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Dealing with prime numbers I.: On the Goldbach conjecture  [PDF]
Fausto Martelli
Mathematics , 2013,
Abstract: In this paper we present some observations about the well-known Goldbach conjecture. 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.
About certain prime numbers  [PDF]
Diana Savin
Mathematics , 2009,
Abstract: We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers.
The Bouniakowsky conjecture and the density of polynomial roots to prime moduli  [PDF]
Timothy Foo
Mathematics , 2009, DOI: 10.4064/aa144-1-1
Abstract: We establish a result linking the Bouniakowsky conjecture and the density of polynomial roots to prime moduli.
Some history about Twin Prime Conjecture  [PDF]
Sadegh Nazardonyavi
Mathematics , 2012,
Abstract: This article is a collected information from some books and papers, and in most cases the original sentences is reserved about twin prime conjecture.
Prime numbers, quantum field theory and the Goldbach conjecture  [PDF]
Miguel-Angel Sanchis-Lozano,J. Fernando Barbero G.,Jose Navarro-Salas
Mathematics , 2012, DOI: 10.1142/S0217751X12501369
Abstract: Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\dag}$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.
Formulas giving prime numbers under Cramér's conjecture  [PDF]
Bakir Farhi
Mathematics , 2006,
Abstract: Under Cram\'er's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$. Under the same conjecture, we also prove that for any $\epsilon>0$, there exists a positive real number $B$ for which the formula $[B.{n!}^{2+\epsilon}]$ gives a prime number for any sufficiently large positive integer $n$.
The Proof of Goldbach’s Conjecture on Prime Numbers  [PDF]
Silviu Guiasu
Natural Science (NS) , 2019, DOI: 10.4236/ns.2019.119029
Abstract: Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.
Remarks on Goldbach’s Conjecture on Prime Numbers  [PDF]
Silviu Guiasu
Natural Science (NS) , 2019, DOI: 10.4236/ns.2019.1112036
Abstract: The oldest Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. The recent proof [1] connected Goldbach’s Conjecture with the fact that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes. The present paper contains explicit additional and complementary details of the proof, insisting on the existence and the number of Goldbach’s representations of even positive integers as sums of pairs of primes.
Two statements that are equivalent to a conjecture related to the distribution of prime numbers  [PDF]
Germán Paz
Mathematics , 2014,
Abstract: Let $n\in\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every $n\leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\lfloor\sqrt{n}\rfloor-1]$. Let $\pi[n+g(n),n+f(n)+g(n)]$ denote the amount of prime numbers in the interval $[n+g(n),n+f(n)+g(n)]$. Here we show that the conjecture described in [8] is equivalent to the statement that $$\pi[n+g(n),n+f(n)+g(n)]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$ where $$f(n)=\left(\frac{n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta}{|n-\lfloor\sqrt{n}\rfloor^2-\lfloor\sqrt{n}\rfloor-\beta|}\right)(1-\lfloor\sqrt{n}\rfloor)\text{, }g(n)=\left\lfloor1-\sqrt{n}+\lfloor\sqrt{n}\rfloor\right\rfloor\text{,}$$ and $\beta$ is any real number such that $1<\beta<2$. We also prove that the conjecture in question is equivalent to the statement that $$\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]\ge 1\text{, }\forall n\in\mathbb{Z}^+\text{,}$$ where $$S_n=n+\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor^2-\frac{1}{2}\left\lfloor\frac{\sqrt{8n+1}-1}{2}\right\rfloor+1\text{.}$$ We use this last result in order to create plots of $h(n)=\pi[S_n,S_n+\lfloor\sqrt{S_n}\rfloor-1]$ for many values of $n$.
A proof of a simple conjecture about harmonic numbers  [PDF]
Jacopo D'Aurizio
Mathematics , 2011,
Abstract: We prove that, for any prime number $p\geq 5$, the set of natural numbers $n$ such that $p\mid H_n$ is finite.
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