BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-72-274
       ENTRY:: October 16, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Linear combinations of sets of consecutive integers.
        TYPE:: Technical Report
      AUTHOR:: Klarner, David A.
      AUTHOR:: Rado, Richard
        DATE:: March 1972
       PAGES:: 11
    ABSTRACT:: Let k-1,$m_1, \ldots ,m_k$ denote non-negative integers, and
               suppose the greatest common divisor of $m_1, \ldots ,m_k$ is
               1. We show that if $S_1, \ldots ,S_k$ are sufficiently long
               blocks of consecutive integers, then the set $m_1 S_1 +
               \ldots\ m_k S_k$ contains a sizable block of consecutive
               integers. For example, if m and n are relatively prime
               natural numbers, and u, U, v, V are integers with U-u $\geq$
               n-1, V-v $geq$ m-1, then the set m{u,u+1, $\ldots$,U} +
               n{v,v+1, $\ldots$,V} contains the set {mu + nv -
               $\sigma$(m,n), $\ldots$,mU + nV - $\sigma$(m,n)} where
               $\sigma${m,n) = (m-1)(n-1) is the largest number such that
               $\sigma$(m,n)-1 cannot be expressed in the form mx + ny with
               x and y non-negative integers.
       NOTES:: [Adminitrivia V1/Prg/19951016]
         END:: STAN//CS-TR-72-274