Chapter Two. Number Theory


 Florence Ashlie Martin
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1 Chapter Two Number Theory 2.1 INTRODUCTION Number theory is that area of mathematics dealing with the properties of the integers under the ordinary operations of addition, subtraction, multiplication and division. It is one of the oldest and, without dispute, one of the most beautiful branches of mathematics. Its problems and theorems have been studied by mathematicians, both amateur and professional, for well over 2000 years. In a large measure, the subject is characterized by the simplicity with which difficult problems can be stated and the ease with which they can be understood and appreciated by persons without much mathematical background. Thus it should come as no surprise that such problems have attracted the attention of professional mathematicians and amateurs alike. Many of the most basic and interesting problems in number theory involve prime numbers. Here is an example of one such problem: We prove in Section 2.4 that there are infinitely many prime numbers; but Are there infinitely many primes of the form n 2 + 1, where n Z +? For instance, the primes 2, 5, and 17 are of this form, since 2 = , 5 = , and 17 = This question is certainly easy to understand and yet, to this day, no one has determined the answer to it. Another famous problem, one with an intriguing history, is due to the famous French mathematician Pierre de Fermat ( ). To begin our discussion of it, recall from plane geometry the Pythagorean theorem, which says that the side lengths a, b, and c of a right triangle (where c is the length of the hypotenuse) satisfy the relation c 2 = a 2 + b 2 Triples (a, b, c) of positive integers that satisfy this relation and are called Pythagorean triples; the smallest and most wellknown is (3, 4, 5). Are there infinitely many Pythagorean triples? Well, of course! Once we have one triple (a, b, c) we can get infinitely many others just by taking multiples of it; that is, look at (na, nb, nc) where n is any positive integer. Starting with the triple (3, 4, 5), for instance, we obtain (6, 8, 10), (9, 12, 15), and so on. But multiples of a given triple are not very interesting. So let s call a Pythagorean triple primitive provided it is not simply a multiple of some smaller triple. We then get a revised, and more interesting, question: Are there infinitely many primitive Pythagorean triples? It turns out that the answer is yes. In fact, Pythagoras himself is credited with the following result.
2 84 Chapter 2 Number Theory Theorem 2.1 (Pythagoras): If n is an odd integer, n 3, then is a primitive Pythagorean triple. (n, n2 1 2, n2 + 1 ) 2 Note that Pythagoras formula yields the following triples: (3, 4, 5), (5, 12, 13), (7,24,25),(9, 40, 41), (11, 60, 61), (13,84, 85),... However, it does not give us all the primitive triples, for example, (8, 15, 17) does not fit Pythagoras formula. For more on the problem of finding primitive Pythagorean triples, see Chapter Problems 19 and 20. Are you starting to think like a mathematician yet? Seeing that the equation z 2 = x 2 + y 2 has infinitely many solutions in the positive integers, it may seem natural to wonder about similar equations of higher degree. What about the equation z 3 = x 3 + y 3? Does this equation have any solutions in the positive integers? That is, are there any triples (a, b, c) of positive integers such that c 3 = a 3 + b 3? In general, let d be a positive integer, and consider the equation z d = x d + y d For d > 2, are there any solutions to equation in the positive integers? Fermat couldn t find any; in fact, he claimed, in 1637, to have proved the following assertion. Fermat s Conjecture: For d > 2, no solutions to equation exist in the positive integers. Now here s where the story gets interesting. Fermat had the practice of making notes in his copy of the works of the Greek mathematician Diophantus (circa A.D. 300). He would quite often write down, without proof, a result he had discovered. The preceding conjecture is one of these discoveries. In fact, it is the only one that mathematicians had, until very recently, been unable to prove. Tantalizing us even further, Fermat himself wrote, For this I have discovered a truly wonderful proof, but the margin is too small to contain it. Because of this claim the conjecture has been called Fermat s last theorem, or FLT for short. Many famous mathematicians worked on the Fermat conjecture. Euler, for example, proved the conjecture for the case d = 3. Fermat himself proved it for d = 4 and, in 1825, Legendre and Dirichlet independently proved it for d = 5. More recently, in 1983, Gerd Faltings proved a conjecture of Mordell which implies that, for each d > 2, there are only finitely many (possibly none!) solutions to equation in the positive integers. But, though many tried, no one was able to prove Fermat s conjecture until recently, that is. As is often the case in mathematics, failed attempts to prove the general Fermat conjecture were far from fruitless; they gave rise to a wealth of important mathematics, including a good portion of abstract algebra. Let us now fastforward to the summer of A 40yearold mathematics professor at Princeton University, Andrew Wiles, had just spent the last seven years working alone and in secrecy on the world s most famous unsolved math problem. Finally, a shout of Eureka! In fact, Wiles had managed to prove (he thought) an important special case of a very general conjecture known as
3 2.1 Introduction 85 the ShimuraTaniyama conjecture, and from this result Fermat s last theorem follows as a corollary. Wiles decided to unveil his results by giving a series of three lectures in June at a number theory conference at Cambridge University in England. Maintaining suspense to the very end, Wiles gave his lecture series the very general title of Modular Forms, Elliptic Curves, and Galois Representations. By the third lecture, many in the audience had guessed what Wiles was up to. When he wrote his main theorem on the blackboard, there was an audible gasp in the room, and when he then wrote that Fermat s last theorem followed as a corollary, the audience of mathematicians (usually a fairly staid bunch) broke into applause! messages flashed the news across the world that Wiles had proven FLT. The news even made front page headlines in many newspapers, including the New York Times. But wait, a few skeptics cautioned, shouldn t the celebration be put on hold until the details of the proof have been checked? Haven t other people claimed to have proven FLT, only to have errors found in their proofs upon closer examination? Indeed, FLT is generally considered to hold the record for incorrect proofs; in fact, several purported proofs were at first judged to be correct and were actually published in mathematics journals. But following the Cambridge conference, even though no one had as yet read the 200 or so handwritten pages of Wiles manuscript, most of the experts believed that Wiles had indeed proven FLT this based on Wiles excellent reputation, the outline of the proof he had given in his series of lectures, and the fact that his approach just seemed right to experts in the field. Ever cautious, Wiles initially refused to circulate his manuscript broadly, preferring instead to have a small number of close associates check it. He knew it was inevitable that a number of minor errors would be uncovered errors that, hopefully, could be fixed easily. Unfortunately, one seemingly small gap turned out to be rather large. In December of 1993, Wiles sent out an message acknowledging that a gap had been found, but expressing the hope that it could be bridged using the ideas explained in his Cambridge lectures. Fortunately, the story has a happy ending. With the help of a colleague, Richard Taylor, Wiles was able to fix his proof of FLT as Faltings puts it, Taylor and Wiles did not bridge the gap, but rather circumvented it. A set of two manuscripts, a long one by Wiles alone, and another shorter, joint paper by Taylor and Wiles, were released in late October, Having been checked already by several leading experts in the field, they were accepted for publication, and the articles appeared in 1995 in the Annals of Mathematics [A. Wiles, Modular elliptic curves and Fermat s Last Theorem, Ann. Math. 141 (1995), ; R.L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), ]. Thus, for the record, let us formally state Fermat s last theorem. Fermat s Last Theorem (A. Wiles and R.L. Taylor, 1995): For any positive integer d > 2, the equation z d = x d + y d has no solution (x, y, z) such that each of x, y, and z is a positive integer. The primary aim of this chapter is to provide some basic information from elementary number theory. This includes a treatment of several numbertheoretic algorithms. In addition, we provide additional practice with mathematical induction, which provides an important technique for proving statements about the positive integers. Many of the ideas and results presented in this chapter are used in succeeding chapters of this book, and will be encountered again by the student taking subsequent course work in the mathematical sciences.
4 86 Chapter 2 Number Theory One of the most basic principles used in mathematics, especially in number theory, is the principle of wellordering (PWO). This was introduced in Chapter 1, and we restate it now. Principle of Wellordering: Every nonempty subset of Z + has a smallest element. It is not possible to prove the principle of wellordering using the familiar properties satisfied by the integers under addition and multiplication. However, a little thought should convince you of its selfevident nature. Hence, the principle of wellordering is adopted as an axiom, or basic assumption. To get a better grasp of the principle of wellordering (or, wellordering principle), let s find the smallest element of several nonempty subsets of Z +. Example 2.1: Find the smallest element of each of these nonempty subsets of Z +. (a) S 1 = {n Z + n is prime} (b) S 2 = {n Z + n is a multiple of 7} (c) S 3 = {n Z + n = m for some m Z} (d) S 4 = {n Z + n = 12s + 18t for some s, t Z} Solution: (a) The set S 1 is the set of primes, and the smallest prime is 2. (b) The set S 2 is the set of positive multiples of 7, and the smallest positive multiple of 7 is 7. (c) Here we must find the smallest positive integer n of the form m, where m is an integer. The number 110 = (0) is of this form and, as m increases, n decreases. In fact, as m takes on the values 0, 1, 2, 3,..., the values of n form the sequence 110, 93, 76, 59,..., 8, 9,... Hence, the smallest element of S 3 is 8. The number 8 just happens to be the remainder when 110 is divided by 17. This is more than just a coincidence, as is shown in the next section where the division algorithm is discussed. (d) In this part we are looking for the smallest positive number n of the form 12s + 18t, where s and t are integers. Note that 12s + 18t = 6(2s + 3t); thus, any element of S 4 must be a multiple of 6. Moreover, 6 = 12( 1) + 18(1), so that 6 S 4. This shows that 6 is the smallest element of S 4. The number 6 happens to be the greatest common divisor of 12 and 18, an idea that is explored in Section 2.3. We often make use of the following slight extension of the principle of wellordering.
5 2.1 Introduction 87 Theorem 2.2: Any nonempty subset of the set {0, 1, 2, 3,...} of nonnegative integers has a smallest element. Proof: Let S be an arbitrary nonempty subset of the set of nonnegative integers. We consider two cases, depending on whether or not 0 S. In the first case, if 0 S, then clearly 0 is the smallest element of S (because 0 is the smallest nonnegative integer). In the second case, if 0 / S, then S is a nonempty subset of Z +. In this case the principle of wellordering implies that S has a smallest element. In either case, then, S has a smallest element, and this completes the proof. Exercise Set Plato is credited with the following result: If n is a positive integer, n 3, then is a Pythagorean triple. (2n, n 2 1, n 2 + 1) (a) Verify this result. (b) Find the Pythagorean triples given by Plato s formula for n {3, 4, 5,..., 12}. Which of them are primitive? (c) Give a necessary and sufficient condition (on n) for Plato s formula to yield a primitive Pythagorean triple. (d) In what sense do the formulas of Plato and Pythagoras (Theorem 2.1) complement each other? 2. Using Fermat s last theorem, show that the equation has no solution (x, y, z) in the positive integers. 3. Prove Theorem 2.1. z 3 = 8x y 3 4. In general, a subset T of R is said to be wellordered provided every nonempty subset of T has a smallest element. Determine whether these subsets of R are wellordered. (a) (b) { 9, 6, 3, 0, 1, 2,3} (c) {0} Q + (d) 2Z (e) { 9, 8, 7, 6,...} 5. Find the smallest element of each subset of Z +. (a) A = {n Z + n = m 2 10m + 28 for some integer m} (b) B = {n Z + n = 5q + 2 for some integer q} (c) C = {n Z + n = m for some integer m} (d) D = {n Z + n = 5s + 8t for some integers s and t} 6. Let T, T 1, and T 2 denote arbitrary subsets of R. Referring to the definition given in Exercise 4, prove each of the following:
6 88 Chapter 2 Number Theory (a) If T is a finite subset of R, then T is wellordered. (b) If T is wellordered, then c + T is wellordered for any real number c. (c) If T is wellordered, then ct is wellordered for any nonnegative real number c. (d) If T is a subset of Z and T itself has a smallest element, then T is wellordered. (e) If T 1 T 2 and T 2 is wellordered, then T 1 is wellordered. 7. Verify that (20, 21, 29) is a primitive Pythagorean triple that results neither from the formula of Pythagoras nor from the formula of Plato. (Is it the smallest Pythagorean triple that is missed by both of these formulas? See Chapter Problem 20.) 2.2 DIVISION ALGORITHM One of the fundamental concepts included in any introduction to number theory is that of factoring integers. In particular, given an integer n > 1, we are interested in expressing n as a product of primes. For example, if n = 132, then n = Is it always possible to do this? Can it, for some n, be done in more than one way? Before these questions can be answered, it is necessary to define and work with certain fundamental terms, like factor and prime. Definition 2.1: Let a and b be integers, with a 0. We say that a divides b, denoted a b, provided there is an integer q such that b = aq. In this case we also say that a is a factor (or divisor) of b and we call b a multiple of a. Example 2.2: (a) 2 6, since 2 3 = 6. (b) 3 27, since ( 3)( 9) = 27. (c) 12 ( 72), since 12( 6) = 72. (d) 4 does not divide 7, since there is no integer q such that 4q = 7. (e) 8 does not divide 28, since there is no q Z such that ( 8)q = 28. (f) For which integers m is it true that 0 is a multiple of m? In order for m 0 to hold, there must exist an integer q such that mq = 0. Note that q = 0 works, since m 0 = 0. Thus, 0 is a multiple of m for every integer m. Now, if m is not zero and m is a factor of the integer n, then n/m is also a factor of n; in fact, n m m = n However, this won t work if m = 0. For this and other technical reasons, we do not allow 0 to be a factor. Note that, for any integer b > 1, both of the numbers 1 and b are positive factors of b. For some positive integers b these are the only positive factors of b.
7 2.2 Division Algorithm 89 Definition 2.2: An integer p > 1 is called a prime number (or, simply, a prime) provided the only positive factors of p are 1 and p. An integer n > 1 that is not prime is called a composite number (or, simply, a composite). Suppose that the integer n is composite. Then n > 1 and n is not prime. This means that n has a factor d such that 1 < d < n. Thus it follows that n = dq, where q is an integer and 1 < q < n. In general, we refer to factors such as d and q as proper factors (or proper divisors) of n, and we call 1 and n the trivial factors (or trivial divisors) of n. Example 2.3: The numbers 2 and 3 are prime, 4 = 2 2 is composite, 5 is prime, 6 = 2 3 is composite, 7 is prime, 8 = 2 4 is composite, and 9 = 3 3 is composite. The primes less than 100 are: 2, 3, 5, 7, 11,13,17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,67,71,73,79,83,89,97 Primes are discussed further in Section 2.4. Example 2.4: Find the factors of 126. Solution: Note that so the set of positive factors of 126 is 126 = 2 63 = 3 42 = 6 21 = 7 18 = 9 14 S = {1, 2, 3, 6, 7,9,14,18,21,42,63,126} Moreover, for any integers a and b (a 0), if a is a factor of b then a is also a factor of b. It follows that the set of negative factors of 126 is 1S = { 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126} A number of basic properties of the relation divides are used in this and subsequent chapters. The next theorem lists several of these properties. Theorem 2.3: The following implications hold for any integers a, b, and c, with a 0: 1. If a b, then a (bx) for any integer x. 2. If a b and b c, then a c. (Here, b 0.) 3. If a b and a c, then a (bx + cy) for any integers x and y. 4. If a b and b a, then a = b or a = b. (Here, b 0.) 5. If a b and b is nonzero, then a b. Proof: We give direct proofs of parts 2 and 4 and leave the remaining parts to Exercise 2; the proofs of parts 2 and 4 provide a good illustration of how the remaining parts are proved.
8 90 Chapter 2 Number Theory For part 2, assume a b and b c. Then there exist integers q 1 and q 2 such that b = aq 1 and c = bq 2. It follows (using substitution for b) that c = bq 2 = (aq 1 )q 2 = a(q 1 q 2 ) Thus, there exists an integer q, namely q = q 1 q 2, such that c = aq. Therefore, a c. For part 4, assume a b and b a. Then there exist integers q 1 and q 2 such that b = aq 1 and a = bq 2. Hence, we have that a = bq 2 = (aq 1 )q 2 = a(q 1 q 2 ) Since a 0, it follows that 1 = q 1 q 2 and, since q 1 and q 2 are integers, we have that either q 1 = q 2 = 1 or q 1 = q 2 = 1. Hence, a = ±b. Let us emphasize again the meaning of the statement a is a factor of b. This means that there is some integer q such that b = aq. You probably remember learning about long division in elementary school. In this process, one integer b, called the dividend, is divided by another integer a, called the divisor, to obtain a quotient q and a remainder r. For example, when b = 23 is longdivided by a = 7, we obtain a quotient of 3 and a remainder of 2. One then checks this by noting that 23 = 7(3) + 2. However, note that 23 can be expressed in the form 7q + r in other ways; for example, 23 = 7(4) + ( 5) = 7(2) + 9. Is it wrong to say that the quotient is 4 and the remainder is 5, or that the quotient is 2 and the remainder is 9? Well, yes it is, because, in longdividing 23 by 7, one is taught to determine the largest quotient q for which the remainder r is nonnegative. This implies that the remainder must be less than the divisor. In fact, it should be remarked that, among all integer expressions of the form 23 7q, one chooses the smallest nonnegative one for the remainder. In general, given integers a and b with a > 0, there exist unique integers q and r such that b = aq + r, where 0 r < a. Analogous to what is stated above regarding 23 and 7, the remainder r will turn out to be the smallest nonnegative value of the expression b aq, where q Z. This property is known as the division algorithm, and in this context the integers a, b, q, and r are called the divisor, dividend, quotient, and remainder, respectively. Theorem 2.4 (Division Algorithm): Given integers a and b with a > 0, there exist integers q and r such that b = aq + r and 0 r < a. Moreover, q and r are uniquely determined by a and b. Proof: Let integers a and b be given with a > 0. We first show that there exist integers q and r such that b = aq + r and 0 r < a. In order to do this, we apply the extended version of the principleof wellordering (Theorem 2.2) to the set S = {b ax x Z and b ax 0} So S is a set of nonnegative integers. In order to apply Theorem 2.2, we must show that S is nonempty. If b 0, then b S, since, letting x = 0, we obtain b = b a(0). Suppose, on the other hand, thatb < 0. Then b ab S, since letting x = b yields b ab = b(1 a) 0. Thus, in either case, S is nonempty. Therefore, by Theorem 2.2, S has a smallest element; call it r. Since r S, there is somex Z, say x = q, such that r = b aq. Thus, b = aq + r and, since r S, we have that r 0.
9 2.2 Division Algorithm 91 It remains to show that r < a. To do this, we proceed by contradiction and suppose that r a. Let t = r a. Then t 0 and, since a > 0, we have that t < r. Moreover, t = r a = (b aq) a = b (aq + a) = b a(q + 1) But this means that t S (let x = q + 1) and t < r, contradicting the fact that r is the smallest element of S. This completes the proof of the existence of q and r. We next show that the quotient q and the remainder r are uniquely determined by a and b. To show that there is a unique mathematical object with a given property, a standard technique is to suppose that there are two objects with the given property, and then show that the two objects must, in fact, be equal. So, suppose that aq 1 + r 1 = b = aq 2 + r 2, where q 1, r 1, q 2, and r 2 are integers and both 0 r 1 < a and 0 r 2 < a. We wish to show that q 1 = q 2 and r 1 = r 2. Assume, without loss of generality, that r 1 r 2 ; hence, r 2 r 1 0. Since aq 1 + r 1 = aq 2 + r 2, we obtain a(q 1 q 2 ) = r 2 r 1 Thus, a (r 2 r 1 ). Since 0 r 2 r 1 < a, it must be the case that r 2 r 1 = 0. Therefore, r 2 = r 1. Then, since a(q 1 q 2 ) = r 2 r 1 = 0 and a 0, we obtain that q 1 q 2 = 0, so that q 1 = q 2. This completes the proof. The proof of Theorem 2.4 is an existence proof. It concentrates on verifying the existence of integers q and r satisfying the properties stated in the theorem, rather than on giving a method for finding q and r. However, the proof implicitly suggests an algorithm for finding q and r from a and b, using only the operations of addition and subtraction. We give an informal description of this algorithm here, leaving further investigation of it to the exercises. If b 0, then consider the sequence b a(0), b a(1), b a(2),... of numbers obtained by starting with b and then repeatedly subtracting a. Since a > 0, the numbers in this sequence are eventually negative; r = b aq is the last nonnegative term in this sequence. On the other hand, if b < 0, then consider the sequence b a(0), b a( 1), b a( 2),... of numbers obtained by starting with b and then repeatedly adding a. The numbers in this sequence are eventually nonnegative; r = b aq is the first nonnegative term. (Note in this case that q 0.) The following corollary to Theorem 2.4 extends the division algorithm to handle the case of a negative divisor. Its proof is left to Exercise 10. Corollary 2.5: Given integers a and b with a 0, there exist uniquely determined integers q and r such that b = aq + r, where 0 r < a.
10 92 Chapter 2 Number Theory Example 2.5: In the context of Corollary 2.5, find q and r for the given a and b. (a) a = 17, b = 110 (b) a = 7, b = 59 (c) a = 11, b = 41 (d) a = 5, b = 27 (e) a = 13, b = 7 (f) a = 13, b = 7 Solution: (a) Since 110 = 17(6) + 8, we have that q = 6 and r = 8. (b) Since 59 = 7( 9) + 4, we have that q = 9 and r = 4. In this problem it is easy to make the mistake of saying q = 8 and r = 3, since 59 = 7( 8)+( 3). However, remember that any remainder is required to be nonnegative, so r = 3 can t be right. (c) In this part we find that q = 3 and r = 8. (d) Here we find that q = 6 and r = 3. Parts (e) and (f) are meant to illustrate the following general problem: Given integers a and b with 0 b < a, find q and r. This is actually an easy problem, since b = a(0) + b and b is a valid remainder; see Exercise 24. In part (e), for example, note that 7 = 13(0) + 7 and 0 7 < 13 Thus, q = 0 and r = 7. Similarly, in part (f), 7 = ( 13)(0)+7 and 0 7 < 13. Hence it follows that q = 0 and r = 7. Example 2.6: Show that any integer m is expressible in precisely one of the forms 3q, 3q + 1, or 3q + 2, where q is an integer. Solution: Apply Corollary 2.5 with dividend m and divisor 3 it states that there exist unique integers q and r such that m = 3q + r, where 0 r < 3. Hence r = 0, r = 1, or r = 2. Since r is uniquely determined, it follows that m is expressible in precisely one of the forms 3q, 3q + 1, or 3q + 2. Example 2.7: Show that the product of any two consecutive integers is even. Solution: According to the division algorithm (Corollary 2.5), every integer m is uniquely expressible in the form m = 2q + r, where 0 r < 2. Thus m is expressible in precisely one of the forms 2q or 2q + 1. If m = 2q, we call m an even integer, whereas if m = 2q + 1, we call m an odd integer. Now, consider two arbitrary consecutive integers, n and n + 1, say. We want to show that n(n + 1) is even. This means that we must show that n(n + 1) = 2k for some integer k. We consider two cases, depending on whether n itself is even or odd. In the first case, suppose that n is even, say n = 2q. Then n(n + 1) = 2q(2q + 1) = 2(2q 2 + q) This shows that n(n + 1) is even. (Here, k = 2q 2 + q.)
11 2.2 Division Algorithm 93 In the second case, suppose that n is odd, say n = 2q + 1. Then n(n + 1) = (2q + 1)(2q + 2) = 2(2q + 1)(q + 1) so again n(n + 1) is even. (In this case, k = (2q + 1)(q + 1).) In general, given an arbitrary positive integer n, the division algorithm tells us that every integer is expressible in precisely one of the following forms: nq, nq + 1, nq + 2,..., nq + (n 1) In much of the discussion that follows you need to make use of this idea, so be ready for it. The following example illustrates this point. Example 2.8: Show that the product of any two integers of the form 6k + 5 has the form 6k + 1. Solution: Let m 1 and m 2 be two integers of the form 6k + 5. This means that m 1 = 6k and m 2 = 6k for some integers k 1 and k 2. Thus, m 1 m 2 = (6k 1 + 5)(6k 2 + 5) = 36k 1 k k k = 36k 1 k k k = 6(6k 1 k 2 + 5k 1 + 5k 2 + 4) + 1 Therefore, m 1 m 2 has the form 6k + 1 (with k = 6k 1 k 2 + 5k 1 + 5k 2 + 4), as was to be shown. As already noted, if the division algorithm is applied to an integer m and the divisor 3, then the remainder r is precisely one of the numbers 0, 1, or 2. Define the sets S 0, S 1, and S 2 by S 0 = {3q q Z} = 3Z S 1 = {3q + 1 q Z} = 1 + 3Z S 2 = {3q + 2 q Z} = 2 + 3Z Then, for r {0, 1, 2}, S r is the set of all those integers m that yield a remainder of r when divided by 3. For instance, 11 = , so 11 S 2, whereas 11 = 3( 4) + 1, so 11 S 1. By the uniqueness of r, each integer m belongs to exactly one of the sets S 0, S 1, or S 2. It follows that: 1. Z = S 0 S 1 S 2 ; 2. The sets S 0, S 1, and S 2 are pairwise disjoint. Because these two properties hold, we say that {S 0, S 1, S 2 } is a partition of the set Z. important concept of partition is explored further in Chapter 4. In explicit form, S 0 = {... 9, 6, 3, 0, 3,6,9,...} S 1 = {... 8, 5, 2, 1, 4,7,10,...} S 2 = {... 7, 4, 1, 2, 5,8,11,...} Note that two integers are in the same set S r if and only if they differ by a multiple of 3. suggests the following general and important result. The This
12 94 Chapter 2 Number Theory Theorem 2.6: Let m 1, m 2, and n be integers, with n 0, and let the remainders upon division of m 1 and m 2 by n be r 1 and r 2, respectively. Then r 1 = r 2 if and only if n (m 2 m 1 ) Proof: Let m 1, m 2, and n be integers, with n 0, and let the remainders upon division of m 1 and m 2 by n be r 1 and r 2, respectively. To prove this result, we must prove the two implications: (1) If r 1 = r 2, then n (m 2 m 1 ). (2) If n (m 2 m 1 ), then r 1 = r 2. According to the division algorithm, for some integers q 1 and q 2. Thus, m 1 = nq 1 + r 1 and m 2 = nq 2 + r 2 m 2 m 1 = (nq 2 + r 2 ) (nq 1 + r 1 ) = n(q 2 q 1 ) + (r 2 r 1 ) We first prove (1) directly. Assume r 1 = r 2. Then r 2 r 1 = 0, and so m 2 m 1 = n(q 2 q 1 ) which shows that n (m 2 m 1 ). To complete the proof, we must prove implication (2). This is left to Exercise 8. Exercise Set In the context of Corollary 2.5, find q and r for the given a and b. (a) a = 11, b = 297 (c) a = 8, b = 77 (e) a = 5, b = 35 (b) a = 9, b = 63 (d) a = 6, b = 71 (f) a = 6, b = Prove Theorem 2.3, parts 1, 3, and 5. Also, prove part 4 as a corollary to part Let a, b, and c be integers, with a 0. Prove each of these implications: (a) If a b and a c, then a 2 (bc). (b) If a b, then a ( b) and ( a) b. 4. Prove that the following implication holds for any integers a, b, c, and d, with a and c nonzero: 5. Prove each of the following facts: If a b and c d, then (ac) (bd). (a) The square of any odd integer is of the form 4k + 1 (for some integer k). (b) The square of any integer is of the form 3k or 3k + 1.
13 2.2 Division Algorithm Let a, b, and c be arbitrary integers with a and c nonzero. Prove: If (ac) (bc), then a b. 7. Apply the result of part (a) of Exercise 5 to show that none of the numbers 11, 111, 1111, and is a perfect square. (Hint: Apply the division algorithm, with a divisor of 4.) Based on the result of this exercise, make a general conjecture regarding numbers of the form Complete the proof of Theorem 2.6. (Hint: Show that r 1 r 2 is a multiple of n and that 0 r 1 r 2 < n. It follows that r 1 r 2 = 0. Why?) 9. Let a, b, and c be arbitrary integers, with a 0. Prove or disprove: If a (bc), then either a b or a c. 10. Prove Corollary 2.5. (Hint: If a < 0, then a > 0, and we can apply Theorem 2.4 to find integers q and r such that b = ( a)q + r, where 0 r < a.) 11. Apply the result of part (b) of Exercise 5 to show that, for any integer m, 3m 2 1 is not a perfect square. 12. Prove that: (a) The sum of any two even integers is even. (b) The sum of any two odd integers is even. (c) The sum of any even integer and any odd integer is odd. 13. Prove: Given any three consecutive integers, (exactly) one of them is a multiple of 3. (Hint: Denote the three consecutive integers by m, m + 1, m + 2 and use the fact that m is expressible in exactly one of the forms 3q, 3q + 1, or 3q + 2.) 14. Prove that, for any integer m, m 3 m is a multiple of 3. (Hint: Note that m 3 m = m(m 2 1); if m is not a multiple of 3, what can be said about m 2 1?) 15. Prove: For any integer m, (exactly) one of the integers m, m + 4, m + 8, m + 12, m + 16 is a multiple of Let m represent an arbitrary integer. Prove: If m has the form 6q + 5 for some integer q, then mhas the form 3k + 2 for some integer k. What about the converse of this implication? 17. Prove the following results as corollaries to Theorem 2.3. (a) For any integer m, if m is even, then mx is even for any integer x. (b) For any positive integers m and n, if m n, then m n. 18. Other than 2, show that no positive integer of the form n is prime. (Hint: Apply the standard formula for factoring the sum of two cubes.) 19. Let p represent an arbitrary prime. Prove: If p has the form 3q + 1 for some integer q, then p has the form 6k + 1 for some integer k. (Hint: If p has the form 3q + 1, then p must be odd, so what can be said about q?) 20. Prove: If (a, b, c) is a Pythagorean triple, then one of a, b, or c is divisible by 3, one is divisible by 4, and one is divisible by 5. (For example, if a = 5, b = 12, and c = 13, then 3 b, 4 b, and 5 a.)
14 96 Chapter 2 Number Theory 21. In this exercise, we introduce the div and mod notation. Given integers a and b with a 0, Corollary 2.5 states that there exist uniquely determined integers q and r such that b = aq + r, where 0 r < a. In this context, we define the operators div and mod as follows: b div a = q b mod a = r Find b div a and b mod a for the pairs of integers a and b given in Exercise Given positive integers a and b, explain how to use a standard pocket calculator to compute 23. Let m 1 and m 2 be integers such that Find: b div a and b mod a m 1 div 5 = q 1 m 1 mod 5 = 2 m 2 div 5 = q 2 m 2 mod 5 = 3 (a) (m 1 + m 2 ) div 5 (b) (m 1 + m 2 ) mod 5 (c) (m 1 m 2 ) div 5 (d) (m 1 m 2 ) mod Given integers a and b with 0 b < a, find b div a and b mod a. 25. Compute each of the following. (a) 47 div 10 (b) 47 mod 10 (c) 47 div ( 10) (d) 47 mod ( 10) (e) ( 47) div 10 (f) ( 47) mod 10 (g) ( 47) div ( 10) (h) ( 47) mod ( 10) 26. Show, for a > 0, that b div ( a) = (b div a) and b mod ( a) = b mod a 27. Given that m 1 div 7 = q 1, m 1 mod 7 = 2, m 2 div 7 = q 2, and m 2 mod 7 = 6, find: (a) (m 1 + 5) div 7 (b) (m 1 + 5) mod 7 (c) (2m 1 ) div 7 (d) (2m 1 ) mod 7 (e) ( m 2 ) div 7 (f) ( m 2 ) mod 7 (g) (m 1 + m 2 ) div 7 (h) (m 1 + m 2 ) mod 7 (i) (2m 1 + 3m 2 ) div 7 (j) (2m 1 + 3m 2 ) mod 7 (k) (m 1 m 2 ) div 7 (l) (m 1 m 2 ) mod Let a and b be positive integers with 1 a b. (a) Find and verify a formula for ( b) div a in terms of b div a. (b) Find and verify a formula for ( b) mod a in terms of b mod a. 29. Let a and b be positive integers. (a) What is the smallest possible value for b div a? (b) What is the largest possible value for b div a?
15 2.3 Euclidean Algorithm 97 Given a and b, suppose one guesses a value q for b div a in the range of possible values. Note that, if 0 b aq < a, then b div a = q and b mod a = b aq. However: (c) If b aq < 0, what does this indicate about the guess q? (d) If b aq a, what does this indicate about the guess q? 30. Describe (and implement as a computer program) an algorithm that inputs integers a and b with a 0 and outputs b div a and b mod a. Base your algorithm on the results of Exercises 26, 28, and the remarks following the proof of Theorem Several computer programming languages, such as Ada and C++, have built in operators to compute b div a and b mod a for integers a and b. However, the results do not always agree with Definition 2.3. If you are familiar with a language that has such operators, write a short program to test them. 2.3 EUCLIDEAN ALGORITHM In this section we define the greatest common divisor of two integers and describe an efficient method for finding it, given the integers. Definition 2.3: Given integers a and b, the integer c 0 is called a common divisor (or common factor) of a and b provided both c a and c b. If a and b are not both zero, then we define the greatest common divisor (or greatest common factor) of a and b to be the largest common factor of a and b. The greatest common divisor of a and b is denoted by gcd(a, b). Let us make a few observations about gcd(a, b). First, since 1 a and 1 b, we have that Second, 1 gcd(a, b) gcd(b, a) = gcd(a, b) so we may, without loss of generality, assume that a b. Third, since gcd( a, b) = gcd(a, b) = gcd(a, b) we can assume that 0 a b. Finally note that, for b > 0, gcd(0, b) = b = gcd(b, b) Thus, in seeking gcd(a, b), it suffices to consider the case when 1 a < b; in this case, 1 gcd(a, b) a. In particular, this last statement implies that gcd(a, b) exists.
16 98 Chapter 2 Number Theory Example 2.9: Find: (a) gcd(12, 36) (b) gcd(18, 42) (c) gcd(15, 28) Solution: (a) Note that 12 is a factor of 36; hence gcd(12, 36) = 12. (b) The positive factors of 18 are 1, 2, 3, 6, 9, and 18; of these, only 1, 2, 3, and 6 are factors of 42. Therefore, gcd(18, 42) = 6. (c) The positive factors of 15 are 1, 3, 5, and 15. Of these, only 1 is a factor of 28. Thus, gcd(15, 28) = 1. Generalizing the result of part (a) of the preceding example, note that, for 1 a < b, gcd(a, b) = a if and only if a b For 1 a < b, a simpleminded method for finding d = gcd(a, b) is to search the list of numbers a, a 1, a 2,..., 2, 1, looking for the largest one that is a common factor of a and b. In the worst case (when gcd(a, b) = 1), this method would take a steps to find d, where each step consists of determining whether a given positive integer is a common factor of a and b. In a number of practical applications (e.g., data encryption) in which gcd(a, b) must be computed, a might be a number on the order of Even with a fast computer that performs, say, steps per second, finding gcd(a, b) by this method could take seconds in the worst case. Since there are less than 10 8 seconds in a year, this is a very long time far longer than the estimated age of the universe! Fortunately, there is a much faster method, which goes way back to Euclid ( 300 B.C.), and is based on repeated application of the division algorithm. For this method, assume we are given integers a and b with 0 a < b; we wish to find gcd(a, b). First of all, let s handle the easy case; namely, if a = 0, then gcd(a, b) = b. To handle the case when a > 0, we make use of the following lemma. Lemma 2.7: For integers a and b with 0 < a b, let r = b mod a. Then gcd(a, b) = gcd(r, a) Proof: Let d 1 = gcd(a, b) and d 2 = gcd(r, a) We wish to show that d 1 = d 2. We do this by showing that d 1 d 2 and d 2 d 1. Let q = b div a. Then b = aq + r. Since b = a(q) + r(1), by Theorem 2.3, part 3, any common factor of a and r is also a factor of b. Hence, d 2 is a factor of b, and thus d 2 is a common factor of a and b. Therefore, d 2 d 1. (Why?)
17 2.3 Euclidean Algorithm 99 Similarly, since r = b aq = a( q) + b(1), any common factor of a and b is also a factor of r. Hence, d 1 is a factor of r, and thus d 1 is a common factor of a and r. It follows that d 1 d 2. Now consider finding gcd(a, b) when 0 < a b. By the division algorithm, there exist (unique) integers q and r such that b = aq + r, with 0 r < a. By Lemma 2.7, we see that gcd(a, b) = gcd(r, a) This observation forms the basis for a procedure known as the Euclidean algorithm. Note that, for a > 0, we replace the problem of finding gcd(a, b) with the problem of finding gcd(r, a). In the sense that r < a and a < b, this new problem constitutes a reduced form of the original problem. But, you may ask, How do I now find gcd(r, a)? The answer is, Apply the same reasoning again! That is, if r = 0, then gcd(r, a) = a. Otherwise, let r be the remainder when a is divided by r; then gcd(r, a) = gcd(r, r). The Euclidean algorithm is an example of a recursive algorithm, because it operates by reducing a (nontrivial) instance of a given type of problem to a smaller instance of the same type of problem. Euclidean Algorithm: Given integers a and b with 0 a b: 0. If a = 0, then gcd(a, b) = b; 1. Otherwise, let r be the remainder when b is divided by a; then gcd(a, b) = gcd(r, a). Example 2.10: Use the Euclidean algorithm to compute gcd(64, 148). Solution: Since 64 > 0, we apply the division algorithm to 64 and 148, obtaining 148 = 64(2) + 20, namely, a quotient of 2 and a remainder of 20. By step 1 of the algorithm, then, gcd(64, 148) = gcd(20, 64) Next 20 > 0, so now we divide 64 by 20, obtaining a quotient of 3 and a remainder of 4. So, by step 1 of the algorithm, gcd(20, 64) = gcd(4, 20) Still 4 > 0, so we apply the recursive step again. Dividing 20 by 4 yields a quotient of 5 and a remainder of 0, so that gcd(4, 20) = gcd(0, 4) Finally, gcd(0, 4) = 4. Therefore, putting all of the steps together, we see that gcd(64, 148) = gcd(20, 64) = gcd(4, 20) = gcd(0, 4) = 4 It is the repeated application of Lemma 2.7 that indicates the general form of the Euclidean algorithm. Let s look at this form more carefully. Suppose that a and b are positive integers with a < b. We begin by setting r 0 = b and r 1 = a. We then successively apply the division algorithm as follows:
18 100 Chapter 2 Number Theory r 0 = r 1 q 1 + r 2 0 r 2 < r 1 r 1 = r 2 q 2 + r 3 0 r 3 < r 2.. r k 1 = r k q k + r k+1 0 r k+1 < r k. Consider the values r 0, r 1, r 2,..., r k 1, r k, r k+1,.... In view of the requirement in the division algorithm that any remainder be less than its corresponding divisor, we see that these numbers form a strictly decreasing sequence of integers; namely, that r 0 > r 1 > r 2 > > r k 1 > r k > r k+1 > However, the above sequence of remainders can t go on forever, because each remainder is nonnegative, and it s impossible to have an infinite, strictly decreasing sequence of nonnegative integers. Hence, there must be some positive integer n + 1 such that r n+1 = 0, and so the above list of relations can be rewritten as follows: Then we obtain r 0 = r 1 q 1 + r 2 0 r 2 < r 1 r 1 = r 2 q 2 + r 3 0 r 3 < r 2.. r n 2 = r n 1 q n 1 + r n 0 r n < r n 1 r n 1 = r n q n + r n+1 r n+1 = 0 gcd(a, b) = gcd(r 2, a) = gcd(r 3, r 2 ) = = gcd(r n+1, r n ) = gcd(0, r n ) = r n It can be shown (see Chapter Problem 30) that the Euclidean algorithm requires not more than 2 log 2 a divisions to compute gcd(a, b). For a on the order of , this bound is on the order of 200(log 2 10) 665. So the Euclidean algorithm is very efficient! Given integers a and b, a linear combination of a and b (over Z) is any expression of the form as + bt with s, t Z. Our next result provides an important characterization of gcd(a, b), showing that it is the smallest positive integer that can be expressed as a linear combination of a and b. The proof of the theorem applies the division algorithm in a strong way, and also makes use of the principle of wellordering.. Theorem 2.8: Let a and b be integers, not both 0. Then gcd(a, b) is the smallest positive integer expressible as a linear combination of a and b. Proof: Consider the set S = {ax + by x, y Z and ax + by > 0} If we let x = a and y = b, then ax + by = a 2 + b 2 > 0 (since not both a and b are zero). Thus, the set S is nonempty. By the principle of wellordering, S has a smallest element; call it d. So d is the smallest positive integer expressible as a linear combination of a and b, say, d = as + bt, where s, t Z. To show that d = gcd(a, b), we must verify the following:
19 2.3 Euclidean Algorithm d is a common divisor of a and b; 2. If c is any common divisor of a and b, then c d. To show 1, we first apply the division algorithm to a and d, obtaining integers q and r such that a = dq + r, with 0 r < d. To show that d a, it suffices to show that r = 0. Since d = as + bt, we have that r = a dq = a (as + bt)q = a(1 sq) + b( tq) where both 1 sq and tq are integers. So r is a linear combination of a and b. But r < d, and d is the smallest positive linear combination of a and b, so r can t be positive. Hence, r = 0, as we wished to show. In a completely analogous way, it can be shown that d b. Next, to show 2, let c be any common divisor of a and b. If c < 0, then clearly c d, so we may assume that c > 0. Since c is a common divisor of a and b, it follows from Theorem 2.3, part 3, that c is a divisor of any linear combination of a and b; in particular, c d. Then, since both c and d are positive, it follows from part 5 of Theorem 2.3 (more directly, from Exercise 17, part (b) of Exercise Set 2.2), that c d. This completes the proof. So, given integers a and b, not both zero, there exist integers s and t such that gcd(a, b) = as + bt We have an efficient algorithm, namely, the Euclidean algorithm, for finding gcd(a, b). Is there some way to extend this algorithm so that it also finds the integers s and t? Indeed there is, and it is called the extended Euclidean algorithm. We illustrate the general form of the extended Euclidean algorithm by looking at a particular example. In particular, let s compute d = gcd(141, 486) and find integers s and t such that d = 141s + 486t. Recall that d is the last nonzero remainder obtained in the process of applying the Euclidean algorithm. There is an especially nice way to display the remainders and quotients obtained along the way. In the general case, where 0 < a < b, if r n is the last nonzero remainder, then d = r n, and we can display the results in the following table (recall that r 0 = b and r 1 = a): b a r 2 r 3 r 4 r n 0 q 1 q 2 q 3 q 4 q n It is easy to see how the relations obtained from our successive application of the division algorithm give rise to the entries in this table. Namely, for 0 k < n, we have the relation r k = r k+1 q k+1 + r k+2 and in the table this information is entered into successive columns as follows: r k r k+1 r k+2 q k+1 In our particular example, with a = 141 and b = 486, you should verify that the following table is obtained:
20 102 Chapter 2 Number Theory Thus, we see that d = gcd(141, 486) = 3. The method for determining the values of s and t such that 3 = 141s + 486t makes use of the above table. The idea is to express each remainder r k, 0 k n, as a linear combination of a and b. That is, for each k, 0 k n, we wish to find integers s k and t k such that r k = as k + bt k Then, when k = n, we obtain the desired relation expressing d as a linear combination of a and b. So the method generates two additional sequences: s 0, s 1, s 2,..., s n and t 0, t 1, t 2,..., t n. These two sequences are added as rows to the table, so that in general the table looks like this: b a r 2 r n 0 q 1 q 2 q n s 0 s 1 s 2 s n t 0 t 1 t 2 t n To get things started, we need to find values for s 0, t 0, s 1, and t 1 such that b = r 0 = as 0 + bt 0 a = r 1 = as 1 + bt 1 That s easy! Simply let s 0 = 0, t 0 = 1, s 1 = 1, and t 1 = 0. It s also easy to obtain the values of s 2 and t 2. By the division algorithm, r 2 = b aq 1 = a( q 1 ) + b(1) Hence, s 2 = q 1 and t 2 = 1. So far, then, our general table looks like this: b a r 2 r 3 r n 0 q 1 q 2 q 3 q n 0 1 q 1 s 3 s n t 3 t n In our particular example, when a = 141 and b = 486, we have the following so far: s 3 s t 3 t 4 It is important to understand that each column of this table indicates how to express the remainder in that column as a linear combination of a and b. For example, in the above table, the column corresponding to k = 2 indicates that 63 = 141( 3) + 486(1)
21 2.3 Euclidean Algorithm 103 Next, we need to determine values for s 3 and t 3 such that 15 = r 3 = as 3 +bt 3. To do this, we first make use of the division algorithm to express r 3 in terms of r 1 and r 2. Recall that r 1 = r 2 q 2 + r 3 ; hence: 15 = r 3 = r 1 r 2 q 2 = (2) We then use the values already found for s 1, s 2, t 1, and t 2 to replace each of r 1 and r 2 in the above expression by a linear combination of a and b. In the general case, this gives us: r 3 = r 1 r 2 q 2 = (as 1 + bt 1 ) (as 2 + bt 2 )q 2 = a(s 1 s 2 q 2 ) + b(t 1 t 2 q 2 ) and so we see that s 3 = s 1 s 2 q 2 and t 3 = t 1 t 2 q 2. In our particular example, we find that r 3 = 15 = (2) = [ 141(1) + 486(0) ] [ 141( 3) + 486(1) ] (2) = 141(1 ( 3)2) + 486(0 1(2)) = 141(7) + 486( 2) Thus, s 3 = 7 and t 3 = 2. (Check that 15 = 141(7) + 486( 2).) Try to notice a general pattern in the above expressions for s 3 and t 3. Can you guess what the expressions for s 4 and t 4 are? Let s work it out. Again the division algorithm is the key, because from it we know that r 2 = r 3 q 3 + r 4. Hence, r 4 = r 2 r 3 q 3 = (as 2 + bt 2 ) (as 3 + bt 3 )q 3 = a(s 2 s 3 q 3 ) + b(t 2 t 3 q 3 ) Thus, s 4 = s 2 s 3 q 3 and t 4 = t 2 t 3 q 3. In our example, then, we find that s 4 = s 2 s 3 q 3 = 3 7(4) = 31 t 4 = t 2 t 3 q 3 = 1 ( 2)(4) = 9 Hence, r 4 = 3 = 141( 31) + 486(9). (Check this!) Since d = r 4, our example is complete. In summary, our results indicate that d = 3, s = 31, and t = 9, and here is the complete table: Let us now return to consideration of the extended Euclidean algorithm in the general case. As already noted, the results from an application of the algorithm can be displayed in table form as follows: b a r 2 r k 1 r k r k+1 r n 0 q 1 q 2 q k 1 q k q k+1 q n s 0 s 1 s 2 s k 1 s k s k+1 s n t 0 t 1 t 2 t k 1 t k t k+1 t n Suppose that the above table has been completed through column k, except for the value of q k, and we next wish to find q k and then fill in the values of r k+1, s k+1, and t k+1 in column k + 1. If we understand how this is done, then we understand how the extended Euclidean algorithm works in general. Now then, we know how to obtain q k and r k+1, since these are the quotient and remainder, respectively, obtained by dividing r k 1 by r k. Hence it follows that r k 1 = r k q k + r k+1, and we
22 104 Chapter 2 Number Theory use this relation and the values from columns k 1 and k to find s k+1 and t k+1. This is done as follows: r k+1 = r k 1 r k q k = (as k 1 + bt k 1 ) (as k + bt k )q k = a(s k 1 s k q k ) + b(t k 1 t k q k ) Thus, s k+1 = s k 1 s k q k and t k+1 = t k 1 t k q k. In summary, the sequence s 0, s 1, s 2,..., s n 1 is defined by s 0 = 0 s 1 = 1 s k+1 = s k 1 s k q k, for k = 1, 2,..., n 1 We say that the sequence is defined recursively by the initial values s 0 = 0 and s 1 = 1 and the recurrence relation (or recurrence formula) s k+1 = s k 1 s k q k. Similarly, the sequence t 0, t 1, t 2,..., t n 1 is defined recursively by the following initial values and recurrence relation: t 0 = 1 t 1 = 0 t k+1 = t k 1 t k q k, for k = 1, 2,..., n 1 In words, to obtain the value of s in a given column, multiple the value of s in the preceding column by the quotient in that column, and then subtract this product from the value of s in the column two columns before the given one. Similarly, to obtain the value of t in a given column, multiply the value of t in the preceding column by the quotient in that column, and then subtract this product from the value of t two columns before. Example 2.11: Use the extended Euclidean algorithm to find d = gcd(1407, 3255) and integers s and t such that d = 1407s t. Solution: First, we successively apply the division algorithm to obtain the first two rows of the table: So we see that d = gcd(1407, 3255) = 21. Now we must complete the third and fourth rows of the table using the initial values and recurrence relations for the s and t values. It is recommended that you complete the third row first, and then do the fourth row. Completing the third row, you should get For example, s = s 4 = s 2 s 3 q 3 = 2 (7)5 = 2 35 = 37. We then do the fourth row; see if you get
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