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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