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:
a^{5} + b^{5} + c^{5}
+ d^{5} = e^{5}
The problem remained open until 1966 when Lander and Parkin
found the counterexample:
27^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}
Does the equation:
a^{6} + b^{6} + c^{6}
+ d^{6} + e^{6}
= f^{6}
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.
September 2005