COMPUTING THE FROBENIUS NUMBER

Abdallah Badra (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

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 $A=\{a,b,c\}$.

Mots clés: Frobenius number; Linear Diophantine equation

Domaines: Théorie des nombres [math.NT]

Abdallah Badra. COMPUTING THE FROBENIUS NUMBER. International Journal of Pure and Applied Mathematics, 2010, 60 (1), pp.89-105. ⟨hal-00477408⟩