IJPAM: Volume 60, No. 1 (2010)

COMPUTING THE FROBENIUS NUMBER

Abdallah Badra
Laboratoire de Mathématiques
Université Blaise Pascal
Les Cézeaux, Aubière Cedex, 63177, FRANCE
e-mail: [email protected]


Abstract.The Frobenius number $g(A)$ of a finite subset $A\subset \N$ such that $\gcd(A)=1$ is the largest integer which cannot be expressed as $\sum_{a\in A}ax_{a}$ with non-negative integers $x_a$. We present an algorithm for the computation of $g(A)$. Without loss of generality we suppose that there exist $a,b\in A$ such that $\gcd(a,b)=1$. We give a formula for $g(A)$ in the particular case that for all $c,d\in A$, $c+d$ can be written in the form $c+d=xa+yb$ with $x,y\geq 0$ (e.g. $c+d>ab-a-b$). Using Euler polynomials we give a formula for $g(A)$ in the case that $A=\{a,b,c\}$.

Received: February 19, 2010

AMS Subject Classification: 11D04

Key Words and Phrases: Frobenius number, linear Diophantine equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 1