Frobenius Problem and dead ends in integers
Number Theory
2011-11-09 v2 Group Theory
Abstract
Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In the appendix we show that every finitely generated group has a generating set with respect to which dead ends exist.
Cite
@article{arxiv.math/0612271,
title = {Frobenius Problem and dead ends in integers},
author = {Zoran Sunic},
journal= {arXiv preprint arXiv:math/0612271},
year = {2011}
}