Given integer-valued relatively prime `coins' a1; a2; : ak, the Frobenius number is the largest integer n such that the linear diophantine equation a1m1 + a2m2 +::: + akmk = n has no solution in non-negative integers m1;m2; : mk. We denote by g(a1; : ak) the largest integer value not attainable by this coin system. That is to say that any integer x greater than the Frobenius number g(a1; : ak) has a representation x = a1x1 + a2x2 +::: + akxk by a1; a2; : ak for some non-negative integers x1; x2; : xk. We say x is representable by a1; a2; : ak. While it is obvious that there are representable positive integers and non-representable positive integers, must there be a largest non-representable integer? Maybe there are indefinitely large non-representable integers for a1; a2; : ak with gcd (a1; a2; : ak) = 1. This notion of whether or not the Frobenius number is well-defined will be the first bit of mathematics we look at in this paper. Proposition 1.1. The Frobenius number g(a1; : ak) is well-defined. Proof. Given a1; a2; : ak with gcd (a1; a2; : ak) = 1, the extended Euclidean algorithm gives that there exist m1;m2; : mk 2 Z such that...