# This Collection

By Defense Date By Authors By Titles By Subject

# Statistics

View Statistics All RECERCAT

# My RECERCAT

 Title: The Frobenius problem: a geometric approach Barrière Figueroa, Eulalia; Miralles de la Asunción, Alicia Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV; Universitat Politècnica de Catalunya. COMBGRAF - Combinatòria, Teoria de Grafs i Aplicacions 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. Representations of semigroupsDiophantine equationsFrobenius probleminteger latticesemigroupSemigrupsEquacions diofàntiquesClassificació AMS::11 Number theory::11D Diophantine equationsClassificació AMS::20 Group theory and generalizations::20M Semigroups Attribution-NonCommercial-NoDerivs 2.5 Spain http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Article

# Related documents

## Other documents of the same author

Comellas Padró, Francesc de Paula; Miralles de la Asunción, Alicia
Comellas Padró, Francesc de Paula; Miralles de la Asunción, Alicia
Batlle Arnau, Carles; Fossas Colet, Enric; Merillas Santos, Iván; Miralles de la Asunción, Alicia
Aguiló Gost, Francisco de Asis L.; Miralles de la Asunción, Alicia
Aguiló Gost, Francisco de Asis L.; Miralles de la Asunción, Alicia; Zaragoza Monroig, M. Luisa

Coordination

Supporters