THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS

If S = {a(1), a(2),..., a(n)} is a set of relatively prime positive integers, it is well known that any sufficiently large integer can be expressed as a nonnegative integral combination of the elements of S. The Frobenius problem consists of determining how large is sufficiently large. That is, find the smallest possible integer L(a(1), a(2),..., a(n)) with the property that any number greater than or equal to it can be expressed as a nonnegative integral combination of a(1), a(2),..., a(n) We review two classical approaches to the problem, and offer a third one. We then apply this latter approach to obtain simplified proofs for several known results and to obtain some new results.