Elementary. Number. Theory and lts. Applications. Kenneth H. Rosen. AT&T Informo Elementary Number Theory and Its Applications, 5th edition, Instructor's. Elementary Number Theory and Its Applications (5th Edition). Home · Elementary Author: Kenneth H. Rosen Elementary Number Theory, Sixth Edition. Elementary Number Theory, 6th Edition. Kenneth H. Rosen, AT&T Laboratories. © |Pearson | Available. Share this page. Elementary Number Theory, 6th.

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Sorry, this document isn't available for viewing at this time. In the meantime, you can download the document by clicking the 'Download' button above. (See Problem 33 of Section l.2of the text.) Library of Congress Cataloging in Publication Data. Rosen, Kenneth H. Elementary number theory and its applications. Elementary Number Theory - 6th Edition - Kenneth H. Rosen - Ebook download as PDF File .pdf), Text File .txt) or read book online. Number Theory by Rosen.

The Greatest Integer Function In number theory. What integers do the representations in Exercise 19 represent if each is the two's complement representation of an integer? Decide which of the following integersare divisible by Find the following Fibonacci numbers. Many number theoretic problems.

Proceed with the left hand side, that is,. For every integer there is an integer solution to the equation.

The integer is called the additive inverse of and is denoted by. By we mean. Chegg Solution Manuals are written by vetted Chegg 1 experts, and rated by students - so you know you're getting high quality answers. Solutions Manuals are available for thousands of the most popular college and high school textbooks in subjects such as Math, Science Physics , Chemistry , Biology , Engineering Mechanical , Electrical , Civil , Business and more.

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A manual providing resources supporting the computations and explorations can be found on the Web site for this book. This manual provides worked-out solutions or partial solutions to many of these computational and exploratory exercises, as well as hints and guidance for attacking others. An extensive collection of applets are provided on the Web site. These applets can be used by students for some common computations in number theory and to help understand concepts and explore conjectures.

Besides algorithms for comptutions in number theory, a collection of cryptographic applets is also provided. A useful collection of suggested projects can also be found on the Web site for this book. These projects can serve as final projects for students and for groups of students. Worked solutions to all exercises in the text, including the even-numbered execises, and a variety of other resources can be found on the Web site for instructors which is not available to students.

Among these other resources are sample syllabi, advice on planning which sections to cover, and a test bank.

This book can serve as the text for elementary number theory courses with many different slants and at many different levels. Consequently, instructors will have a great deal of flexibility designing their syllabi with this text. Most instructors will want to cover the core material in Chapter 1 as needed , Section 2. To fill out their syllabi, instructors can add material on topics of interest.

Generally, topics can be broadly classified as pure versus applied. P ure topics include Mobius inversion Section 7. Some instructors will want to cover accessible applications such as divisibility tests, the perpetual calendar, and check digits Chapter 5. Those instructors who want to stress computer applications and cryptography should cover Chapter 2 and Chapter 8. They may also want to include Sections 9. After deciding which topics to cover, instructors may wish to consult the following figure displaying the dependency of chapters: Although Chapter 2 may be omitted if desired, it does explain the big-0 notation used throughout the text to describe the complexity of algorithms.

Chapter 12 depends only on Chapter 1, as shown, except for T heorem Section Chapter 11 can be studied without covering Chapter 9 if the optional comments involving primitive roots in Section 9. For further assistance, instructors can consult the suggested syllabi for courses with different emphases provided in the Instructor's Resource Guide on the Web site. Acknowledgments I appreciate the continued strong support and enthusiam of Bill Hoffman, my editor at Pearson and Addison-Wesley far longer than any of the many editors who have preceded him, and Greg Tobin, president of the mathematics division of Pearson.

Special thanks go to Bart Goddard who has prepared the solutions of all exercises in this book, including those found at the end of the book and on the Web site, and who has reviewed the entire book. Thanks also goes to Larry Washington and Keith Conrad for their helpful suggestions concerning congruent numbers and elliptic curves. Many of their ideas have been incorporated in this edition. My profound thanks go to the reviewers who helped me prepare the sixth edition: They have helped improve this book throughout its life.

Finally, I thank in advance all those who send me suggestions and corrections in the future. You may send such material to me in care of Pearson at math pearson. Kenneth H. Thousands of people work on communal number theory problems over the Internet W hat is this subject, and why are so many people interested in it today?

Number theory is the branch of mathematics that studies the properties of, and the relationships between, particular types of numbers. Of the sets of numbers studied in number theory, the most important is the set of positive integers. More specifically, the primes, those positive integers with no positive proper factors other than 1, are of special importance.

A key result of number theory shows that the primes are the multiplicative building blocks of the positive integers. This result, called the fundamental theorem of arithmetic, tells us that every positive integer can be uniquely written as the product of primes in nondecreasing order. Interest in prime numbers goes back at least years, to the studies of ancient Greek mathematicians. Perhaps the first question about primes that comes to mind is whether there are infinitely many.

In The Elements, the ancient Greek mathematician Euclid provided a proof, that there are infinitely many primes. This proof is considered to be one of the most beautiful proofs in all of mathematics.

Interest in primes was rekindled in the seventeenth and eighteenth centuries, when mathematicians such as Pierre de Fermat and Leonhard Euler proved many important results and conjectured approaches for generating primes.

The study of primes progressed substantially in the nineteenth century; results included the infinitude of primes in arithmetic progressions, and sharp estimates for the number of primes not exceeding a positive number. An example of a notorious unsolved question is whether there are infinitely many twin primes, which are pairs of primes that differ by 2.

New results will certainly follow in the coming decades, as researchers continue working on the many open questions involving primes. The development of modem number theory was made possible by the German mathematician Carl Friedrich Gauss, one of the greatest mathematicians in history, who in the early nineteenth century developed the language of congruences. We say that two integers a and b are congruent modulo m, where m is a positive integer, if m divides a.

Gauss developed many important concepts in number theory; for example, he proved one of its most subtle and beautiful results, the law of quadratic reciprocity. This law relates whether a prime p is a perfect square modulo 1. Gauss developed many different proofs of this law, some of which have led to whole new areas of number theory. Distinguishing primes from composite integers is a key problem of number theory. Work on this problem has produced an arsenal of primality tests.

The simplest primality test is simply to check whether a positive integer is divisible by each prime not exceeding its square root. Unfortunately, this test is too inefficient to use for extremely large positive integers. Many different approaches have been used to determine whether an integer is prime. For example, in the nineteenth century, Pierre de Fermat showed that p divides 2P - 2 whenever p is prime.

Some mathematicians thought that the converse also was true that is, that if n divides 2n - 2, then n must be prime. However, it is not; by the early nineteenth century, composite integers n, such as , were known for which n divides 2n - 2. Such integers are called pseudoprimes.

Though pseudoprimes exist, primality tests based on the fact that most composite integers are not pseudoprimes are now used to quickly find extremely large integers which are are extremely likely to be primes. However, they cannot be used to prove that an integer is prime.

Finding an efficient method to prove that an integer is prime was an open question for hundreds of years.

In a surprise to the mathematical community, this question was solved in by three Indian computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena.

Their algorithms can prove that an integer n is prime in polynomial time in terms of the number of digits of n. Factoring a positive integer into primes is another central problem in number theory. The factorization of a positive integer can be found using trial division, but this method is extremely time-consuming. Fermat, Euler, and many other mathematicians devised imaginative factorization algorithms, which have been extended in the past 30 years into a wide array of factoring methods.

Using the best-known techniques, we can easily find primes with hundreds or even thousands of digits; factoring integers with the same number of digits, however, is beyond our most powerful computers. The dichotomy between the time required to find large integers which are almost certainly prime and the time required to factor large integers is the basis of an extremely important secrecy system, the RSA cryptosystem.

The RSA system is a public key cryptosystem, a security system in which each person has a public key and an associated private key. Messages can be encrypted by anyone using another person's public key, but these messages can be decrypted only by the owner of the private key.

Concepts from number theory are essential to understanding the basic workings of the RSA cryptosystem, as well as many other parts of modem cryptography.

The overwhelming importance of number theory in cryptography contradicts the earlier belief, held by many mathematicians, that number theory was unimportant for real-world applications. It is ironic that some famous mathematicians, such as G.

Hardy, took pride in the notion that number theory would never be applied in the way that it is today. The search for integer solutions of equations is another important part of number theory.

An equation with the added proviso that only integer solutions are sought is called diophantine, after the ancient Greek mathematician Diophantus. As Wiles's proof shows, number theory is not a static subject!

New discoveries continue steadily to be made, and researchers frequently establish significant theoretical results. The fantastic power available when today's computers are linked over the Internet yields a rapid pace of new computational discoveries in number theory. Everyone can participate in this quest; for instance, you can join the quest for the new Mersenne primes, P primes of the form 2 1, where p itself is prime.

In August , the first prime with more than 10 million decimal digits was found: A concerted effort is under way to find a prime with more than 1 00 million digits, with a. After learning about some of the topics covered in this text, you may decide to join the hunt yourself, putting your idle computing resources to good use.

Students who plan to continue the study of mathematics will learn about more advanced areas of number theory, such as analytic number theory which takes advantage of the theory of complex variables and algebraic number theory which uses concepts from abstract algebra to prove interesting results about algebraic number fields.

It is pure mathematics with the greatest intellectual appeal, yet it is also applied mathematics, with crucial applications to cryptography and other aspects of computer science and electrical engineering. I hope that you find the many facets of number theory as captivating as aficionados who have preceded you, many of whom retained an interest in number theory long after their school days were over. Experimentation and exploration play a key role in the study of number theory.

The results in this book were found by mathematicians who often examined large amounts of numerical evidence, looking for patterns and making conjectures. They worked diligently to prove their conjectures; some of these were proved and became theorems, others were rejected when counterexamples were found, and still others remain unresolved. As you study number theory, I recommend that you examine many examples, look for patterns, and formulate your own conjectures.

You can examine small examples by hand, much as the founders of number theory did, but unlike these pioneers, you can also take advantage of today's vast computing power and computational engines. Working through examples, either by hand or with the aid of computers, will help you to learn the subject-and you may even find some new results of your own!

In this chapter, we will discuss some particularly important sets of numbers,. We will briefly introduce the notion of approximating real numbers by rational numbers.

We will also introduce the concept of a sequence, and particular sequences of integers, including some figurate numbers studied in ancient Greece. A common problem is the identification of a particular integer sequence from its initial terms; we will briefly discuss how to attack such problems.

Using the concept of a sequence, we will define countable sets and show that the set of rational numbers is countable. We will also introduce notations for sums and products, and establish some useful summation formulas. One of the most important proof techniques in number theory and in much of mathematics is mathematical induction.

We will discuss the two forms of mathematical induction, illustrate how they can be used to prove various results, and explain why mathematical induction is a valid proof technique.

Continuing, we will introduce the intriguing sequence of Fibonacci numbers, and describe the original problem from which they arose. We will establish some identities and inequalities involving the Fibonacci numbers, using mathematical induction for some of our proofs. The final section of this chapter deals with a fundamental notion in number theory, that of divisibility.

We will establish some of the basic properties of division of integers, including the "division algorithm. Numbers and Sequences In this section, we introduce basic material that will be used throughout the text.

In particular, we cover the important sets of numbers studied in number theory, the concept of integer sequences, and summations and products. The proof that e is integers to show that irrational is left as Exercise We refer the reader to [HaW r08] for a proof that rr is irrational.

S has a smallest element. J2 were rational. J2 is irrational.. J2 is a member of S. Another important class of numbers in the study of number theory is the set of numbers that can be written as a ratio of integers.

J2 is irrational. We say that the set of positive integers is well ordered. Then there would exist positive integers a and b such that. The proof that we provide. See Appendix A for axioms for the set of integers. The integers play center stage in the study of number theory. If Example 1. You may prefer the proof that we will give in Chapter The well-ordering property can be taken as one of the axioms defining the set of positive integers or it may be derived from a set of axioms in which it is not included.

Theorem 1. Examples of irrational We can use the well-ordering property of the set of positive. One property of the positive integers deserves special mention. The real number Definition.

It is not easy. Suppose that. The well-ordering property may seem obvious. The Well-Ordering Property Every nonempty set of positive integers has a least element. Note that every integer numbers are. The number a is called transcendental if it is not algebraic.

Because s. The following example establishes. In Chapter The Greatest Integer Function In number theory. That is. For example.

The numbers e and rr are also transcendental. The sets of integers. Such notation will be used occasionally in this book. We briefly mention several other types of numbers here. The ceiling function of a real number x' denoted by rxl. It is less than s. This contradicts the choice of s as the smallest positive integer in S. J2 is algebraic. The irrational number. Instead of [x] for this function. Besides being important in number theory. The proof will depend on the famous pigeonhole principle.

The adjective diophantine comes from the Greek mathematician Diophantus. The Pigeonhole Principle. Although this seems like a particularly simple idea.. Dirichlet called it the Schubfachprinzip in German. The Integers 8 a useful property of this function.. We now state and prove this important fact.

Show that if n is an integer. Example 1. In particular. But can we show that one of the first k multiples of a real number must be much closer to an integer? An important part of number theory called diophantine approximation studies questions such as this.

Here we will show that among the first n multiples of a real number a. To show that this property holds.. This implies that m The fractional part of a real number x. Additional properties of the greatest integer function are found in the exercises at the end of this section and in [GrKnPa94]. A biography of Dirichlet can be found in Section 3.

Recall that IxI. Also recall that Ix.

The proof we give illustrates the utility of the pigeonhole principle. It follows that there exist integers j and k with Dirichlet's Approximation Theorem. Note that in the proof we make use of the absolute value function. We will return to this topic in Chapter 1 2. This contradiction shows that one of the boxes contains at least two or more of the objects. To see this. J2 has the smallest fractional part.. If a is a real number and n is a positive integer. Proving a stronger version of Theorem 1.

See [Ro07] for more applications of the pigeonhole principle. A common problem in number theory is finding a formula or rule for constructing the terms of a sequence. Conjecture a formula for an.. We note that each term is approximately 3 n n times the previous term. A g e ome tric progression is a sequence of the form a. We introduce several useful sequences in the following examples. This is the sequence of powers of 2.

We will consider many particular integer sequences in our study of number theory. This is the sequence ofthe squares ofintegers. The particular sequence in Example 1. Even though the initial terms of a sequence do not determine the sequence. There are many sequences in which each successive term is obtained from the previous term by multiplying by a common factor.

We note that each term. This leads to the following definition.

Consider the following examples.. The set of rational numbers is countable. Among the sequences we will study are the Fibonacci numbers. This is an example of a recursive definition of a sequence. We arrange these by placing the fraction with a particular numerator in the position this numerator occupies in the list of. After examining this sequence from different perspectives. Note that in early We can list the rational numbers as the terms of a sequence.

We put all fractions with a denominator of 1 in the first row. This site provides a program for finding sequences that match initial terms provided as input. You may find this a valuable resource as you continue your study of number theory as well as other subjects.

We now define what it means for a set to be countable. At first glance. A set that is not countable is called uncountable. An infinite set is countable if and only if its elements can be listed as the terms of a sequence indexed by the set of positive integers. Neil Sloane has amassed a fantastically diverse G collection of more than This produces the sequence 0. This collection is available on the Web.

The book [S1Pl95] is an earlier printed version containing only a small percentage of the current contents of the encyclopdia. Integer sequences appear in an amazing range of subjects besides number theory. Conjecture a formula for an. The terms listed in this example are the initial terms of the Fibonacci sequence.. Prove that both the sum and the product of two rational numbers are rational. Figure 1. Determine whether each of the following sets is well ordered.

Show that if a and b are positive integers. Such an example is provided by the set of real numbers. Prove or disprove each of the following statements. Either give a proof using the well-ordering property of the set of positive integers. We leave it to the reader to fill in the details.

Find the fractional part of each of these numbers: What is the situation when both xand yare negative? When one of xand yis negative and the other positive? Use the well-ordering propertyto show that. Show that -[-x] is the least integer greater than or equal to x when xis a real number. Find the following values of the greatest integer function. Show that everynonemptyset of negative integers has a greatest element. Show that if m and n are integers. Show that if m is a positive integer.

Show that if xand yare positive real numbers. Prove that if a f3. Using a computational aid.. Show that the set of all rational numbers of the form Show that the union of a countable number of countable sets is countable. Find the first ten terms of the spectrum sequence of each of the following numbers. Show that if a is a real number and n is a positive integer..

Show that the union of two countable sets is countable. Using a computational aid. Prove the following stronger version of Dirichlet's approximation. Show that the set of all integers greater than is countable. Use Dirichlet's approximation theorem to show that if a is an irrational number.

We specify that u1 1 and u2 2. See the preamble to 38 for the definition of spectrum. Ulam had a fabulous memory and was an extremely verbal person.

Ulam the first atomic bomb. Luckily for Ulam. Show that the number whose ith decimal digit is 4 when the ith decimal digit of the ith real number in the list is 5 and is 5 otherwise is not on the list. Find the first ten Ulam numbers.

These numbers are named for Stanislaw Ulam. Suppose it is possible to list the real e is irrational. Computations and Explorations During these years he returned each summer to Poland where he spent time in cafes.

Ulam also developed the Monte Carlo method. At Los Alamos. ULAM was born in Lvov. He wrote several books. Ulam received his Ph. U1am remained at Los Alamos after the war until He decided to learn the mathematics required to understand relativity theory.

In Advanced Study. Show that the set of real numbers is uncountable. He was interested in and contributed to many areas of mathematics. He served on the faculties of the University of Southern California. Show that there are infinitely many IDam numbers. He became interested in astronomy and physics at age His mind was a repository of stories. In What conjectures can you make about the number of Ulam numbers less than an integer n?

Do your computations support these conjectures?

Programming Projects 1. The following notation represents the sum of the numbers ai. Can the sum of any two consecutive Ulam numbers. How many pairs of consecutive integers can you find where both are Ulam numbers? We see that I: How large are the gaps between consecutive Ulam numbers?

Do you think that these gaps can be arbitrarily long? Given a number a. If so. If m and n are integers such that m See the preamble to Exercise 3 8 for the definition of spectrum. Find as many terms as you can of the spectrum sequence of rr.

For k instance. Find the first Ulam numbers.. For instance. Find the first n Ulam numbers. This use of notation is illustrated in the following example. The following three properties for summations are often useful. We leave their proofs to the reader. We can use summation notation to specify the particular property or properties the index must have for a term with that index to be included in the sum. We often need to evaluate sums of consecutive terms of a geometric series.

The following example shows how a formula for such sums can be derived. Example Telescoping sums are easily evaluated because n L aj.

It follows that n rS.. Let n be a positive integer. To find the sum n I: When we isolate the factor k. The following example illustrates one such sequence of numbers. See Exercise 7 for another way to find tn. How can we find a formula for the nth triangular number? The triangular numbers ti..

The product of the numbers ai. We also n! Find and prove a formula for I: By putting together two triangular arrays. Find each of the following sums. To illustrate the notation for products. Then n! Use the 6. The Integers 20 The letter j above is a "dummy variable. The pentagonal numbers Pi. Recall that a heptagon is a seven-sided polygon. Recall that a hexagon is a six-sided polygon. The tetrahedral numbers Ti. Prove that the sum of the n..

From this. Find n! Find all positive integers x. Given the terms of a geometric progression. Justify your answer.

Find as many triangular numbers that are perfect squares as you can. Let ba2. Without multiplying all the terms. Find the values of the following products.

What are the largest values of n for which n! Given the terms of a sequence ai.