Number Theory: A Very Short Introduction

2020 ◽  
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Short Introduction ◽  
New Developments ◽  
Leonhard Euler

Number Theory: A Very Short Introduction explains the branch of mathematics primarily concerned with the counting numbers, 1, 2, 3, …. Long considered one of the most ‘beautiful’ areas of mathematics, number theory dates back over two millennia to the Ancient Greeks, but has seen some startling new developments in recent years. Trailblazers in the field include mathematicians Euclid of Alexandria, Pierre de Fermat, Leonhard Euler, and Carl Friedrich Gauss. Number theory has intrigued and attracted amateurs and professionals alike for thousands of years, appearing in both recreational contexts (puzzles) and practical concerns (cryptography). Some problems in number theory are easy, whereas others are notorious with no known solutions to date.

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

scholarly journals Major Advances and Challenges in Heterogeneous Catalysis for Environmental Applications: A Review

2018 ◽  
Vol 25 (1) ◽  
pp. 9-34 ◽  
Author(s):  
Stanisław Wacławek ◽  
Vinod V.T. Padil ◽  
Miroslav Černík
Keyword(s):  
Heterogeneous Catalysis ◽  
Sustainable Energy ◽  
Soil Treatment ◽  
Environmental Applications ◽  
Environmental Sciences ◽  
Short Introduction ◽  
New Developments ◽  
Historical Aspects ◽  
Small Section ◽  
Strongly Connected

Abstract Heterogeneous catalysis is one of the fastest developing branches of chemistry. Moreover, it is strongly connected to popular environment-related applications. Owing to the very fast changes in this field, for example, numerous discoveries in nanoscience and nanotechnologies, it is believed that an update of the literature on heterogeneous catalysis could be beneficial. This review not only covers the new developments of heterogeneous catalysis in environmental sciences but also touches its historical aspects. A short introduction to the mechanism of heterogeneous catalysis with a small section on advances in this field has also been elaborated. In the first part, recent innovations in the field of catalytic air, water, wastewater and soil treatment are presented, whereas in the second part, innovations in the use of heterogeneous catalysis for obtaining sustainable energy and chemicals are discussed. Catalytic processes are ubiquitous in all branches of chemistry and there are still many unsolved issues concerning them.

Emotion: A Very Short Introduction

2019 ◽  
Author(s):  
Dylan Evans
Keyword(s):  
Artificial Intelligence ◽  
Decision Making ◽  
Human Nature ◽  
Middle Ages ◽  
Emotional Impact ◽  
Neural Basis ◽  
Short Introduction ◽  
New Developments ◽  
Biological Basis ◽  
Wide Range

Was love invented by European poets in the Middle Ages or is it part of human nature? Will winning the lottery really make you happy? Is it possible to build robots that have feelings? Emotion: A Very Short Introduction explores the latest thinking about emotions, drawing on a wide range of scientific research, from anthropology and psychology to neuroscience and artificial intelligence. It discusses the evolution of emotions and their biological basis, the science of happiness, and the role that emotions play in memory and decision-making. This new edition has been updated to incorporate new developments in our understanding of emotions, including the neural basis of empathy and the emotional impact of films.

scholarly journals A review of new developments in the Friedel–Crafts alkylation – From green chemistry to asymmetric catalysis

2010 ◽  
Vol 6 ◽  
Author(s):  
Magnus Rueping ◽  
Boris J Nachtsheim
Keyword(s):  
Green Chemistry ◽  
Asymmetric Catalysis ◽  
Lewis Acid ◽  
Environmentally Benign ◽  
Short Introduction ◽  
New Developments ◽  
Wide Range ◽  
Benzyl Halides ◽  
Allyl Alcohols ◽  
Definition Of

The development of efficient Friedel–Crafts alkylations of arenes and heteroarenes using only catalytic amounts of a Lewis acid has gained much attention over the last decade. The new catalytic approaches described in this review are favoured over classical Friedel–Crafts conditions as benzyl-, propargyl- and allyl alcohols, or styrenes, can be used instead of toxic benzyl halides. Additionally, only low catalyst loadings are needed to provide a wide range of products. Following a short introduction about the origin and classical definition of the Friedel–Crafts reaction, the review will describe the different environmentally benign substrates which can be applied today as an approach towards greener processes. Additionally, the first diastereoselective and enantioselective Friedel–Crafts-type alkylations will be highlighted.

scholarly journals Considerations for Goldbach's Strong Conjecture

2021 ◽  
Author(s):  
Carleilton Severino Silva
Keyword(s):  
Number Theory ◽  
Prime Numbers ◽  
Weak Version ◽  
Strong Version ◽  
Leonhard Euler

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

Consciousness: A Very Short Introduction

2017 ◽  
Author(s):  
Susan Blackmore
Keyword(s):  
Neural Correlates ◽  
Altered States Of Consciousness ◽  
Altered States ◽  
Short Introduction ◽  
New Developments ◽  
Brain Science ◽  
States Of Consciousness ◽  
The World ◽  
Construction Of Self ◽  
The Brain

Consciousness is ‘the last great mystery for science’. How can a physical brain create our experience of the world? What creates our identity? Do we really have free will? Could consciousness itself be an illusion? Exciting new developments in brain science are continuing these debates, and the field has now expanded to include biologists, neuroscientists, psychologists, and philosophers. Consciousness: A Very Short Introduction clarifies the potentially confusing arguments, and the major theories, whilst also outlining the amazing pace of neuroscience discoveries. Covering areas such as construction of self in the brain, mechanisms of attention, neural correlates of consciousness, and physiology of altered states of consciousness, it highlights the latest findings.

Globalization: A Very Short Introduction

2020 ◽  
Author(s):  
Manfred B. Steger
Keyword(s):  
Climate Change ◽  
New Technologies ◽  
Short Introduction ◽  
Cultural Economic ◽  
New Developments ◽  
Mass Migration ◽  
The World ◽  
The Future ◽  
The Impact ◽  
Economic Deregulation

Globalization: A Very Short Introduction looks at the interconnected and accelerated processes changing how we see and experience the world. Is globalization really a new phenomenon? Is increased connection between people and nations inevitable, or are we witnessing the beginning of an era of ‘deglobalization’ or ‘anti-globalization’? Updated with new developments including advancing climate change, the Trump presidency, and the Mexico–USA border, this VSI explores the history and impact of globalization. Chapters on the cultural, economic, political, and ecological dimensions of globalization investigate the impact of new technologies, economic deregulation, and mass migration on our world and consider what we might expect from the future of globalization.

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