This book is a country walk through the magical world of numbers. Most people will have recognised some of the fascinating patterns exhibited by many numbers; some of these indicate a deep and complex structure which is revealed in this book in a way that is accessible to all, from amateur to expert. The author focusses on powers of numbers, which have been studied from the time of Pythagoras until the present day, with the proof of Fermat's Last Theorem. Indeed some of the results described by the author were only established quite recently, giving the book a very contemporary flavour. In sum, this will make a stimulating resource for teachers of mathematics, and will be as well a fund of knowledge for amateurs.
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American Mathematical Monthly arithmetic progression base Catalan numbers Chapter column of Table conjecture consecutive cubes consecutive integers consecutive squares consecutive terms coprime corresponding cubic deduce Determine difference digits diophantine equation elements entries equation x2 example fcth Fibonacci sequence fifth powers fn+i fourth powers given greatest common divisor Hence induction Journal of Recreational L. E. Dickson magic square Mathematical Gazette Mathematical Monthly 53 Mathematics Magazine multiple natural numbers nonzero Note number equal number theory obtain odd number Pell's equation perfect square polynomial positive integer possible power sums prime Prove pythagorean triples quadratic reader may wish Recreational Mathematics result root running total satisfies second-order recursion sets of numbers Show side smallest numbers square sums sum of cubes sum of three Suppose three squares triangle triangular numbers triples whose smallest values vector Verify whence written xn+i yields