The multidimensional Frobenius problem

The Parametric Frobenius Problem

Given relatively prime positive integers a1, . . . , an, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the ai. We examine the parametric version of this problem: given ai = ai(t) as functions of t, compute the Frobenius number as a function of t. A function f : Z+ → Z is a quasi-polynomial if there exists a period m and polynomials f0...

The Diophantine Frobenius Problem

The Multi-Dimensional Frobenius Problem

Consider the problem of determining maximal vectors g such that the Diophantine system Mx = g has no solution. We provide a variety of results to this end: conditions for the existence of g, conditions for the uniqueness of g, bounds on g, determining g explicitly in several important special cases, constructions for g, and a reduction for M .

The Frobenius Coin Problem Upper Bounds on The Frobenius Number

In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal t...

The Frobenius problem on lattices

It is widely known that if p and q are relatively prime positive integers then (a) the set of linear combinations of p and q with nonnegative integer coefficients includes all integers greater than pq − p − q, (b) exactly half the integers between 0 and pq − p − q belong to this set and (c) an integer m belongs to this set if and only if pq−p−q−m does not. A multidimensional version of statemen...