Related papers: Short Proof: Exact Solution to the Finite Frobeniu…
In this paper, we give convenient formulas in order to obtain explicit expressions of a generalized Frobenius number called the $p$-Frobenius number as well as its related values. Here, for a non-negative integer $p$, the $p$-Frobenius…
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…
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 2007, a new variety of the well-known problem of identifying a counterfeit coin using a balance scale was introduced in the sixth International Kolmogorov Math Tournament. This paper offers a comprehensive overview of this new problem by…
This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency…
The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation $a_1 x_1+\cdots+a_k x_k=n$ ($a_1,\dots,a_k$ are given positive integers with $\gcd(a_1,\dots,a_k)=1$)…
We discuss several coin-weighing problems in which coins are known to be of three different weights and only a balance scale can be used. We start with the task of sorting coins when the pans of the scale can fit only one coin. We prove…
We show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split.
For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. The it is well known that only finitely many positive integers…
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…
An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest…
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion…
A numerical semigroup is irreducible if it cannot be obtained as intersection of two numerical semigroups containing it properly. If we only consider numerical semigroups with the same Frobenius number, that concept is generalized to atomic…
There are many papers written on the Two Envelopes Problem that usually study some of its variations. In this paper we will study and compare the most significant variations of the problem. We will see the correct decisions for each player…
For a real set $A$ consider the semigroup $S(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen…
For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. Then it is well known that only finitely many positive integers…
Consider $n$ independent, biased coins, each with a known probability of heads. Presented with an ordering of these coins, flip (i.e., toss) each coin once, in that order, until we have observed both a *head* and a *tail*, or flipped all…
Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to…
Fake coin problems using balance scales to identify one fake coin and its type among n coins (n > 2) were solved by Dyson in 1946. Dyson gave adaptive solutions with the minimum number of weighings where later weighings may be dependent on…