Bounds on generalized Frobenius numbers

Abstract

Let $N\ge 2$ and let $1<a_1<...<a_N$ be relatively prime integers. The Frobenius number of this $N$-tuple is defined to be the largest positive integer that has no representation as ${\sum }_{i=1}^N{a}_i{x}_i$ where $x_1,...,x_N$ are non-negative integers. More generally, the $s$-Frobenius number is defined to be the largest positive integer that has precisely $s$ distinct representations like this. We use techniques from the Geometry of Numbers to give upper and lower bounds on the $s$-Frobenius number for any nonnegative integer $s$.

We include an appendix with an erratum and addendum to the published version of this paper: two inaccuracies in the statement of Theorem 2.2 are corrected and additional bounds on s-Frobenius numbers are derived

References in corpus (1)

• On Feasibility of Integer Knapsacks

Cited by in corpus (2)

• On the number of integer points in translated and expanded polyhedra
• Positive semigroups and generalized Frobenius numbers over totally real number fields