Elementary number theory and its applications. Bibliography: p. Includes index. l. Numbers, Theory of. I. Title. QA24l.R67 rsBN ', Rosen Elementary Number Theory and Its Applications 5th Edition. Uploaded by. 다 룽. Download with Google Download with Facebook or download with email. Elementary Number Theory and Its Applications (5th Edition) Advanced Number Theory with Applications (Discrete Mathematics and Its Applications).

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Rosen - Elementary number theory and its applications. Pages Rachel_Carson,_Al_Gore,_Lois_and_Louis_darling_S(zlibraryexau2g3p_onion) .pdf. Rights Reserved. Download as PDF, TXT or read online from Scribd Rosen, Kenneth H. Elementary number theory and its applications I Kenneth H. Rosen. shed light on analytic number theory, a subject that is rarely seen or approached careful choice of topics and its importance in the theory of numbers. The freedom is given in the Many interesting theorems, applications and conjectures.

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Philpot Mechanics of Materials 6th Ed. Neamen Microelectronic Circuit Design 3rd Ed. Pozar Microwave Engineering, 3rd Ed. 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. You may prefer the proof that we will give in Chapter 4. It is not easy. The well-ordering property may seem obvious. J2 is irrational.. The proof that we provide.

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. The proof that e is integers to show that irrational is left as Exercise Then there would exist positive integers a and b such that..

Theorem 1. J2 is irrational. S has a smallest element. The real number Definition. We say that the set of positive integers is well ordered. One property of the positive integers deserves special mention. If Example 1. Note that every integer numbers are. The integers play center stage in the study of number theory. Suppose that. Examples of irrational We can use the well-ordering property of the set of positive.

J2 is a member of S.. J2 were rational. 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.

See Appendix A for axioms for the set of integers. The Well-Ordering Property Every nonempty set of positive integers has a least element. We refer the reader to [HaW r08] for a proof that rr is irrational.

The number a is called transcendental if it is not algebraic. The ceiling function of a real number x' denoted by rxl. That is. In Chapter J2 is algebraic. The following example establishes.. The sets of integers. Besides being important in number theory.

The numbers e and rr are also transcendental.. The Greatest Integer Function In number theory. Because s. The irrational number. We briefly mention several other types of numbers here.

For example. It is less than s. Instead of [x] for this function. This contradicts the choice of s as the smallest positive integer in S. Such notation will be used occasionally in this book. This implies that m The fractional part of a real number x. Here we will show that among the first n multiples of a real number a. Although this seems like a particularly simple idea. The proof will depend on the famous pigeonhole principle.

Additional properties of the greatest integer function are found in the exercises at the end of this section and in [GrKnPa94]. The adjective diophantine comes from the Greek mathematician Diophantus.. 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. We now state and prove this important fact. The Pigeonhole Principle.

Number Theory

A biography of Dirichlet can be found in Section 3. The Integers 8 a useful property of this function. Dirichlet called it the Schubfachprinzip in German. To show that this property holds. In particular.

Example 1. Show that if n is an integer. That is.. To see this.

The proof we give illustrates the utility of the pigeonhole principle. This contradiction shows that one of the boxes contains at least two or more of the objects.. Proving a stronger version of Theorem 1. See [Ro07] for more applications of the pigeonhole principle. If a is a real number and n is a positive integer. It follows that there exist integers j and k with Recall that IxI.

Also recall that Ix. We will return to this topic in Chapter 1 2. Note that in the proof we make use of the absolute value function. Dirichlet's Approximation Theorem. J2 has the smallest fractional part..

We note that each term.. This is the sequence ofthe squares ofintegers. Even though the initial terms of a sequence do not determine the sequence. A g e ome tric progression is a sequence of the form a. Consider the following examples.

We introduce several useful sequences in the following examples. There are many sequences in which each successive term is obtained from the previous term by multiplying by a common factor. A common problem in number theory is finding a formula or rule for constructing the terms of a sequence. The particular sequence in Example 1. This leads to the following definition. This is the sequence of powers of 2.

Grupy dyskusyjne

Conjecture a formula for an. We will consider many particular integer sequences in our study of number theory. We note that each term is approximately 3 n n times the previous term..

Note that in early 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. We now define what it means for a set to be countable. You may find this a valuable resource as you continue your study of number theory as well as other subjects. At first glance. We put all fractions with a denominator of 1 in the first row. We arrange these by placing the fraction with a particular numerator in the position this numerator occupies in the list of.

This collection is available on the Web. The terms listed in this example are the initial terms of the Fibonacci sequence.. Neil Sloane has amassed a fantastically diverse G collection of more than This is an example of a recursive definition of a sequence. The book [S1Pl95] is an earlier printed version containing only a small percentage of the current contents of the encyclopdia..

This produces the sequence 0. Integer sequences appear in an amazing range of subjects besides number theory. We can list the rational numbers as the terms of a sequence. After examining this sequence from different perspectives. Among the sequences we will study are the Fibonacci numbers.

A set that is not countable is called uncountable. This site provides a program for finding sequences that match initial terms provided as input.

The set of rational numbers is countable. Show that if a and b are positive integers. Figure 1. Either give a proof using the well-ordering property of the set of positive integers.

Prove that both the sum and the product of two rational numbers are rational. Such an example is provided by the set of real numbers. Determine whether each of the following sets is well ordered. We leave it to the reader to fill in the details. Prove or disprove each of the following statements. Show that if xand yare positive real numbers. Show that if m is a positive integer.

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? Find the following values of the greatest integer function. 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.

Show that if m and n are integers. Show that everynonemptyset of negative integers has a greatest element. Show that the set of all integers greater than is countable. Show that the set of all rational numbers of the form Using a computational aid.

Show that if a is a real number and n is a positive integer.. Prove the following stronger version of Dirichlet's approximation.

Prove that if a f3. Find the first ten terms of the spectrum sequence of each of the following numbers. Use Dirichlet's approximation theorem to show that if a is an irrational number.

Show that the union of a countable number of countable sets is countable. Show that the union of two countable sets is countable. In Advanced Study. 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.

He became interested in astronomy and physics at age He was interested in and contributed to many areas of mathematics. Ulam received his Ph. Ulam had a fabulous memory and was an extremely verbal person. Ulam also developed the Monte Carlo method. He wrote several books. His mind was a repository of stories. Show that the set of real numbers is uncountable.

Show that there are infinitely many IDam numbers. Computations and Explorations 1. He served on the faculties of the University of Southern California. Find the first ten Ulam numbers. In During these years he returned each summer to Poland where he spent time in cafes.

Suppose it is possible to list the real e is irrational. We specify that u1 1 and u2 2. U1am remained at Los Alamos after the war until Luckily for Ulam. These numbers are named for Stanislaw Ulam.

At Los Alamos. Ulam the first atomic bomb. See the preamble to 38 for the definition of spectrum. He decided to learn the mathematics required to understand relativity theory. ULAM was born in Lvov. See the preamble to Exercise 3 8 for the definition of spectrum. 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.

How large are the gaps between consecutive Ulam numbers? Do you think that these gaps can be arbitrarily long? Find the first n Ulam numbers. If so. Can the sum of any two consecutive Ulam numbers. If m and n are integers such that m Find the first Ulam numbers.

How many pairs of consecutive integers can you find where both are Ulam numbers? We see that I: Given a number a.. Given a number a. Find as many terms as you can of the spectrum sequence of rr.

For instance. For k instance. 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. The following three properties for summations are often useful. The following example shows how a formula for such sums can be derived. This use of notation is illustrated in the following example. We often need to evaluate sums of consecutive terms of a geometric series..

To find the sum n I: Let n be a positive integer.. Telescoping sums are easily evaluated because n L aj. It follows that n rS. The triangular numbers ti. The product of the numbers ai.. When we isolate the factor k. How can we find a formula for the nth triangular number?

See Exercise 7 for another way to find tn. The following example illustrates one such sequence of numbers. Find each of the following sums. Find and prove a formula for I: Use the 6. We also n! The Integers 20 The letter j above is a "dummy variable.

By putting together two triangular arrays. To illustrate the notation for products. Then n! From this. The tetrahedral numbers Ti. Recall that a hexagon is a six-sided polygon. Prove that the sum of the n.

Recall that a heptagon is a seven-sided polygon. The pentagonal numbers Pi. Find as many tetrahedral numbers that are perfect squares as you can. Find as many triangular numbers that are perfect squares as you can. Show that the nth tetrahedral number is the sum of the first n triangular numbers.

Find n! Given the terms of a geometric progression. Given the terms of a sequence ai. Find the values of the following products. Without multiplying all the terms.. Justify your answer. Find and prove a closed formula for the nth tetrahedral number. W e will study this question in the Exercises in Section Find all positive integers x. What are the largest values of n for which n!

List the integers !. Let ba2. Letting S be the set of positive integers for which we claim the statement to be true. We must accomplish two things to prove by mathematical induction that a particular statement holds for every positive integer. Given a positive integer n. Once these two steps are completed. We illustrate the use of mathematical induction by several examples. We will use the principle of mathematical induction. This is called the basis step. The Principle ofMathematical Induction.

A set of positive integers that contains the integer l. We begin with the basis step. The Integers 24 Example 1.

Maurolico presented various properties of the integers. The first use of mathematical induction in his book was in the proof that the sum of the first positive integers equals n2. Because both the basis and the inductive steps have been completed. Using the inductive hypothesis. He devised the method of mathematical induction so that he could complete some of the proofs. We can show by mathematical induction that n! We will use mathematical induction to show that n L 2j.

The basis step. In his book Arithmeticorum Libri Duo. To complete the proof. By the way. The Second Principle of Mathematical Induction. Assume for the sake of contradiction that S is not the set of Let S be a set of positive integers containing the integer whenever it contains all positive integers.

Example By the well-ordering property.

We will show that any amount of postage more than one cent can be formed using just two-cent and three-cent stamps. But because S contains n.


This completes the proof.. A set of positive integers that contains the integer 1. For the basis step. For the inductive step. This shows that S must be the set of all positive integers.

S must be the set of all positive integers. Then 1 is in S. Instead of explicitly specifying the value of the function at n.

The principle of mathematical induction can be used to show that a function that is defined recursively is defined uniquely at each positive integer see Exercise 25 at the end of this section. Let S be the set of all positive integers n such that all the positive integers less than or equal to n are in T.

We can define a function whose domain is the set of positive integers by specifying its value at 1 and giving a rule. Proof 1 and such that for every positive integer n. We illustrate how to define a function recursively with the following definition. The second principle of mathematical induction also serves as a basis for recursive definitions. Use mathematical induction to prove that L:: This will be the basis for the definition of the sequence of Fibonacci numbers discussed in Section 1.

Prove that your induction. Prove your result using 1. Show that any amount of postage that is an integer number of cents greater than 1 1 cents can be formed using just 4-cent and 5-cent stamps. Hn be the nth partial sum of the harmonic series. Compare this to Example 1.

Use mathematical induction to prove that 2. Use mathematical induction to prove that n is a positive integer. Compare this to Exercise 17 in Section 5. Conjecture a formula for the sum of the first n even positive integers. Use mathematical induction to prove that 7. Use mathematical induction to show that H n Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and cent stamps.

Prove your answer using mathematical induction. Show that exactly n. This completes the induction argument. A jigsaw puzzle is solved by putting its pieces together in the correct way..

Use the principle of mathematical induction to show that a set of integers that contains the integer k.. Use the second principle of mathematical induction.

We define a function recursively for all positive integers for n Because these two sets of horses have common members. The puzzle includes three pegs and eight rings of different sizes placed in order of size. Explain what is wrong with the following proof by mathematical induction that all horses are 1 horse are all the same color. Showthatan n n n for every nonnegative integer 0 Now assume that all horses in any set of be the same color.

They started moving the rings.. By the induction hypothesis. When they finish transferring the rings to the second peg. Use the principle of mathematical induction to show that the value at each positive integer of a function defined recursively is uniquely determined..

Clearly all horses in any set of the basis step. Academic Press. Published Date: Page Count: Flexible - Read on multiple operating systems and devices. Easily read eBooks on smart phones, computers, or any eBook readers, including site. When you read an eBook on VitalSource Bookshelf, enjoy such features as: Access online or offline, on mobile or desktop devices Bookmarks, highlights and notes sync across all your devices Smart study tools such as note sharing and subscription, review mode, and Microsoft OneNote integration Search and navigate content across your entire Bookshelf library Interactive notebook and read-aloud functionality Look up additional information online by highlighting a word or phrase.

Institutional Subscription. Instructor Ancillary Support Materials. Free Shipping Free global shipping No minimum order. Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle East.

Advanced undergraduates in Number Theory courses, sophomore, juniors, and seniors. English Copyright:In our subsequentanalysis of the Euclidean algorithm, we wil! We will describe a procedure for finding the base bexpansion of an integer. We have a dedicated site for Ukraine. Show that if n is a pqsitive integer, then un: Prove that whenever n is a nonegative integer. Kulakowski , F. We can list the rational numbers as the terms of a sequence.

Using the inductive hypothesis. A sequence ai.