a D i v i s i b i l i t y statement which asserts that all numbers possess a certain property cannot be proved in this manner. The assertion, "Every prime number of the form An + 1 is a sum of two squares," is substantially more difficult to establish (see Lemma 2.13 in Section 2.1). Finally, it is presumed that you are familiar with the usual formulation of mathematical propositions. In particular, if A and B are two assertions, the following statements are logically equivalent they are just different ways of saying the same thing. A implies B. If A is true, then B is true. In order that A be true it is necessary that B be true. B is a necessary condition for A. A is a sufficient condition of B. If A implies B and B implies A, then one can say that B is a necessary and sufficient condition for A to hold. In general, we shall use letters of the roman alphabet, a, b,c,, m,n,,x,y,z to designate integers unless otherwise specified. We let Z denote the set {, -2, -1,0,1,2, } of all integers, Q the set of all rational numbers, U the set of all real numbers, and C the set of all complex numbers. 1.2 DIVISIBILITY Divisors, multiples, and prime and composite numbers are concepts that have been known and studied at least since the time of Euclid, about 350 b.c. The fundamental ideas are developed in this and the next section. Definition 1.1 An integer b is diuisible by an integer a, not zero, if there is an integer x such that b = ax, and we write a \b. In case b is not divisible by a, we write axb. Other language for the divisibility property a \ b is that a divides b, that a is a divisor of b, and that 6 is a multiple of a. If a\b and 0 < a < b, then a is called a proper divisor of b. It is understood that we never use 0 as the left member of the pair of integers in a \b. On the other hand, not only may 0 occur as the right member of the pair, but also in such instances we always have divisibility. Thus a 10 for every integer a not zero. The notation ak\\b is sometimes used to indicate that ak\b but ak+l/b.
1. 2 D i v i s i b i l i t y 5 Theorem 1.1 (1) a\b implies a \be for any integer c; (2) a\b and b\c imply a\c; (3) a\b and a\c imply a\(bx + cy) for any integers x and y; (4) a\b and b\a imply a = ±b; (5) a\b, a > 0, b > 0, imply a < b; (6) if m * 0, a\b implies and is implied by ma\mb. Proof The proofs of these results follow at once from the definition of divisibility. Property 3 admits an obvious extension to any finite set, thus: a\bua\b2,-,a\bn imply a 52 bjxj for any integers Xj, Property 2 can be extended similarly. To give a sample proof, consider item 3. Since a\b and a\c are given, this implies that there are integers r and s such that b = ar and c = as. Hence, bx + cy can be written as a(rx + sy), and this proves that a is a divisor of bx + cy. The next result is a formal statement of the outcome when any integer b is divided by any positive integer. For example, if 25 is divided by 7, the quotient is 3 and the remainder is 4. These numbers are related by the equality 25 = 7 3 + 4. Now we formulate this in the general case. Theorem 1.2 The division algorithm. Given any integers a and b, with a > 0, there exist unique integers q and r such that b=qa+r,0<r<a. If axb, then r satisfies the stronger inequalities 0 < r < a. Proof Consider the arithmetic progression -, b - 3 a, b - 2 a, b - a, b, b + a, b + 2 a, b + 3 a, - - - extending indefinitely in both directions. In this sequence, select the smallest non-negative member and denote it by r. Thus by definition r satisfies the inequalities of the theorem. But also r, being in the sequence, is of the form b - qa, and thus q is defined in terms of r. To prove the uniqueness of q and r, suppose there is another pair q{ and r, satisfying the same conditions. First we prove that r, = r. For if not, we may presume that r < r, so that 0 < r, r < a, and then we see
6 D i v i s i b i l i t y that r, - r = a(q - qx) and so a\(rx - r), a contradiction to Theorem 1.1, part 5. Hence r = r,, and also q = qv We have stated the theorem with the assumption a > 0. However, this hypothesis is not necessary, and we may formulate the theorem without it: given any integers a and b, with a * 0, there exist integers q and r such that b " qa + r, 0 < r < d. Theorem 1.2 is called the division algorithm. An algorithm is a mathe matical procedure or method to obtain a result. We have stated Theorem 1.2 in the form "there exist integers q and r," and this wording suggests that we have a so-called existence theorem rather than an algorithm. However, it may be observed that the proof does give a method for obtaining the integers q and r, because the infinite arithmetic progression,b - a,b,b + a, need be examined only in part to yield the smallest positive member r. In actual practice the quotient q and the remainder r are obtained by the arithmetic division of a into b. Remark on Calculation Given integers a and b, the values of q and r can be obtained in two steps by use of a hand-held calculator. As a simple example, if b = 963 and a = 428, the calculator gives the answer 2.25 if 428 is divided into 963. From this we know that the quotient q = 2. To get the remainder, we multiply 428 by 2, and subtract the result from 963 to obtain r = 107. In case b = 964 and a = 428 the calculator gives 2.2523364 as the answer when 428 is divided into 964. This answer is approximate, not exact; the exact answer is an infinite decimal. Nevertheless, the value of q is apparent, because q is the largest integer not exceeding 964/428; in this case q = 2. In symbols we write q = [964/428]. (In general, if x is a real number then [x] denotes the largest integer not exceeding x. That is, [x] is the unique integer such that [x] < x < [x] + 1. Further proper ties of the function [x] are discussed in Section 4.1.) The value of r can then also be determined, as r = b - qa = 964-2 428 = 108. Because the value of q was obtained by rounding down a decimal that the calculator may not have determined to sufficient precision, there may be a question as to whether the calculated value of q is correct. Assuming that the calculator performs integer arithmetic accurately, the proposed value of q is confirmed by checking that the proposed remainder b - qa = 108 lies in the interval 0 < r < a = 428. In case r alone is of interest, it would be tempting to note that 428 times 0.2523364 is 107.99997, and then round to the nearest integer. The method we have described, though longer, is more reliable, as it depends only on integer arithmetic.
2. 2 D i v i s i b i l i t y ' Definition 1.2 The integer a is a common divisor ofb and c in case a\b and a\c. Since there is only a finite number of divisors of any nonzero integer, there is only a finite number of common divisors ofb and c, except in the case b = c = 0. If at least one of b and c is not 0, the greatest among their common divisors is called the greatest common divisor of b and c and is denoted by (b,c). Similarly, we denote the greatest common divisor g of the integers bx, b2,-, b, not all zero, by (fe,, b2,- -,bn). Thus the greatest common divisor (b, c) is defined for every pair of integers b, c except b = 0, c = 0, and we note that (b, c) > 1. Theorem 1.3 If g is the greatest common divisor of b and c, then there exist integers x0 and y such that g = (b, c) = bx() + cy0. Another way to state this very fundamental result is that the greatest common divisor (abbreviated g.c.d.) of two integers b and c is expressible as a linear combination of b and c with integral multipliers x0 and y0. This assertion holds not just for two integers but for any finite collection, as we shall see in Theorem 1.5. Proof Consider the linear combinations bx + cy, where x and y range over all integers. This set of integers (bx + cy) includes positive and negative values, and also 0 by the choice x = y = 0. Choose jc and y0 so that bxt) + cy0 is the least positive integer / in the set; thus / = bxa + cy{). Next we prove that l\b and l\c. We establish the first of these, and the second follows by analogy. We give an indirect proof that l\b, that is, we assume l/b and obtain a contradiction. From l/b it follows that there exist integers q and r, by Theorem 1.2, such that b = Iq + r with 0 < r < I. Hence we have r = b - Iq = b - q(bx{) + cy0) = Ml - qx0) + c(-qyf)), and thus r is in the set {bx + cy}. This contradicts the fact that / is the least positive integer in the set {bx + cy}. Now since g is the greatest common divisor of b and c, we may write b = gb, c = gc, and I = bxn + cy{) = g(bx() + Cy0). Thus g\l, and so by part 5 of Theorem 1.1, we conclude that g < /. Now g < I is impossible, since g is the greatest common divisor, so g = / = bx() + cy0. Theorem 1.4 The greatest common divisor g ofb and c can be characterized in the following two ways: (1) It is the least positive value of bx + cy where x and y range over all integers; (2) it is the positive common divisor of b and c that is divisible by every common divisor.
c o D i v i s i b i l i t y Proof Part 1 follows from the proof of Theorem 1.3. To prove part 2, we observe that if d is any common divisor of b and c, then d\g by part 3 of Theorem 1.1. Moreover, there cannot be two distinct integers with prop erty 2, because of Theorem 1.1, part 4. If an integer d is expressible in the form d = bx + cy, then d is not necessarily the g.c.d. (b, c). However, it does follow from such an equation that {b, c) is a divisor of d. In particular, if bx + cy = 1 for some integers x and y, then (b,c) = 1. Theorem 1.5 Given any integers />,, b2,- -,bn not all zero, with greatest common divisor g, there exist integers xt,x2,---, xn such that n g = {bx,b2,---,b ) = bjxj. /-i Furthermore, g is the least positive value of the linear form E"=iV, where the y range over all integers; also g is the positive common divisor of b1,b2,- -,b that is divisible by every common divisor. Proof This result is a straightforward generalization of the preceding two theorems, and the proof is analogous without any complications arising in the passage from two integers to n integers. Theorem 1.6 For any positive integer m, (ma, mb) = m(a, b). Proof By Theorem 1.4 we have (ma,mb) = least positive value of max + mby = m {least positive value of ax + by} = m(a, b). Theorem 1.7 If d\a and d\b and d > 0, then I a b \ 1 If (a, b) = g, then a b -, - 1 - J. g 8
1. 2 D i v i s i b i l i t y 9 Proof The second assertion is the special case of the first obtained by using the greatest common divisor g of a and b in the role of d. The first assertion in turn is a direct consequence of Theorem 1.6 obtained by replacing m, a, b in that theorem by d, a/d, b/d respectively. Theorem 1.8 // (a, m) = (b,m) = 1, then (ab, m) = 1. Proof By Theorem 1.3 there exist integers x0, y0, xx, y, such that 1 = ax0 + myq = bx{ + myx. Thus we may write (axqxbxx) = (1 - my0) (1 - myx) = 1 - my2 where y2 is defined by the equation y2 = yn + y) - my0y{. From the equation abx0xt + my2 = 1 we note, by part 3 of Theorem 1.1, that any common divisor of ab and m is a divisor of 1, and hence (ab, m) = 1. Definition 1.3 We say that a and b are relatively prime in case (a, b) = 1, and that ax, a2,-,an are relatively prime in case (a{, a2,-, an) = 1. We say that av a2,-, an are relatively prime in pairs in case (a,, a ) = 1 for all i = \,2,-,n and j = 1,2,, n with i + j. The fact that (a, b) = 1 is sometimes expressed by saying that a and b are coprime, or by saying that a is prime to b. Theorem 1.9 For any integer x, (a, b) = (b, a) = (a, - b) = (a,b + ax). Proof Denote (a, b) by d and (a, b + ax) by g. It is clear that (b, a) = (a,-b) = d. By Theorem 1.3, we know that there exist integers x{) and yn such that d = ax0 + by0. Then we can write d = a(x{] - xy0) + (b + ax)y{]. It follows that the greatest common divisor of a and b + ax is a divisor of d, that is, g\d. Now we can also prove that d\g by the following argument. Since d\a and d\b, we see that d\(b + ax) by Theorem 1.1, part 3. And from Theorem 1.4, part 2, we know that every common divisor of a and b + ax is a divisor of their g.c.d., that is, a divisor of g. Hence, d\g. From d\g and g\d, we conclude that d = ±g by Theorem 1.1, part 4. However, d and g are both positive by definition, so d = g.