comments: 2 Pages.

Based on Dudek’s proof that assumed the truth of the Riemann’s hypothesis, that there exists a prime between {x – (4/pi)( x^ 1/2)(log x)} and x, we determine the size of prime gaps that must exist between successive primes, so that we can be sure that there is atleast one prime number between their squares.
Category: Number Theory

Euler Numbers , Catalan's Constant , Number pi

Authors: Edgar Valdebenito
Comments: 2 Pages.

We give some formulas involving Catalan's constant G=0.915965...
Category: Number Theory

Numbers : Part 1

Authors: Edgar Valdebenito
Comments: 1 Page.

This note presents two Elementary integrals.
Category: Number Theory

A Note on Ramanujan's Integral

Authors: Edgar Valdebenito
Comments: 2 Pages.

We give some remarks on Ramanujan's integral: int(f(x),x=0..infinite)=(2/3)sqrt(pi).
Category: Number Theory

The Information Paradox

Authors: Andrea Berdondini
Comments: 4 Pages.

ABSTRACT: The following paradox is based on the consideration that the value of a statistical datum does not represent a useful information, but becomes a useful information only when it is possible to proof that it was not obtained in a random way. In practice, the probability of obtaining the same result randomly must be very low in order to consider the result useful. It follows that the value of a statistical datum is something absolute but its evaluation in order to understand whether it is useful or not is something of relative depending on the actions that have been performed. So two people who analyze the same event, under the same conditions, performing two different procedures obviously find the same value, regarding a statistical parameter, but the evaluation on the importance of the data obtained will be different because it depends on the procedure used. This condition can create a situation like the one described in this paradox, where in one case it is practically certain that the statistical datum is useful, instead in the other case the statistical datum turns out to be completely devoid of value. This paradox wants to bring attention to the importance of the procedure used to extract statistical information; in fact the way in which we act affects the probability of obtaining the same result in a random way and consequently on the evaluation of the statistical parameter.
Category: Number Theory

Fermat's Last Theorem (Excluding the Case of N=2^t). Unified Method

Authors: Victor Sorokine
Comments: 4 Pages. English version

IN THE FIRST CASE every number (A) is replaced by the sum (A'+A°n) of the last digit and the remainder. After binomial expansion of the Fermat's equality, all the members are combined in two terms: E=A'^n+B'^n-C'^n with the third digit E''', which in one of the n-1 equivalent Fermat's equalities is equal to 2, and the remainder D with the third digit D''', which is equal either to 0, or to n-1, and therefore the third digit of the number A^n+B^n-C^n is different from 0.

IN THE SECOND CASE (for example A=A°n^k, but (BС)'≠0), after having transformed the 3kn-digit ending of the number B into 1 and having left only the last siginificant digits of the numbers A, В, С, simple calculations show that the (3kn-2)-th digit of the number A^n+B^n-C^n is not 0 and does not change after the restoration of all other digits in the numbers A, B, C, because it depends only on the last digit of the number A°.