An extension of the Frobenius coin - exchange problem 1 Matthias Beck and Sinai Robins 2 Dedicated to the memory of

Dedekind sums : a combinatorial - geometric viewpoint Matthias Beck and Sinai Robins

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational poly-topes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer...

Dedekind sums: a combinatorial-geometric viewpoint

How to Change Coins, M&m’s, or Chicken Nuggets: the Linear Diophantine Problem of Frobenius

Let’s imagine that we introduce a new coin system. Instead of using pennies, nickels, dimes, and quarters, let’s say we agree on using 4-cent, 7-cent, 9-cent, and 34-cent coins. The reader might point out the following flaw of this new system: certain amounts cannot be exchanged, for example, 1, 2, or 5 cents. On the other hand, this deficiency makes our new coin system more interesting than th...

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, . . . ...

Frobenius Problem and the Covering Radius of a Lattice

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as P N i=1 aixi where x1, ..., xN are non-negative integers. The condition that gcd(a1 , ..., aN ) = 1 implies that such number exists. The general problem of determining the Frobenius number given N and a1, ..., aN is NP-har...