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**1 - 6**of**6**### New experimental results concerning the Goldbach conjecture

- ALGORITHMIC NUMBER THEORY (THIRD INTERNATIONAL SYMPOSIUM, ANTS-III
, 1998

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### A set of new Smarandache functions, sequences and conjectures

- in number theory, American Research Press, 2000: http://www.gallup.unm.edu/~smarandache/Felice-Russo-book1.pdf

"... 2000 To two stars: ..."

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### EXPANSION AND IMPROVEMENT OF SIEVE AND APPLICATION IN GOLDBACH’S PROBLEM

, 904

"... Abstract. This paper expands and improves on the general Sieve method. This expaned and improved Sieve is applied to Goldbach’s problem. A new estimate of the exception set in Goldbach’s number E(X), an improved lower bound D1,2(N) and upper bound D(N) are proposed. The proposed values ..."

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Abstract. This paper expands and improves on the general Sieve method. This expaned and improved Sieve is applied to Goldbach’s problem. A new estimate of the exception set in Goldbach’s number E(X), an improved lower bound D1,2(N) and upper bound D(N) are proposed. The proposed values

### Generalized Riemann Hypothesis

, 2012

"... (Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The p ..."

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(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA]). In this paper, we give a proof of Generalized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach’s weak conjecture (also known as the odd Goldbach conjecture) one of the oldest and bestknown unsolved problems in number theory and in all of mathematics.

### 0.2 +---------::;zl~""""--------------I

"... In this paper some Smarandache conjectures and open questions will be analysed. The first three conjectures are related to prime numbers and formulated by F. Smarandache in [1]. 1) First Smarandache conjecture on primes The equation: where Pn is the n-th prime, has a unique solution between 0.5 and ..."

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In this paper some Smarandache conjectures and open questions will be analysed. The first three conjectures are related to prime numbers and formulated by F. Smarandache in [1]. 1) First Smarandache conjecture on primes The equation: where Pn is the n-th prime, has a unique solution between 0.5 and 1; • the maximum solution occurs for n = 1, i.e. when x = 1; the minimum solution occurs for n = 31, i.e. when x = 0.567148K = aD First of all observe that the function Bnex) which graph is reported in the fig. 5.1 for some values of n, is an increasing function for x> 0 and then it admits a unique solution for 0.5:$; x:$; 1. 172 Smarandache function Bn(x) vs x

### Dealing with prime numbers I.: On the Goldbach

"... In this paper we present some observations about the well-known Goldbach con-jecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also determine very thin and low diverging ranges in which the proba ..."

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In this paper we present some observations about the well-known Goldbach con-jecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also determine very thin and low diverging ranges in which the probability of finding a prime is one.