Prime numbers, friends who give problems, a trialogue with Papa Paulo by Paulo Ribenboim, pp. 320, £32 (paper), ISBN 978-9-814-72581-1, also available as e-book, World Scientific, 2016.

2017 ◽  
Vol 101 (552) ◽  
pp. 559-560
Author(s):  
Peter Macgregor
Keyword(s):  
Prime Numbers ◽  
World Scientific ◽  
Book World

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

Henry Kissinger's book World order: Reflections on the beginnings of nations and the course of history

2019 ◽  
Vol 3 (6) ◽  
pp. 165
Author(s):  
Abdul Hamid Al - Eid Al - Mousawi
Keyword(s):  
New World ◽  
World Order ◽  
New World Order ◽  
Central Idea ◽  
Global System ◽  
The World ◽  
Western European ◽  
Book World

The central idea of Henry Kissinger's latest book, The Global System, is that the world desperately needs a new world order, otherwise geopolitical chaos threatens the world, and perhaps chaos will prevail and settle in the world. According to Kissinger, the world order was not really there at all, but what was closest to the system was the Treaty of Westphalia, which included about twenty Western European states for almost four centuries.

The First 50 Million Prime Numbers

1977 ◽  
Vol 1 (S2) ◽  
pp. 7-19 ◽  
Author(s):  
Don Zagier
Keyword(s):  
Prime Numbers

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

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