(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. 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. 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.
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Elementary Number Theory and Its Applications (5th Edition). Home · Elementary Author: Kenneth H. Rosen Elementary Number Theory: Second Edition. Elementary. Number. Theory and lts. Applications. Kenneth H. Rosen. AT&T Informo. Burton, David M. Elementary number theory I David M. Burton. -7th ed . p. dressed in a course in number theory. Proofs of basic theorems are presented in an interesting and comprehensive way that can be read and understood even.
How long will the world last? Use the principle of mathematical induction to show that the value at each positive integer of a function defined recursively is uniquely determined. Explain what is wrong with the following proof by mathematical induction that all horses are 1 horse are all the same color.
Show that exactly n. Because these two sets of horses have common members. This completes the induction argument. Clearly all horses in any set of the basis step. Prove your answer using mathematical induction. A jigsaw puzzle is solved by putting its pieces together in the correct way. They started moving the rings. We define a function recursively for all positive integers for n Now assume that all horses in any set of be the same color. Use the second principle of mathematical induction.
Showthatan n n n for every nonnegative integer 0 The puzzle includes three pegs and eight rings of different sizes placed in order of size. By the induction hypothesis. Use the principle of mathematical induction to show that a set of integers that contains the integer k.
The goal of the puzzle is to move all of the rings. This completes n horses are the same color. When does equality hold? Find such representation for as many positive integers of three unit fractions. Use mathematical n. Because the ancient Egyptians represented fractions as sums of distinct unit fractions.
List the moves in the tower of Hanoi puzzle see Exercise Explore this conjecture using the greedy algorithm that successively adds the unit fraction with the least positive odd denominator q at each stage. Attempt to prove your conjecture via mathematical induction using numerical and symbolic computation. Using the algorithm in Exercise Use mathematical induction to show that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces.
Paul Erdos and E. Complete the basis and inductive steps. Cover a 2n If you can. The arithmetic mean and the geometric mean of the positive real numbers ai. Use strong induction on the numerator p to show that the greedy algorithm that adds the largest possible unit fraction at each stage always terminates. Examining the initial terms of the Fibonacci sequence will be useful as we study their properties. As this pair does not breed during the second month.
We compute the first ten Fibonacci numbers as follows: The Integers 30 3. In bis Liber Abaci Fibonacci introduced Arabic notation for numerals and their algorithms for arithmetic into the European world. This problem can be phrased as follows: A young pair of rabbits. The answer to Fibonacci's question is that there are f.
The termsoftbis sequenceare called theFibonaccinumbers.. Let In be the number of pairs of rabbits after n months. It was in this book that bis famous rabbit problem appeared.
Fibonacci also wrote Practica geometriae. The mathematician Edouard Lucas named this sequence after Fibonacci in the nineteenth century when he established many of its properties.
To find on the island the previous month.
Fibonacci was a merchant who traveled extensively throughout the Mideast. Assuming that rabbits do not breed until they are two months old and after they are two months old. G Definition. The Fibonacci numbers also satisfy an extremely large number of identities.
They occur in the solution of a tremendous variety of counting problems. We can also define fn where n is a negative number so that the equality in the recursive definition is satisfied see Exercise We provide two different demonstrations. C The Fibonacci numbers occur in an amazing variety of applications. Looking at these numbers.
We will show. Using the formula for a telescoping sum found in Section 1. The inductive hypothesis consists of assuming that ak k. The following inequality. The inductive We must show that. By the inductive hypothesis. We can also prove this identity using mathematical induction. We have presented a few important results involving the Fibonacci numbers. Find each of the following Fibonacci numbers. The advantage of the first two approaches is that they can be used to fi n d the formula. Prove that 6.
There is a vast literature concerning these numbers and their many applications to botany. Find the following Fibonacci numbers. Prove that 5. There is even a scholarly journal. We will not provide a proof in the text. Exercise 40 asks that you prove this identity by showing that the terms satisfy the same recursive definition as the Fibonacci numbers do. The Fibonacci Quarterly. Prove that 4. Prove that L: G computer science. Find and prove a simple formula for the sum of the first n Fibonacci numbers with odd indices when n is a positive integer.
Prove that Show that every positive integer has a unique Zeckendorf representation. Show that Ln. Show that Find and prove a formula for the sum of the first n Lucas numbers when n is a positive integer. Find and prove a formula for the sum of the first n Lucas numbers with odd indices when n Prove that! They satisfy the same recurrence relation as the Fibonacci numbers.
Find the first are different.. Find and prove a formula for the sum of the first n Lucas numbers with even indices when n is a positive integer. Show that fn The Zeckendoif representation of a positive integer is the unique expression of this integer as the sum of distinct Fibonacci numbers.
Give a recursive definition of the Fibonacci number fn when n is a negative integer. The values of these constants can be found using the two initial terms of the sequence By taking determinants of both sides of the result of Exercise Where is the extra square unit?
It is not difficult to show see [Ro0 7] that if the 2 equation r.. Prove that whenever n is a nonegative integer.: Prove this conjecture using mathematical induction.
Use the results of Exercise 3 7 to formulate a conjecture that relates the values of f-n and fn when n is a positive integer. Find an explicit formula for fn. The generating fanction for the sequence a0. What is wrong with the claim that an 8 reassembled to form a 5 x x 8 square can be broken into pieces that can be 13 rectangle as shown? Look at the identity in Exercise Give a positive integer n.
Find the Fibonacci numbers 2. Find an explicit formula for the Lucas numbers using the technique of Exercise Computations and Explorations! Formulate a conjecture based on your evidence. Use mathematical induction to prove Theorem 1. Exercise If a and that bis a multiple of a.
Examine as manyFibonacci numbers as possible to determine which are triangular numbers. Find the largest Fibonacci number less than Examine as many Fibonacci numbers as possible to determine which are perfect cubes. See [Ro07] for information on using generating functions. The Integers 36 1: Examine as many Fibonacci numbers as possible to determine which are perfect squares.
The Division Algorithm. Because a I band b I c. As and The following theorem states an important fact about division. We discuss algorithms in Section 2.
In subsequent chapters. We also call a the dividend and b the divisor.
We use the traditional In the equation given in the division algorithm. We now prove the division algorithm using the well-ordering property.
Because Theorem 1. Be careful not to confuse the notations a I b. Before we prove the division algorithm. We note that is a is divisible by b if and only if the remainder in the division algorithm 0.
The following examples display the quotient and remainder of a division. T has a least element values for To show that these values for q and r are unique. These are the q and r specified in the theorem. Show that if is a real number. By subtracting the second of these equations from the first.
Because [x] Let T be the set of all nonnegative integers in S. To prove this identity. T is nonempty. By Equation 1. This tells us that b divides We now use the greatest integer function defined in Section 1.
Because Because the quotient q is the largest integer such that bq By the division algorithm. The integers a and b. Note that n. The greatest common divisor of a and b is written as a.
Greatest Common Divisors If a and b are integers. We will study greatest common divisors at length in Chapter 4.
Note that the notation gcd a. We will use the traditional notation 0. We will also prove many important results about them that lead to key theorems in number theory. We are interested in the largest integer among the common divisors of the two integers. In that chapter.
This leads to the following definition of some common terminology. Such pairs of integers are called Definition.
The greatest common divisor of two integers a and b. Decide which of the following integers are divisible by 7. Find all positive integers less than 11 thatare relatively prime to it. The Integers 40 1. Are there integers a. What can you conclude if a and bare nonzero integers such that a Ib and b Ia? Find these greatest common divisors by finding all positive integers that divide each integer in the pair and selecting the largest that divides both. Show that if a and b are positive integers and a I b.
Decide which of the following integersare divisible by Find all positive integers that divide each of these integers. Show that the sum of two even or of two odd integers is even.
Show that if a. Find the quotient and remainder in the division algorithm. Find all pairs of positive integers not exceeding 10 that are relatively prime. Find all positive integers less than 10 that are relatively prime to it. Show that is divisible by 7. Show that if a and b are odd positive integers and bla.
Show that if a and b are integers such that a Ib. Show that the product of two odd integers is odd. Find all pairs of positive integers between 10 and Find a formula involving the greatest integer function for the cost of mailing a letter in early In early Show that the number of positive integers less than or equal to number.
Show that if b la. This result is called the modified division algorithm. Use mathematical induction t o show that n5.
Show that if a is an integer. Find the number of positive integers not exceeding that are not divisible by 3. When the integer a is divided by the integer b. Use mathematical induction to show that the sum of the cubes of three consecutive integers is divisible by 9. How many integers between and are divisible by 7?
Show that the product of any three consecutive integers is divisible by 6. Find the number of positive integers not exceeding that are divisible by 5.
Show that the integer n is even if and only if n. Find the number of positive integers not exceeding that are not divisible by 3 or 5. Find the number of positive integers not exceeding that are divisible by 3 but not by 4. Show that fn is divisible by 3 if and Find the sequence obtained by iterating T starting SO.
Show that fn is even if and only if n is divisible by Show that fn is divisible by 4 if and only if n is divisible by Verify the Collatz conjecture described in the preamble to Exercise exceeding The Integers 42 In Exercises Find the quotient and remainder when Use with n Im.
A well-known conjecture. Jj ] is odd whenever n is a nonnegative integer. Begin by Using numerical evidence. Verify that there is a term in the sequence obtained by iterating integer n.. Prove the divison algorithm using the second principle of mathematical induction. Computations and Explorations 1. Decide whether an integer is divisible by a given integer. Find the quotient. Compute the terms of the sequence n.
Integer Representations 2 and Operations T he way in which integers are represented has a major impact on how easily people and computers can do arithmetic with these integers. There is no special reason for using ten as the base in a fixed positional number system.
Babylonian mathematicians who lived more than years ago expressed integers using sixty as a base. The ancient Mayans used a positional notation with twenty as a base.
With the invention and proliferation of computers. The Romans employed Roman numerals. In Section The purpose ofthis chapter is to explain how integers are represented using base bexpansions. We write out numbers using digits to represent powers often. We will describe a procedure for finding the base bexpansion of an integer. As we will see. Throughout ancient and modem history. Many other systems ofinteger notation have been invented and used over time. In this section. See the exercise set at the end of this section to learn about one's and two's complement notations.
We first divide n If ai ak 0. To see that we must reach such a step. We obtain an expression of the desired type by successively applying the division algorithm in the following way. From the first equation above. Theorem 2. For more information about the fascinating history of positional number systems. We now show that every positive integer greater than 1 may be used as a base. Subtracting one expansion from the other..
We conclude that our base b expansion of n is unique. If the two expansions are different. Corollary 2. The proof of Theorem 2.
We then read up the list of remainders to find the base b expansion. The remainder is the digit a0. Example 2. To obtain the base 2 expansion of From Theorem 2. We continue this process. We illustrate this procedure in Example 2. To distinguish representations of integers with different bases. Base 2 expansions are called binary expansions.
In other words. We write akak-l. Binary digits are called bits binary digits in computer terminology. The process stops once a quotient of 0 is obtained.. In the expansions described in Theorem To find the base 2 expansion of We call base 10 notation. The coefficients ai are called the digits of the expansion.
The remainder is the digit a1. To convert A35BOF 16 from hexadecimal to decimal notation. The blocks are. To convert from binary to hex. We can write each hex digit as a block of four binary digits according to the correspondences given in Table 2. Computers use base 8 or base 16 for display purposes. We break this into blocks of four.
The following example demonstrates the conversion from hexadecimal to decimal notation. The letters A. Translating each block to hex. Each hex digit is converted to a block of four binary digits the initial zeros in the initial block h corresponding to the digit 2 i6 are omitted.
We can use "on" to represent the digit 1 and "off" to represent the digit O. Find the base -2 representations of the decimal numbers Convert h from base 7 to decimal notation. Use Exercise 12 to show that any weight not exceeding k 3.
Convert 1 1 0 from decimal to base 7 notation. Convert h from binary to decimal notation and 10 from decimal to binary notation. Convert i0 from decimal to base 8 notation.
Show that any weight not exceeding 2. Convert 8 from base 8 to decimal notation. Integer Representations and Operations 50 We note that a conversion between two different bases is as easy as binary-hex conversion whenever one of the bases is a power of the other. Explain how to convert from base r to base rn notation. Show that if b is a negative integer less than Explain how to convert from base 3 to base 9 notation.
Convert h and lO h from binary to hexadecimal. Explain why we really are using base notation when we break large decimal integers into blocks of three digits.
What integers do the representations in Exercise 19 represent if each is the two's complement representation of an integer? How is the one's complement representation of -m obtained from the one's complement of m. Find the one's complement representations. Find the two's complement representations.. For a negative integer. The leftmost bit represents the sign. What integer does each of the following one's complement representations of length five represent?
How is the two's complement representation of -m obtained from the two's complement representation of m. For positive integers. For a positive integer. The leftmost bit is used to represent the sign. Sometimes integers are encoded by using four-digit binary expansions to represent each decimal digit.
This produces the binary coded decimal form of the integer. How many bits are required to represent a number with n decimal digits using this type of encoding? For negative integers. A 0 in this position is used for positive integers..
To represent n l n l an integer x with Find Cantor expansions of Show that a position is a winning one if and only if the number of ls in each column is even. The players take turns. For each arrangement of matches into piles. Show that there is a move from any nonwinning position to a winning one.
Show that every positive integer has a unique Cantor expansion. Show that the position in nim where there are two piles. Let a' be the integer with a decimal expansion obtained by writing the digits of a in descending order. To make a move. For am.
Show that any move from a winning position produces a nonwinning one. There are several piles of matches. Let a be an integer with a four-digit decimal expansion. For example: Three piles of 3. Let b be a positive integer and let a be an integer with a four-digit base b expansion.
The Chinese game of nim is played as follows. An example is the position where there are two piles. A winning p osition is an arrangement of matches in piles such that if a player can move to this position. The integer is called Kaprekar's constant. He published extensively. Use numerical evidence to make conjectures about the behavior of the sequence D..
He received his secondary school education in Thana and studied at Ferguson College in Poona. From until bis retirement in Evaluate each of the following sums. Kaprekar discovered many interesting properties in recreational number theory. Show that no Kaprekar constant exists for four-digit numbers to the base 6.
Find the decimal expansion of a i c each of the following integers. A sequence ai. Kaprekar attended the Universiy t of Bombay.
Find the Cantor expansions of the integers Prove that your answer is correct. Use Theorem 2. Determine whether there is a Kaprekar constant for three-digit integers to the base See the preamble to Exercise 28 for the definition of Cantor expansions.
Find the binary. Verify the result described in Exercise 33 for not all digits are the same. This upper limit is called the word size. Find the base -2 notation of an integer from its decimal notation see Exercise 8. Convert from binary notation to hexadecimal notation.
Play a winning strategy in the game of nim see the preamble to Exercise Also note that to find the base expansion of an integer. Either way.
Find the binary expansion of an integer from the decimal expansion of this integer. Explore the behavior for different bases b of the sequence a. To do arithmetic with integers larger than the word size. We have mentioned that computers internally represent numbers using bits.
Computers have a built-in limit on the size of integers that can be used in machine arithmetic. Find the Cantor expansion of an integer from its decimal expansion see the preamble to Exercise Investigate the sequence a. In the following section. What conjectures can you make? Repeat your exploration using four-digit and then five-digit integers in base b notation.
Many number theoretic problems. Convert from base b1 notation to base b2 notation. Find the balanced ternary expansion of an integer from its decimal expansion see Exercise T he word size is usually a power of 2. These methods are examples of algorithms. To add llOlh and lOOlh. The algorithms described are used for both binary arithmetic with integers less than the word size of a computer. W hen we add a and h.
We will describe algorithms for performing addition. W hen performing baser addition by hand. C0 is the carry to the next place..
Proceeding inductively. Consider n-i n-i n-i i i i a.. B0 is the borrow from the next place of the base r expansion of a. With growing interest in computing machines. Abu Ja'far Mohammed ibn Masa al-Khwarizmi see his biography included on the next page. We proceed inductively to find integers Bi and di. The word "algorism" originally referred only to the rules of performing arithmetic using Hindu-Arabic numerals. We use the division algorithm again to find integers B1 and di such that When ai.
To multiply h by People in the West first learned about algebra from his works. The name al-KhwSrizmt means '1i'om the town of Kowarzizm.
To an-l. To multiply an-l. We found the binary digits 1. Al-Khwarizmt was the author of books on mathematics. In general.. To subtract h from l lO l l i. Another book describes procedures for arithmetic operations using Hindu-Arabic numerals. T here1ore. To multiply l lOlh and 11 lOh.
W hen multiplying two integers with baser expansions.. By mathematical induction. Ri Example 2. To find the other digits of q. This is how we find the digits of q. Now assume that Then establishing 2. Subtract lh from ll llOOOOll z. Multiply 5 and 5. Write algorithms for the basic operations with integers in base -2 notation see Exercise 8 of Section 2.
Multiply lllOl z and llOOOl z. A well-known rule used to find the square of an integer with decimal expansion anan-l. How is the one's complement representation of the difference of two integers obtained from the one's complement representations of those integers? Find the quotient and remainder when 5 is divided by Explain how to add. Find the quotient and remainder when lh is divided by 11lOl z.
How is the one's complement representation of the sum of two integers obtained from the one's complement representations of those integers? Multiply lh and l z. Subtract 5 from 5.
Subtract z from z. Find the quotient and remainder when lh is divided by llOl z. Add z and l z. Add 5 and 5. Integer Representations and Operations 60 2. Add z and z. Give an algorithm for adding and an algorithm for subtracting Cantor expansions see the preamble to Exercise 28 of Section 2. A dozen equals Using base In this exercise.. If f and g are functions taking positive values. Because the number of 3 3 less than a constant times n. The actual amount of time required to carry out a bit operation on a computer varies depending on the computer architecture and capacity.
In general. By a bit operation we mean the addition.. W hen we describe the number of bit operations needed to perform an algorithm. To motivate the definition of this notation. Multiply two arbitrarily large integers using the conventional algorithm. Big-0 notation provides an upper bound on the size of a function in terms of a particular well-known reference function whose size at large values is easily understood..
In describing the number of bit operations needed to perform calculations. We will measure the amount of time in terms of bit operations. Perform subtraction with arbitrarily large integers. Divide arbitrarily large integers.
Show that the base 2Bexpansion of anan-l.. Suppose that to perform a specified operation on an integer bit operations. Verify the rules given in Exercises 23 and 24 for examples of your choice.
Perform addition with arbitrarily large integers. Integer Representations and Operations 62 Big-0 notation is used extensively throughout number theory andin the analysis 0 Paul Bachmannintroduced big-0notationin [Ba94]. Bachmann introduced big-0 notation in After recovering from tuberculosis. Landau first taught at the University of Berlin and then moved to GOttingen. His talent for mathematics was discovered by one of his early teachers..
He authored a three-volwne work on number theory and many other books on mathematical analysis and analytic number theory. The use of big-0 notation in the analysis of algorithms was popularized by renowned computer scientist Donald Knuth. The big-0 notation is sometimes called a Landau symbol. We 8 in the definition. We can show on the set of positiveintegers that n4 To do this. His writings include a five-volume survey of number theory..
We illustrate this concept of big-0notation with several examples. Noting that each Ej. Hf is O g and c is a positive constant. His main contributions to mathematics were in the field of analytic number theory. He received his doctorate in under the direction of Frobenius. Knuth has written for a wide range of professionaljournals in computer science and mathematics.
This series has had a profound influence on the development of computer science. He was an excellent student who also applied his intelligence in unconventional ways. The Art of Computer Programming. He popularized the big-0 notation in his work on the analysis of algorithms. Knuth has also invented the widely used TeX and Met. He is especially interested in updating and adding to his famous series.
At Case. ML and the Internet. Hence Theorem 2. TeX played an important role in the development of HT. Knuth is the founder of the modem study of computational complexity and has made fundamental contributions to the theory of compilers. Knuth graduated from Case Institute of Technology in with B. A0 B0. We will now estimate the number of bit operations required to multiply two n-bit integers by using 2.
This is illustrated by the following example. Well-known reference functions used in n big-0 estimates include 1. By Theorem 2. If we let M n denote the number of bit operations needed to. Note that more complicated functions than these occur in big-0 estimates. Ao Bo. Calculus can be used to show that each function in this list is smaller than the next function in the list. We can use 2. Using 2. To develop one such algorithm. To carry out 1 1 the induction argument.
Iog2 3 is approximately 1. Proofs may be found in [Kn97] or [Kr79]. As the induction hypothesis. From 2. M n is O n10g Using inequality 2. Multiplication of two n-bit integers can be performed using 0 n10g23 bit operations.
Proof From 2. This establishes that 2. Because log2 n and log2 log2 n are much smaller than nE for large numbers n. Show that if f is 0 g. Show that a function only if f is O log7 n. Show that n log n is 0 log n! We state the following theorem. Let r f k is 0 gk for all positive integers k.
Analyzing the conventional algorithms for subtraction and addition. Show that if 9. Determine whether each of the following functions is 0 n on the set of positive integers. Suppose that mis a positive real number. The conventional algorithm described in Section 2. There is an algorithm to multiply two n-bit integers using O n Iog2 n log2 log2 n bit operations. Although we know that M n is 2 O n log2 n log2 log2 n.
Integer Representations and Operations 66 Note that Theorem 2. Show that n! Use identity 2. If A and Bare then x n x n aij and bij for Show that it is possible to multiply two 2 2 x matrices using only seven multiplications of integers. Give an estimate of the number of bit operations needed to find the binary expansion of an integer from its decimal expansion. Show that to multiply an n-bit and an m-bit integer in the conventional manner requires 0 nm bit operations.
Conclude from Exercise 22 that two n x n. Give an estimate for the number of bit operations needed to find each of the following quantities. Complete the multiplication using only nine multiplications of one-digit integers.
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You have successfully signed out and will be required to sign back in should you need to download more resources. Elementary Number Theory, 6th Edition. Description Elementary Number Theory, Sixth Edition , blends classical theory with modern applications and is notable for its outstanding exercise sets.
A full range of exercises, from basic to challenging, helps students explore key concepts and push their understanding to new heights. Computational exercises and computer projects are also available. Reflecting many years of professor feedback, this edition offers new examples, exercises, and applications, while incorporating advancements and discoveries in number theory made in the past few years.
Extensive and diverse exercise sets include exercises to develop basic skills, intermediate exercises to help students put several concepts together and develop new results, exercises designed to be completed with technology tools, and challenging exercises to expand understanding.
Answers are provided to all odd-numbered exercises within the text, and solutions to all odd-numbered exercises are in the Student Solutions Manual, which is hosted on the Companion Website. Applications of number theory are well integrated into the text, illustrating the usefulness of the theory. Computer exercises and projects in each section of the text cover specific concepts or algorithms from that section, guiding students on combining the mathematics with their computing skills.
Cryptography and cryptographic protocols are covered in depth. This is the first number theory text to cover cryptography, and results important for cryptography are developed with the theory in the early chapters. The flexible organization allows instructors to choose from a wealth of topics when designing a course. Historical content and biographies illustrate the human side of number theory, both ancient and modern.
Careful proofs explain and support a number of the key results of number theory, helping students develop their understanding. The Companion Website www. The Instructor's Solution Manual available for download from the Pearson Instructor Resource Center provides complete solutions to all exercises, material on programming projects, and an extensive test bank.
Applets on the Companion Website involve some common computations in number theory and help students understand concepts and explore conjectures. Additionally, a collection of cryptographic applets is also provided. New to This Edition. Many new discoveries, both theoretical and numerical, are introduced. Coverage includes four Mersenne primes, numerous new world records, and the latest evidence supporting open conjectures.