To many laymen, mathematicians appear to be problem solvers, people who do "hard sums". Even inside the profession we dassify ourselves as either theorists or problem solvers. Mathematics is kept alive, much more than by the activities of either dass, by the appearance of a succession of unsolved problems, both from within mathematics-itself and from the in creasing number of disciplines where it is applied. Mathematics often owes more to those who ask questions than to those who answer them. The solu tion of a problem may stifte interest in the area around it. But "Fermat's Last Theorem", because it is not yet a theorem, has generated a great deal of "good" mathematics, whether goodness is judged by beauty, by depth or byapplicability. To pose good unsolved problems is a difficult art. The balance between triviality and hopeless unsolvability is delicate. There are many simply stated problems which experts tell us are unlikely to be solved in the next generation. But we have seen the Four Color Conjecture settled, even ifwe don't live long enough to leam the status of the Riemann and Goldbach hypotheses, of twin primes or Mersenne primes, or of odd perfeet numbers. On the other hand, "unsolved" problems may not be unsolved at all, or may be much more tractable than was at first thought. Mathematics is kept alive by the appearance of new unsolved problems, problems posed from within mathematics itself, and also from the increasing number of disciplines where mathematics is applied. This book provides a steady supply of easily understood, if not easily solved, problems which can be considered in varying depths by mathematicians at all levels of mathematical maturity. For this new edition, the author has included new problems on symmetric and asymmetric primes, sums of higher powers, Diophantine m-tuples, and Conway's RATS and palindromes. The author has also included a useful new feature at the end of several of the sections: lists of references to OEIS, Neil Sloane's Online Encyclopedia of Integer Sequences. About the First Edition: "...many talented young mathematicians will write their first papers starting out from problems found in this book." - András Sárközi, MathSciNet. Front Matter....Pages i-xviii Introduction....Pages 1-2 Prime Numbers....Pages 3-24 Divisibility....Pages 25-57 Additive Number Theory....Pages 58-78 Some Diophantine Equations....Pages 79-109 Sequences of Integers....Pages 110-131 None of the Above....Pages 132-148 Back Matter....Pages 149-161 Number theory has fascinated both the amateur and the professional for a longer time than any other branch of mathematics, so that much of it is now of considerable technical difficulty. Second Edition Sold 2241 Copies In N.a. And 1600 Row. New Edition Contains 50 Percent New Material.
