An extension of the Frobenius coin - exchange problem

Given a set of positive integers A = {a1, . . . , ad} with gcd(a1, . . . , ad) = 1, we call an integer n representable if there exist nonnegative integers m1, . . . ,md such that n = m1a1 + · · ·+mdad . In this paper, we discuss the linear diophantine problem of Frobenius: namely, find the largest integer which is not representable. We call this largest integer the Frobenius number g(a1, . . . , ad). One fact which makes this problem attractive is that it can be easily described, for example, in terms of coins of denominations a1, . . . , ad; the Frobenius number is the largest amount of money which cannot be formed using these coins. The following “folklore” theorem has long been known (probably at least since Sylvester [9]).