Old and New Unsolved Problems in Plane Geometry and Number Theory

Cambridge University Press, 1991 - Mathematics - 333 pages
Victor Klee and Stan Wagon discuss some of the unsolved problems in number theory and geometry, many of which can be understood by readers with a very modest mathematical background. The presentation is organized around 24 central problems, many of which are accompanied by other, related problems. The authors place each problem in its historical and mathematical context, and the discussion is at the level of undergraduate mathematics. Each problem section is presented in two parts. The first gives an elementary overview discussing the history and both the solved and unsolved variants of the problem. The second part contains more details, including a few proofs of related results, a wider and deeper survey of what is known about the problem and its relatives, and a large collection of references. Both parts contain exercises, with solutions. The book is aimed at both teachers and students of mathematics who want to know more about famous unsolved problems.

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Victor Klee and Stan Wagon's book is out-of-date with respect to an Egyptian Fractions chapter, citing the conversion of 5/121 by a greedy algorithm solution and an improved but un-concise solution ( published by Bleicher):
5/121 = 1/35 + 1/1225 + 1/3477 + 1/7081 + 1/11737
on page 206. In 2002, 2005, 2008 and 2010 an improved RMP 2/n table red number method was broken and published on-line in the context of the EMLR, RMP 36 and RMP 37 such that:
5/121 = 1/11*(5/11)
with 5/11 *(3/3) = 15/33 = (11 + 3 + 1)/33 = 1/3 + 1/11 + 1/33
reaching
5/121 = 1/11*(1/3 + 1/11 + 1/33) = 1/33 + 1/121 + 1/363
as Ahmes,the RMP scribe, in 1650 BCE solved
2/95 by 1/5*(2/19)*(12/12) = (19 + 3 + 2)/1140 = 1/60 + 1/380 + 1/570
Wikipedia has posted the 2/95 solution on its Rhind Mathematical Papyrus page
http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus
Stan Wagon also discussed Fibonacci's 1202 AD Liber Abaci and the conversion of 4/23 by the un-concise modern solution on page 177:
4/23 = 1/7 + 1/33 + 1/1329 + 1/23536559
rather than following Fibonacci's solution
(4/23 - 1/6) = (24 -23)/138
recorded in one of three unit fraction notations.
Ahmes' arithmetic discussed on
http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
would have solved 4/23 with LCM 6 finding
4/23*(6/6) = 24/138 = (24 + 1)/138 = 1/6 + 1/138
In RMP 36 and RMP 37 Ahmes used a subtraction method that would have reported
(4/23 - 1/6) = 1/138 by recording
4/23 = 1/6 + 1/138
Hence, Greeks, Arabs and medieval scribes like Fibonacci copied Egyptian number theory, dated to 2050 BCE, to record exact rational number' unit fraction mathematics.
Best Regards to the ancient scribes and modern math historians. Modern books on ancient math and modern unsolved problems need to be updated from time to time.
Milo Gardner

Contents

 Forming Convex Polygons 2528 95102 95 Tiling the Plane 364111119 111 Squaring the Circle 5053128131 128 Fixed Points 6670145150 145 Introduction 167 A Perfect Box 173174203205 173 The Riemann Hypothesis 182185 215220 182 The 3n + 1 Problem 191194 225229 191
 Patterns in Pi 240242251254 240 Summing Reciprocals of Powers 248250261264 248 References 265 BvoDimensional Geometry 269 Number Theory 300 Glossary 315 Subject Index 331 Copyright

 References 234

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Page 332 - The first expresses the ratio of the circumference of a circle to the diameter; (Art. 23.) the second, the ratio of the area of a circle to the square of the diameter (Art.