Abstract: We find the matrix representation of the set Δ(d3), where d3=(d1,d2,d3), of integers that are unrepresentable by d1,d2,d3 and develop a diagrammatic procedure for calculating the generating function Φ(d3;z) for the set Δ(d3). We find the Frobenius number F(d3), the genus G(d3), and the Hilbert series H(d3;z) of a graded subring for nonsymmetric and symmetric semigroups \(\mathsf{S}(\mathbf {d}^{3})\) and enhance the lower bounds of F(d3) for symmetric and nonsymmetric semigroups.

Abstract: (2007). Quadratic Residues and the Frobenius Coin Problem. Mathematics Magazine: Vol. 80, No. 1, pp. 64-67.

Abstract: The classical Frobenius problem is to compute the largest number g not representable as a non-negative integer linear combination of non-negative integers x_1, x_2, ..., x_k, where gcd(x_1, x_2, ..., x_k) = 1. In this paper we consider generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for an analogue of g, depending on the particular measure chosen.

Abstract: Given relatively prime positive integers a1, . . . , ak, let S denote the set of all linear combinations of a1, . . . , ak with nonnegative integral coefficients. The Frobenius problem is to determine the largest integer g(S) which is not representable as such a linear combination. A related question is to determine the set B(S) of integers x that are representable as differences x = s1 − a1 = . . . = sk − ak for some si ∈ S. The construction B(S) can be iterated to obtain a chain of numerical semigroups. We compare this chain to the one obtained by iterating the Lipman semigroup construction. In particular, we consider these chains for generalized Suzuki semigroups.

Abstract: Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.

Abstract: An algorithm generating uniform non-adaptive solutions for two different versions of general counterfeit coin problem has been given.The general counterfeit coin problem is the problem of finding a method to determine the set of k counterfeit coins out of n coins with the same appearance(where k and n are positive integers given as inputs),using an unsealed balance.Two different versions of the problem are studied - one requires to identify the comparative weight of the counterfeit coins,and another does not.By extending the notion of"Dyson sets",the models of non-adaptive solutions to both versions of the problem are given.With these models,the existence of uniform non-adaptive solutions has been proven with an algorithm generating them presented.

Abstract: For the well known Frobenius problem, we present a new geometric approach, based on the use of the $n$-dimensional lattice $\mathbb{Z}^n$, where $n$ is the number of generators. Within this approach we are able to study the cases of two and three generators. The main feature of our geometric representation is that we can nicely visualize the set of {\em gaps}, i.e., the non-representable positive integers. In the case of two generators, we give a description of the set of gaps. Moreover, for any positive integer, $m$, we derive a simple expression for the denumerant $d(m;a,b)$. We show that we can use the $2$-dimensional lattice associated to the set of generators $\{ a,b\}$ to study the Frobenius problem with generators $\{ a,b,c\}$. In particular, we give, as for two generators, a graphical representation of the set of gaps. For a large set of possible values of $c$, this representation allows us to simplify the computation of the Frobenius number and compute the number of gaps.

Abstract: The Frobenius number g(A) of a set A = (a1,a2,... ,an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n ! 1 and obtain a fast algorithm for computing it. The algorithm appears to run in average time that is softly quadratic and we prove that this is the case for almost all of the steps. In practice, the algorithm is very fast: examples with n = 4 and the numbers in A having 100 digits take under one second. The running time increases with dimension and we can succeed up to n = 11. We use the geometric structure of a fundamental domain D, having a1 points, related to a lattice constructed from A. The domain encodes information needed to find the Frobenius number. One cannot generally store all of D, but it is possible to encode its shape by a small set of vectors and that is su! cient to get g(A). The ideas of our algorithm connect the Frobenius problem to methods in integer linear programming and computational algebra. A variation of these ideas works when n = 3, where D has much more structure. An integer programming method of Eisenbrand and Rote can be used to design an algorithm for g(a1,a2,a3) that takes soft linear time in the worst case. We present a variation of the method that we have implemented and that can be viewed in two ways: as having provably soft linear time, but not guaranteed to work (we know of no instances in which it fails), or as an algorithm that always works and appears to be softly linear in the worst case. At the other end, when n is beyond 11 we can get upper and lower bounds that are far better than what was known. Our ideas lead to new theoretical results. The first is a simple characterization of triples A such that the ratio of the number of nonrepresentable positive integers to g(A)+1 is exactly 1/2: the condition holds i" some member of A is representable in terms of the other two, reduced by their gcd. We also obtain new Frobenius formulas. Here’s a quadratic formula that is easy to discover experimentally: For a " 16, g(a,a + 1,a + 4,a + 9) = 1 (a 2 + cka) ! dk, where k is the mod-9 residue of a and ck and dk are defined,