The Frobenius Problem and Its Generalizations

  • Authors:

  • Jeffrey Shallit

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, Canada ON N2L 3G1

  • Venue:
  • DLT '08 Proceedings of the 12th international conference on Developments in Language Theory
  • Year:
  • 2008

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Abstract

Let x1, x2, ..., xnbe positive integers. It is well-known that every sufficiently large integer can be represented as a non-negative integer linear combination of the xiif and only if $\gcd(x_1, x_2, \ldots, x_n) = 1$. The Frobenius problemis the following: given positive integers x1, x2, ..., xnwith $\gcd(x_1, x_2, \ldots, x_n) = 1 $, compute the largest integer notrepresentable as a non-negative integer linear combination of the xi. This largest integer is sometimes denoted g(x1,..., xn).