Abstract: During the early part of the last century, F. G. Frobenius raised, in his lectures, the following problem (called the Diophantine Frobenius Problem FP): given relatively prime positive integers a1, . . . , an, find the largest natural number (called the Frobenius number and denoted by g(a1, . . . , an)) that is not representable as a nonnegative integer combination of a1, . . . , an. It turned out that the knowledge of g(a1, . . . , an) has been extremely useful to investigate many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such ‘methods, ideas, viewpoints, and applications’ for as wide an audience as possible. This book aims to provide a comprehensive exposition of what is known today on FP.

Keywords: Frobenius number, denumerants, integer representation, modular change problem, postage stamp problem, semigroups, gaps