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Wednesday, September 7, 2005

Reading List: Unsolved Problems in Number Theory

Guy, Richard K. Unsolved Problems in Number Theory. 3rd ed. New York: Springer, 2004. ISBN 0-387-20860-7.
Your hard-working and overheated CPU chip does not want you to buy this book! Collected here are hundreds of thorny problems, puzzles, and conjectures, many of which, even if you lack the cerebral horsepower to tackle a formal proof, are candidates for computational searches for solutions or counterexamples (and, indeed, a substantial number of problems posed in the first and second editions have been so resolved, some with quite modest computation by today's standards). In the 18th century, Leonhard Euler conjectured that there was no nontrivial solution to the equation:
a5 + b5 + c5 + d5 = e5
The problem remained open until 1966 when Lander and Parkin found the counterexample:
275 + 845 + 1105 + 1335 = 1445
Does the equation:
a6 + b6 + c6 + d6 + e6 = f6
have a nontrivial integer solution? Ladies and gentlemen, start your (analytical) engines! (Problem D1.) There are a large collection of mathematical curiosities here, including a series which grows so slowly it is proportional to the inverse of the Ackermann function (E20), and a conjecture (E16) regarding the esoteric equation "3x+1" about which Paul Erdös said, "Mathematics may not be ready for such problems." The 196 palindrome problem which caused me to burn up three years of computer time some fifteen years ago closes the book (F32). Many (but not all) of the problems to which computer attacks are applicable indicate the status of searches as of 2003, giving you some idea what you're getting into should you be inclined to launch your own.

For a book devoted to one of the most finicky topics in pure mathematics, there are a dismaying number of typographical errors, and not just in the descriptive text. Even some of the LaTeX macros used to typeset the book are bungled, with "@"-form \index entries appearing explicitly in the text. Many of the errors would have been caught by a spelling checker, and there are a number of rather obvious typesetting errors in equations. As the book contains an abundance of "magic numbers" related to the various problems which may figure in computer searches, I would make a point to independently confirm their accuracy before launching any extensive computing project.

Posted at September 7, 2005 22:14