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coin exchange problem
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Suppose our monetary system used only coins of the§values 5 and 8. Which bills could we exchange using§only these coins? How many sums are impossible to§get? More generally, let a,b,...,n be natural numbers§(the values of the coins). The set of all natural§combinations of these numbers is a numerical§semigroup. Our goal is to determine the structure of§this semigroup.§This question, known as the coin exchange problem or§the Frobenius problem, captured the imagination of§many mathematicians ever since it was first proposed§by Sylvester in 1884. §If we use only 2 kinds of coins, the answer is quite§simple, but the case of 3 or more turns out to be§much more complicated and interesting.§This book presents the problem in a clear and vivid§way, and shows an ingenious simple method to solve§it. The main parameters of the problem, including the§Frobenius number and genus, are explained and§discussed. Finally, the book presents the surprising§link between numerical semigroups and cyclotomic§polynomials. The subject, as well as the clever§solution and clear presentation, make this book a§delight to anyone interested in mathematics,§professionals and amateurs alike.
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