Matthias Beck, David Einstein, Shelemyahu Zacks
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 $\sqrt{a_1 a_2 a_3}^{5/4} - a_1 - a_2 - a_3$.
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In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, Simsek et als(see [13]) have given some identities of q-analogue of Frobenius-Euler polynomials related to q-Bernstein polynomials. From the methods of our paper, we can also derive the results and identities of Simsek et als (cf.[13]).
A Frobenius problem suggested by prime $k$-tuplets (extended version)
本文研究由素数k元组（尤其是素数三元组和四元组）提出的Frobenius问题，并分析相关数值半群的性质。
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."
A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)
本文提出一种新的组合方法来解决弗罗贝尼乌斯问题，通过将问题转化为更易处理的优化问题，推导出弗罗贝尼乌斯数、赛尔伯格数和赛尔伯格和的公式。
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,…
The greatest integer that does not belong to $S$ is the Frobenius number of $S$ and denoted by $F(S)$. To solve the Frobenius problem means the study to find $F(S)$. The Frobenius problem have treated steadily for a long time. In this paper, We will introduce the Frobenius problem for generalized Thabit numerical semigroups, which is motivated by the Frobenius problem for Thabit numerical semigroups.
The Frobenius Problem for the Proth Numbers
本文定义了Proth数值半群$P_{k}(n)$，给出了其嵌入维数公式，并通过推导Frobenius数的闭式公式解决了该半群的Frobenius问题，同时证明了其满足Wilf性质。
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 …
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.
The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$ and finding the Frobenius number is called the Frobenius problem. In this paper, we introduce the Frobenius problem for numerical semigroups generated by Thabit number base b and Thabit number of the second kind base b which are motivated by the Frobenius problem for Thabit numerical semigroups. Also, we introduce the Frobenius problem for numerical semigroups generated by Cunningham number and Fermat number base $b$
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 $g_k(a,b)$…
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 …
The Frobenius problem for shifted square sequences
本文解决了以4的倍数开头的移位平方序列的Frobenius问题，即找到不属于该数值半群的最大整数。
The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number
本文研究由斐波那契数列平移一个斐波那契数所得序列的扩展 Frobenius 问题，并证明相应数值半群族满足 Wilf 猜想。
We extend the famous diophantine Frobenius problem to the case of polynomials over a field $k$. Similar to the classical problem, we show that the $n=2$ case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over $\mathbb{Z}$ to $k[t]$ and give an algorithm to solve the Frobenius problem for polynomials over a field $k$ of sufficiently large size.
This paper provides a formula for the minimal relations and the Frobenius number of a numerical semigroup minimally generated by three pairwise coprime positive integers.
The classical Frobenius problem is to find the largest integer that cannot be written as a linear combination of a given set of positive, coprime integers using nonnegative integer coefficients. Prior work has generalized the classical Frobenius problem from integers to Frobenius problems in other rings. This paper explores Frobenius problems in various rings of (upper) triangular $2 \times 2$ matrices with constant diagonal.
This paper presents a new approach to determine the number of solutions of three variable Frobenius related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three variable Frobenius related problems. The first method is used to reduce the equation of order three from a three variable Frobenius related problem to be a system of equations with two fixed…
The Frobenius problem for Numerical Semigroups generated by binomial coefficients
本文研究由二项式系数生成的数值半群的Frobenius问题，给出其求解结果，同时得到二项式系数的非平凡恒等式，并将结论与整数的$(s,s+1,s+p)$-core分拆理论建立关联。
Suppose that m is a positive integer, not a perfect square. We present a formula solution to the 2-variable Frobenius problem in Z[\sqrt m] of the "first kind" ([3]).
In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.
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.
We give upper and lower bounds for the largest integer not representable as positive linear combination of three given integers, disproving an upper bound conjectured by Beck, Einstein and Zacks.
In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special polynomials
In this paper, we consider the new q-extension of Frobenius-Euler numbers and polynomials and we derive some interesting identities from the orthogonality type properties for the new q-extension of Frobenius-Euler polynomials. Finally we suggest one open question related to orthogonality of polynomials.
The purpose of this paper is to give a complete derivation of the limiting distribution of large Frobenius numbers outlined in earlier work of J. Bourgain and Ya. Sinai and fill some gaps formulated there as hypotheses.
It turns out that all instances of the diophantine Frobenius problem for three coprime a_i have a common geometric structure which is independent of arithmetic coincidences among the a_i. By exploiting this structure we easily obtain Johnson's formula for the largest non-representable z, as well as a formula for the number of such z. A procedure is described which computes these quantities in O(log(max a_i)) steps.
We compute the Frobenius number for sequences of triangular and tetrahedral numbers. In addition, we study some properties of the numerical semigroups associated to those sequences.
We study the problem of variation of Frobenius eigenvalues on the cohomology of families of local systems of algebraic curves over finite fields.
Given $N$ positive integers $a_1, ..., a_N$ with $\gcd(a_1, ..., a_N)=1$, let $f_N$ denote the largest natural number which is not a positive integer combination of $a_1, ..., a_N$. This paper gives an optimal lower bound for $f_N$ in terms of the absolute inhomogeneous minimum of the standard $(N-1)$-simplex.
We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order $pq$. The general $t$-di…
A Graphical Approach to the Frobenius Number Problem in Three Variables
本文提出一种图形化方法，通过分析与直线ax+by=ab及其平移相关的格点来求解三变量Frobenius数，将算术问题转化为定位格点并求值的简单任务，并推导出比已知结果更简洁的一般公式。
In the Frobenius problem we are given a set of coprime, positive integers $a_1, a_2,...,a_k$, and are interested in the set of positive numbers NR that have no representation by the linear form $\sum_i a_ix_i$ in nonnegative integers $x_1, x_2,...,x_k$. We give a functional relationship that completely characterizes the set NR, and apply it to the case when the numbers are in an arithmetic progression.
On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II
本文研究了任意起始项与偶数步长下的广义Fibonacci子列集合，确定了其Frobenius数和亏格的计算问题。
Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers
该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。
Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The largest integer not in $S(A)$ is the Frobenius number of $A$, and the number of positive integers not in $S(A)$ is the genus of $A$. Sharp and Sylvester proved in 1884 that the Frobenius number of the set $A = \{a,b\}$ is $ab-a-b$, and that the genus of $A$ is $(a-1)(b-1)/2$. Graded rings and a simple form of Hilbert's syzygy theorem are used to give a commuta…
We study variants of the \emph{Frobenius coin-exchange problem}: given $n$ positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This problem and its siblings can be understood through generating functions with 0/1 coefficients according to whether or not an integer is representable. In the 2-parameter case, this generating function has an elegant closed form, from which many corollaries follow, including a formula for the Frobenius problem. We establish a similar closed form for the gene…
The extended Frobenius problem for Lucas series incremented by a Lucas number
本文研究了增加一个Lucas数的Lucas序列的扩展Frobenius问题，并证明了与这些序列相关的数值半群族满足Wilf猜想。
On the Frobenius number for three variables
本文解决了三个变量弗罗贝尼乌斯数公式中的不一致问题，提供了更准确的计算方法。
Teaching perspectives of the Frobenius coin problem of two denominators
本文以初等方法刻画两个互素正整数经非负整数系数线性组合所能表示的全部正整数，并探讨该方法在Frobenius硬币问题教学中的应用价值。
Inspired by the theory of p-adic differential equations, this paper introduces an analogous theory for q-difference equations over a local field, when |q|=1. We define some basic concepts, for instance the generic radius of convergence, introduce technical tools, such as a twisted Taylor formula for analytic functions, and prove some fundamental statements, such as an effective bound theorem, the existence of a weak Frobenius structure and a transfer theorem in regular singular disks.
We consider the Diophantine problem of Frobenius for semigroup ${\sf S}({\bf d}^3)$ where ${\bf d}^3$ denotes the tuple $(d_1,d_2,d_3)$, $\gcd(d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic representation for the diagonal elements $a_{kk}({\bf d}^3)$ of the Johnson's matrix of minimal relations in terms of $d_1,d_2,d_3$. Bearing in mind the results of the recent paper this gives the analytic representation for the Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and the Hilbert series $H({\bf d}^3;z)$ for the semigroups ${\sf S}({\bf d}^3)$…
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