• Complain

Grigorieva - Methods of Solving Number Theory Problems

Here you can read online Grigorieva - Methods of Solving Number Theory Problems full text of the book (entire story) in english for free. Download pdf and epub, get meaning, cover and reviews about this ebook. year: 2018, publisher: Birkhäuser, genre: Children. Description of the work, (preface) as well as reviews are available. Best literature library LitArk.com created for fans of good reading and offers a wide selection of genres:

Romance novel Science fiction Adventure Detective Science History Home and family Prose Art Politics Computer Non-fiction Religion Business Children Humor

Choose a favorite category and find really read worthwhile books. Enjoy immersion in the world of imagination, feel the emotions of the characters or learn something new for yourself, make an fascinating discovery.

Grigorieva Methods of Solving Number Theory Problems
  • Book:
    Methods of Solving Number Theory Problems
  • Author:
  • Publisher:
    Birkhäuser
  • Genre:
  • Year:
    2018
  • Rating:
    3 / 5
  • Favourites:
    Add to favourites
  • Your mark:
    • 60
    • 1
    • 2
    • 3
    • 4
    • 5

Methods of Solving Number Theory Problems: summary, description and annotation

We offer to read an annotation, description, summary or preface (depends on what the author of the book "Methods of Solving Number Theory Problems" wrote himself). If you haven't found the necessary information about the book — write in the comments, we will try to find it.

Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermats (Pells) equations. It also covers Fermats factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Warings problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.

Grigorieva: author's other books


Who wrote Methods of Solving Number Theory Problems? Find out the surname, the name of the author of the book and a list of all author's works by series.

Methods of Solving Number Theory Problems — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Methods of Solving Number Theory Problems" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

Reset

Interval:

Bookmark:

Make
Springer International Publishing AG, part of Springer Nature 2018
Ellina Grigorieva Methods of Solving Number Theory Problems
1. Numbers: Problems Involving Integers
Ellina Grigorieva 1
(1)
Department of Mathematics and Computer Science, Texas Womans University, Denton, TX, USA
Ellina Grigorieva
Email:
Many important properties of integers were established in ancient times. In Greece, the Pythagorean school (6 Picture 1 century BC) studied the divisibility of numbers and considered various categories of numbers such as the primes, composite, perfect, and amicable. his Elements , Euclid (3 Picture 2 century BC) gives an algorithm for determining the greatest common divisor of two numbers, outlines the main properties of divisibility of integers, and proves the theorem that primes form an infinite set. Also in the 3 Picture 3 century BC, Eratosthenes discovered an algorithm to extract prime numbers from a series of natural numbers now called, The Sieve of Eratosthenes. There are many great names associated with the development of number theory such as Diophantus, Fermat , Descartes, Euler , Legendre, Gauss, Lagrange , to list just a few.
Over the centuries, an interest in numbers has not been entirely lost. Problems involving numbers are often included in math contests, some of which look simple but require a tremendous amounts of nonstandard thinking. Other problems look scary at first glance but allow beautiful and often short solutions.
The idea of this chapter is to develop the mathematical skills of the readers and enable them to solve unusual math problems involving integers and their properties. We will demonstrate how problems involving numbers can develop your creative thinking and how experience in solving some challenging problems will give you confidence in the subject matter.
1.1 Classification of Numbers: Even and Odd Integers
The positive whole numbers: Methods of Solving Number Theory Problems - image 4 are called the natural numbers and are used for counting, e.g., there are five apples in the basket and ordering, e.g., China is the largest country in the world by population. This is the oldest defined category of numbers and the simplest in membership. In mathematical notation, we may describe the sequence of natural numbers as Methods of Solving Number Theory Problems - image 5 .
The second set of numbers in order of composition is the set of integers that include natural numbers and their negatives, such as Methods of Solving Number Theory Problems - image 6 and zero 0 in order to make the operations of addition and subtraction closed on the set. The notation for this set is Picture 7 which comes from the German word, zahlen , for numbers.
The next set is the set of rational numbers , Picture 8 for quotient. We define a number to be rational if it can be written as a fraction Picture 9 where Picture 10 and Picture 11 . This way we do not have to deal with divisibility by zero. Obviously, natural numbers and integers are also rational but not vice versa.
The numbers that cannot be represented by a fraction are the irrational numbers , such as Picture 12 . This set does not have a special notation. Rational numbers and irrational numbers together form the set of real numbers . Descartes and Fermat (17 Picture 13 century) were the first to use the coordinate method to represent real numbers on the number line. Integers were discrete points on the line with whole coordinates while real numbers span a continuum, i.e., they fill out the entire number line without gaps. The introduction of real numbers allows us to perform the operations of addition, subtraction, multiplication, division (except by zero), and to raise real numbers to a power with a resultant that is also a member of the real numbers.
Finally, there is the largest set of numbers, the set of complex numbers Picture 14 . This set contains numbers of the type Picture 15 , where Picture 16 and i is an imaginary unit, such that Picture 17 . Complex numbers geometrically are points in the plane formed by the Real and Imaginary axis. If Methods of Solving Number Theory Problems - image 18 , then the complex number Methods of Solving Number Theory Problems - image 19 is a lattice point. We say that Methods of Solving Number Theory Problems - image 20 . The relationship between sets (excluding is shown in Figure Fig 11 Sets of Numbers All natural numbers can - photo 21 ) is shown in Figure .
Fig 11 Sets of Numbers All natural numbers can be divided into two groups - photo 22
Fig. 1.1
Sets of Numbers
All natural numbers can be divided into two groups: even numbers and odd numbers . Most of us know that 3, 11, 27 are odd numbers and 20, 36, or 100 are even numbers. Let us find the general form for odd and even numbers. What is common to all even numbers such as 2, 4, 20, 2000?
Correct. They are multiples of the number 2. Therefore, we can say that every even number can be written as 2 n , where Methods of Solving Number Theory Problems - image 23 . Different values of n create different even numbers, for example, Methods of Solving Number Theory Problems - image 24 , Methods of Solving Number Theory Problems - image 25 , etc.
Now let us find the general rule for an odd number. If you write all even and odd numbers together in ascending order from 1 to 100 as Methods of Solving Number Theory Problems - image 26
Next page
Light

Font size:

Reset

Interval:

Bookmark:

Make

Similar books «Methods of Solving Number Theory Problems»

Look at similar books to Methods of Solving Number Theory Problems. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.


Reviews about «Methods of Solving Number Theory Problems»

Discussion, reviews of the book Methods of Solving Number Theory Problems and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.