A Sharp Upper Bound for the Frobenius Number

Abstract

Let $0<a_1<\cdots<a_n$ be integers with $n\ge 2$ and $\gcd(a_1,\ldots,a_n)=1$. Their Frobenius number $F(a_1,\ldots,a_n)$ is the largest integer that cannot be expressed as a nonnegative integer combination of $a_1,\ldots,a_n$. Erdős and Graham proved the classical bound $F(a_1,\ldots,a_n)\le 2a_{n-1}\lfloor a_n/n\rfloor-a_n$, which depends on the two largest generators. Dixmier subsequently proved their conjecture $F(a_1,\ldots,a_n)\le a_n^2/(n-1)$, a bound depending only on $a_n$ and $n$ that removes the factor $2$ from the corresponding quadratic estimate. Following the Erdős--Graham approach, we remove the factor $2$ while retaining the dependence on $a_{n-1}$, and prove $F(a_1,\ldots,a_n)\le a_{n-1}\lceil a_n/(n-1)\rceil-a_n-1$. This bound can be substantially smaller than Dixmier's when $a_{n-1}$ is small relative to $a_n$. For every $n\ge3$, we also construct infinitely many instances satisfying the reverse inequality, which shows that the bound is sharp.