Mathematics
Number Theory
Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves the study of prime numbers, divisibility, modular arithmetic, and Diophantine equations. Number theory has applications in cryptography, computer science, and other fields.
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6 Key excerpts on "Number Theory"
eBook - PDF- Peter Grossman(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
Number Theory 12.1 What is Number Theory? No doubt you have been familiar with the natural numbers since childhood. For this reason, it might seem that there could be little more to learn about them, at least in comparison with other types of numbers: fractions, negative numbers, irrational numbers and complex numbers. Nothing could be further from the truth; in fact, some of the most challenging problems in mathematics involve just the natural numbers. Since ancient times, mathematicians have been intrigued by the subtle properties that underlie the apparent simplicity of these numbers. Number Theory is principally the study of the natural numbers. However, the term is often extended to include the study of all the integers or all the rational numbers, since many of the techniques developed in the study of the natural numbers can also be applied to the study of these more general types of numbers. Even the study of irrational numbers can be included, because any problem that involves distinguishing between rational and irrational numbers (for example a proof that a certain number or class of numbers is irrational) invariably requires the use of properties of the natural numbers in order to solve it. For our purposes, the main reason for studying Number Theory lies in its usefulness in many areas of computing. Until quite recently, Number Theory was viewed as a branch of mathematics with limited practical use, but the situation has changed dramatically as a result of the computer revolution. Data encryption, for example, relies heavily on results from Number Theory, as do the design of error-detecting and error-correcting codes and the generation of pseudo-random numbers for simulation programs, to name just some of the areas where Number Theory can be applied.
eBook - PDFDiscrete Mathematics
Proofs, Structures and Applications, Third Edition
- Rowan Garnier, John Taylor(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 9 Introduction to Number Theory Number Theory is concerned with the properties of the integers; it is sometimes referred to simply as ‘arithmetic’. As such, it deals with some of the most familiar mathematical objects. Despite this, Number Theory is certainly not trivial. There are many deep and beautiful results in Number Theory. Gauss was said to have described mathematics as the ‘Queen of the Sciences’ and Number Theory as the ‘Queen of Mathematics’. Number Theory has sometimes been thought of as an archetypal branch of pure mathematics—very interesting in its own right, but with little practical use outside mathematics. Indeed, G H Hardy who was a number theorist and a leading mathematician of the first half of the 20th century, wrote in his book, A Mathematician ’ s Apology , ‘I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.’ However, results in elementary Number Theory have been used to devise encryption algorithms that are very secure. The particular ‘public key’ properties of these algorithms make them ideal for encryption of data sent via the internet and, in so doing, enable e-commerce to be viable. Hardy also claimed that pure mathematics does no harm to the world: ‘no one has yet discovered any warlike purpose to be served by the theory of numbers . . . ’. But with the arrival of encryption systems based on Number Theory, the factorisation of some large positive integers is now considered a military secret. The point here is that applications of even the ‘purest’ part of mathematics can arise in unexpected places. We will consider public key encryption algorithms based on Number Theory in section 9.5. 436 Divisibility 437 9.1 Divisibility We begin with the simple process of dividing one integer by another. Early on in school mathematics, we learn to ‘divide and take remainders’.
eBook - PDFThe Story of Proof
Logic and the History of Mathematics
- John Stillwell(Author)
- 2022(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R 7 Number Theory The natural numbers 0, 1, 2, 3, 4, . . . are the most fundamental mathe- matical objects, understood to some extent by everybody. They are also the subject of the oldest unsolved problems in mathematics. For exam- ple, are there any odd perfect numbers? Infinitely many twin primes? Yet, until recent times Number Theory was often derided as a “bag of tricks,” of little interest or use to most mathematicians. Attitudes changed in recent decades, when the world went digital and numbers became its lifeblood, needing the protection of encryption, which ultimately relies on Number Theory. So the problem today is not to justify Number Theory but to understand it. As we will see in this chapter, Number Theory is hard because its meth- ods of proof draw on virtually all areas of mathematics. They include geometry, algebra, calculus, and some we haven’t discussed yet, such as topology. This is surprising because Number Theory has extremely sim- ple ingredients: 0, the successor function that takes us from one natural number to the next, and the principle of induction, which essentially says that all natural numbers come from 0 by repeated application of the successor function. The fact is, simple ingredients can create extreme complexity, which is why all the resources of mathematics have been recruited to help num- ber theory. In this chapter we will discuss the influence of geometry, algebra, and calculus on proofs in Number Theory—and vice versa, par- ticularly in the case of algebra. Later, when we come to the mathematical study of proof itself, we will see what makes Number Theory capable of such complexity. 146 CHAPTER 7 Number Theory 7.1 ELEMENTARY Number Theory As we saw in section 2.6, Euclid developed the fundamentals of num- ber theory in books 7–9 of the Elements. To do so, he used induction, the fundamental method of proof in Number Theory and other areas of mathematics that study finite objects, such as combinatorics.
eBook - PDFNumber, Shape, & Symmetry
An Introduction to Number Theory, Geometry, and Group Theory
- Diane L. Herrmann, Paul J. Sally Jr.(Authors)
- 2012(Publication Date)
- A K Peters/CRC Press(Publisher)
. . . . . 1 The Beginnings of Number Theory 1.1 Setting the Table: Numbers, Sets, and Functions Games such as the Triangle Game are intriguing to people who enjoy mathematics. These games build on elementary arithmetic skills and provide problems that have multiple solutions and even give rise to more problems than the initial simple question. They presume only a knowledge of basic arithmetic and a facility with elementary calculation. Numbers and Number Systems Before we start with the formal aspects of arithmetic and Number Theory, let’s discuss informally some of the terms and ideas encountered in ele-mentary school. First, we need to identify different number systems so that we know precisely which numbers we are using in our problem solv-ing attempts. The first set of numbers we meet are the natural numbers N . These are the ones we use to count objects: 1 , 2 , 3 , 4 , 5 , .... Next come the integers , which include 0 and the negatives of all the natural numbers. The letter Z stands for the integers, and comes from the German word for number, zahlen . The natural numbers allow us to do addition and multiplication, and the integers allow us to do both of these as well as subtraction. But when we get to division, we need the rational numbers . The rational numbers are a rich source of Number Theory questions. These numbers, sometimes called fractions, are those that can be expressed as the quotient of two integers a and b , written a b , where b is not allowed to be 0 and the fraction is reduced to lowest terms. We label the rational numbers Q , for quotient. 11 12 1 . The Beginnings of Number Theory Finally, we need the real numbers , R , which include the rational num-bers and also irrational numbers such as π and √ 2. These numbers will surface when we do geometry. For now, think of the real numbers as the set of all decimals. Sets Mathematicians use the word set to mean the same thing as the word does in ordinary English, namely, a collection of objects.
- Alasdair McAndrew(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 2 Basic Number Theory This chapter provides the mathematical background for much of the rest of the book. In particular, it investigates: • Prime numbers, their definition and uses. • Factorization. • Modular arithmetic, including powers and inverses. • Fermat’s theorem, Euler’s totient function and Euler’s generalization of Fermat’s theorem. • The Chinese remainder theorem. • The Euclidean algorithm, both standard and extended forms. • Quadratic residues and the Legendre symbol. • Some methods of primality testing. 2.1 Introduction Much modern cryptography is based around the theory of numbers, in par-ticular prime numbers and their properties. This chapter will include inves-tigating some of the basic properties and developing enough theory to enable future investigation of various public-key cryptosystems. Recall from Chap-ter 1 that the availability of a public-key cryptosystem allows neat protocols for digital signatures. 2.2 Some basic definitions In this chapter, and for the remainder of this text, all numbers will be inte-gers. One of the most important properties for this text is that of divisibility . 23 24 Cryptography with Open-Source Software Given two integers n and k , then n is divisible by k , written k | n if there is an integer m for which mk = n. Another way of saying the same thing is that k is a divisor (or factor) of n , or that k divides n . The following properties of divisibility are standard: 1. If k | n and n | m then k | m . 2. If k | n and k | m then k | ( am + bn ) for any integers a and b . More properties will be discussed later. A prime number is an integer greater than 1 whose only divisors are itself and 1. For example: 3, 5, 7, 11, 257, 65537 are all prime numbers; but 391 is not a prime number (because 391 = 17 × 23). It is a fact that there are infinitely many prime numbers, so that there are prime numbers as big as you like.
eBook - PDF- Steven J. Miller, Ramin Takloo-Bighash(Authors)
- 2020(Publication Date)
- Princeton University Press(Publisher)
PART 1 Basic Number Theory Chapter One Mod p Arithmetic, Group Theory and Cryptography In this chapter we review the basic Number Theory and group theory which we use throughout the book, culminating with a proof of quadratic reciprocity. Good in-troductions to group theory are [J, La3]; see [Da1, IR] for excellent expositions on congruences and quadratic reciprocity, and [Sil2] for a friendly introduction to much of the material below. We use cryptographic applications to motivate some basic background material in Number Theory; see [Ga] for a more detailed expo-sition on cryptography and [Lidl, vdP2] for connections with continued fractions. The guiding principle behind much of this chapter (indeed, much of this book and Number Theory) is the search for efficient algorithms. Just being able to write down an expression does not mean we can evaluate it in a reasonable amount of time. Thus, while it is often easy to prove a solution exists, doing the computations as written is sometimes impractical; see Chapter 6 of [BB] and [Wilf] for more on efficient algorithms. 1.1 CRYPTOGRAPHY Cryptography is the science of encoding information so that only certain specified people can decode it. We describe some common systems. To prove many of the properties of these crypto-systems will lead us to some of the basic concepts and theorems of algebra and group theory. Consider the following two password systems. In the first we choose two large distinct primes p and q ; for example, let us say p and q have about 200 digits each. Let N = pq and display the 400 digit number N for everyone to see. The password is any divisor of N greater than 1 and less than N . One very important property of the integers is unique factorization: any integer can be written uniquely as a product of prime powers. This implies that the only factorizations of N are 1 · N , N · 1 , p · q and q · p . Thus there are two passwords, p and q . For the second system, we choose a 5000 digit number.
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