An Optimal Lower Bound for the Frobenius Problem

Abstract

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 of the absolute inhomogeneous minimum of the standard $(N-1)$-simplex.

10 pages