Related papers: An estimate for Frobenius' diophantine problem in …

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,…

In this paper we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integral combination of given relatively prime (strictly) positive integers (known as…

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.…

In this paper we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integral combination of given relatively prime (strictly) positive integers (known as…

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…

For positive integers $a$, $b$, and $c$ which have no common divisor, the Frobenius number of $a$, $b$ and $c$ is defined to be the largest integer that cannot be expressed as a linear combination of $a$, $b$ and $c$ with non-negative…

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…

The Frobenius coin problem in three variables, for three positive relatively prime integers $a_1< a_2< a_3$ asks to find the largest number not representable as $a_1x_1+a_2x_2+a_3x_3$ with non-negative integer coefficients $x_1$, $x_2$ and…

We give an explicit formula for the $p$-Frobenius number of triples associated with Diophantine equations $x^2+y^2=z^r$, that is, the largest positive integer that can only be represented in $p$ ways by combining the three integers of the…

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…

Let $N \geq 2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are…

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…

The Frobenius number for a set of relatively prime positive integers, where the smallest integer in the set is at least 2, is the largest integer that cannot be expressed as a nonnegative linear combination of those integers. We analyze the…

Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been…

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…

Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive…

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…

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent…

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…

The Frobenius number F(a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is…