Bounds on generalized Frobenius numbers

Details Bounds on generalized Frobenius numbers Bounds on generalized Frobenius numbers

2010-08-29

arxiv.org/abs/1008.4937v3

European Journal of Combinatorics, vol. 32 no. 3 (2011) pg. 361--368

Mathematics

Number Theory

We include an appendix with an erratum and addendum to the published version of this paper: two inaccuracies in the statement

Scientific paper

10.1016/j.ejc.2010.11.001

Let $N \geq 2$ 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 $s$.

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