The Art and Craft of Problem Solving

244 CHAPTER 7 NUMBER THEORY so we can conclude that n > m. That certainly helps. Let's consider one example, with, say, m = 1 ...

7.4 DIOPHANTINE EQUATIONS 245 • Now assume that r = s. Then (7) becomes (a^2 +b^2 )^3 = 22 r a^4 b^4. (8) It is true that both s ...

246 CHAPTER 7 NUMBER THEORY infinitely many solutions. If m = 2(, and n = m + 1, then it is easy to check that x = y = 2 t indee ...

7.5 MISCELLANEOUS INSTRUCTIVE EXAMPLES 247 (c) You discovered that if (u, v) is a solution to .xl -2y^2 = I, then so is (3u+4v,2 ...

248 CHAPTER 7 NUMBER THEORY we are ready to make a tentative guess that f(7 + 3) will also be a multiple of 3. This seems almost ...

7.5 MISCELLANEOUS INSTRUCTIVE EXAMPLES 249 If You Can Count It, It's an Integer Example 7.5.2 Let kEN. Show that the product of ...

250 CHAPTER 7 NUMBER THEORY Sums of Two Squares We shall end the chapter with an exploration of the diophantine equation Y-+i =n ...

7.5 MISCELLANEOUS INSTRUCTIVE EXAMPLES 251 This is promising. Here is an outline of a possible "algorithm" for solving x^2 + y^2 ...

252 CHAPTER 7 NUMBER THEORY The small ticks are the integers from 0 to 29, while the long ticks are placed at the locations y'p, ...

7.5 MISCELLANEOUS INSTRUCTIVE EXAMPLES 253 This concludes our exploration of the equation x^2 + y^2 = p. We have succeeded in pr ...

254 CHAPTER 7 NUMBER THEORY 7.5.25 Does there exist an x such that J.l(x) = J.l(x+ 1) = J.l(x+2) = ... = J.l(x + 1996)? 7.5.26 ( ...

7.5 MISCELLANEOUS INSTRUCTIVE EXAMPLES 255 the smallest positive integer such that Sn = 1. For ex­ ample, the 4th roots of unity ...

Chapter 8 Geometry for Americans 8.1 Three "Easy" Problems 256 We call this chapter "Geometry for Americans" instead of "Geometr ...

8.1 THREE "EASY" PROBLEMS 257 with the basic facts, we will explore a few more advanced ideas, many of them con­ nected by the d ...

258 CHAPTER 8 GEOMETRY FOR AMERICANS 8.1.2 The Angle Bisector Theorem. Let ABC be a triangle and let the angle bisector of LA in ...

8.2 SURVIVAL GEOMETRY I 259 Please do not read passively. The more of these simple problems that you do now, the better off you ...

260 CHAPTER 8 GEOMETRY FOR AMERICANS but in practice we omit the f':.. symbol unless the context is ambiguous. Fact 8.2. 1 Congr ...

8.2 SURVIVAL GEOMETRY I 261 There is one critical fact about parallel lines that leads to many interesting lemmas.^2 Fact 8.2.6 ...

262 CHAPTER 8 GEOMETRY FOR AMERICANS Now we search for equal angles. Since the diagonal BC is a transversal intersecting paralle ...

8.2 SURVIVAL GEOMETRY I 263 The next example presents a simple result that can easily be generalized, using powerful methods of ...