Given relatively prime positive integers a_1,...,a_n, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the a_i. We examine the parametric version of this problem: given a_i=a_i(t) as functions of t, compute the Frobenius number as a function of t. A function f is a quasi-polynomial if there exists a period m and polynomials f_0,...,f_{m-1} such that f(t)=f_{t mod m}(t) for all positive integers t. We conjecture that, if the a_i(t) are polynomials (or quasi-polynomials) in t, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large t. We prove this in the case where the a_i(t) are linear functions, and also prove it in the case where n (the number of generators) is at most 3.
The Frobenius number of relatively prime positive integers $a_1, \ldots, a_n$ is the largest integer that is not a nononegative integer combination of the $a_i.$ Given positive integers $a_1, \ldots, a_n$ with $n \ge 2,$ the set of multiples of $\gcd(a_1, \ldots, a_n)$ which have less than $m$ distinct representations as a nonnegative integer combination of the $a_i$ is bounded above, so we define $f_{m, \ell}(a_1, \ldots, a_n)$ to be the $\ell^{\text{th}}$ largest multiple of $\gcd(a_1, \ldots, a_n)$ with less than $m$ distinct representations (which generalizes the Frobenius number) and $g_m…
Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\mbox{ Sg}_{\geq k}(A)$ of all vectors $b \in \mbox{ Sg}(A)$ such that $P_A(b) \cap {\mathbb Z}^n $ has at least $k$ solutions. …
We study the Frobenius problem: given relatively prime positive integers $a_1,...,a_d$, find the largest value of t (the Frobenius number) such that $\sum_{k=1}^d m_k a_k = t$ has no solution in nonnegative integers $m_1,...,m_d$. Based on empirical data, we conjecture that except for some special cases the Frobenius number can be bounded from above by $\sqrt{a_1 a_2 a_3}^{5/4} - a_1 - a_2 - a_3$.
Let $A$ be a finite subset of $\mathbb{Z}^n$, which generates $\mathbb{Z}^n$ additively. We provide a precise description of the $N$-fold sumsets $NA$ for $N$ sufficiently large, with some explicit bounds on "sufficiently large."
We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d. We introduce a method to compute upper bounds for g(a_1,a_2,a_3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.
Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. The condition that $gcd(a_1,...,a_N)=1$ implies that such number exists. The general problem of determining the Frobenius number given $N$ and $a_1,...,a_N$ is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relati…
The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However,…
Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit formula for the polynomial part of p_A.
Given positive integers $a_1,...,a_n$ with $\gcd(a_1,...,a_n) = 1$, we call an integer t representable if there exist nonnegative integers $m_1,...,m_n$ such that $t = m_1 a_1 + ... + m_n a_n$. 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 $g(a_1,...,a_n)$. We extend this problem to asking for the smallest integer $g_k(a_1,...,a_d)$ beyond which every integer is represented more than k times. We concentrate on the case d=2 and prove statements about $g_k(a,b)$…
Conjecturing formulas and other symbolic relations occurs frequently in number theory and combinatorics. If we could automate conjecturing, we could benefit not only from speeding up, but also from finding conjectures previously out of our grasp. Grammatical evolution, a genetic programming technique, can be used for automated conjecturing in mathematics. Concretely, this work describes how one can interpret the Frobenius problem as a symbolic regression problem, and then apply grammatical evolution to it. In this manner, a few formulas for Frobenius numbers of specific quadruples were found a…
We study the number of lattice points in integer dilates of the rational polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$, where $a_1,...,a_n$ are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers $a_1,...,a_n$, find the largest value of t (the Frobenius number) such that $m_1 a_1 + ... + m_n a_n = t$ has no solution in positive integers $m_1,...,m_n$. This is equivalent to the problem of finding the largest dilate tP such that the facet $\sum_{k=1}^n x_k a_k = t$ contains no …
A Grobner basis-based algorithm for solving the Frobenius Instance Problem is presented, and this leads to an algorithm for solving the Frobenius Problem that can handle numbers with thousands of digits. Connections to irreducible decompositions and Hilbert functions are also presented.
In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating 0-1 vectors with as many component as the Frobenius number.
Let S be a map from a language L to the integers satisfying S(vw)=S(v)+S(w) for all words v,w from the language. The classical Frobenius problem asks whether the complement of S(L) in the natural numbers will be infinite or finite, and in the latter case the value of the largest element in this complement. This is also known as the 'coin'-problem, and L is the full language consisting of all words over a finite alphabet. We solve the Frobenius problem for the golden mean language, any Sturmian language and the Thue-Morse language. We also consider two-dimensional embeddings.
Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors …
We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_0$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_s$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_0, \ g_1, \ g_2, ...$ exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \ g_{\binom{k+2}{k-1}}, ...$ form an arithmetic progression, a…
In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even i…
The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an adiabatic quantum algorithm which solves it, using the so-called Apéry set of a numerical semigroup, via a translation into a QUBO problem. The algorithm has been specifically designed to run in a D-Wave 2X machine.
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.
Let $f_1(n), \ldots, f_k(n)$ be polynomial functions of $n$. For fixed $n\in\mathbb{N}$, let $S_n\subseteq \mathbb{N}$ be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)$. As $n$ varies, we show that many invariants of $S_n$ are eventually quasi-polynomial in $n$, such as the Frobenius number, the type, the genus, and the size of the $Δ$-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups $S_n\subseteq \mathbb{N}^m$ generated be vectors whose coo…
We study the following problem, first introduced by Dekking. Consider an infinite word x over an alphabet {0,1,...,k-1} and a semigroup homomorphism S:{0,1,...,k-1}* -> N. Let L_x denote the set of factors of x. What conditions on S and the abelian complexity of x guarantee that S(L_x) contains all but finitely many elements of N? We examine this question for some specific infinite words x having different abelian complexity functions.
Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Let ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. The largest nonrepresentable integer $\max{\rm NR}$, the number of nonrepresentable positive integers $\sum_{n\in{\rm NR}}1$ and the sum of nonrepresentable positive integers $\sum_{n\in{\rm NR}}n$ have been widely studied for a long time as related to the famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas for the weighted sum $\sum_{n\in{\rm NR}}λ^{n}n^μ$,…
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 m…
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have multiple inputs and outputs. If such a construction is possible, then determining whether the studied semigroup is symmetric, or finding explicit forms of its Apéry set and Hilbert series, is reduced to straightforward computations assisted by a MAPLE program we made available on arXiv. This approach applies to many of the cases considered in literature, inc…
Let consider $n$ natural numbers $a\_1 ,\ldots , a\_{n} $. Let $S$ be the numerical semigroup generated by $a\_1 ,\ldots , a\_{n} $. Set $A=K[t^{a\_1}, \ldots , t^{a\_n}]=K[{x\_1}, \ldots , {x\_n}]/I$. The aim of this paper is: \begin{enumerate}\item Give an effective pseudo-polynomial algorithm on $a\_1$, which computes The Ap{é}ry set and the Frobenius number of $S$. As a consequence it also solves in pseudo-polynomial time the integer knapsack problem : given a natural integer b, b belongs to $S$?\item The \gbb of $I$ for the reverse lexicographic order to $x\_n,\ldots ,x\_1$, without using…
