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\vskip 1cm{\LARGE\bf On a Generalization of the Coin Exchange \\
\vskip .05in
Problem for Three Variables
}
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\large
Amitabha Tripathi \\
Department of Mathematics \\
Indian Institute of Technology \\
Hauz Khas \\
New Delhi - 110016 \\
India \\
\href{mailto:atripath@maths.iitd.ac.in}{\tt atripath@maths.iitd.ac.in} \\
\ \\
Sujith Vijay\footnote{This work was done while the second
author was at the Department of
Mathematics, Indian Institute of Technology, Delhi.}\\
Department of Mathematics \\
Rutgers University -- New Brunswick \\
Piscataway, NJ 08854 \\
USA  \\
\href{mailto:sujith@math.rutgers.edu}{\tt sujith@math.rutgers.edu} \\
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\begin{abstract}
Given relatively prime and positive integers $a_1,a_2,\ldots,a_k$,
let ${\Gamma}$ denote the set of nonnegative integers representable
by the form $a_1x_1+a_2x_2+\cdots+a_kx_k$, and let
${\Gamma}^{\star}$ denote the positive integers in ${\Gamma}$. Let
${\cal S}^{\star}(a_1,a_2,\ldots,a_k)$ denote the set of all
positive integers $n$ not in ${\Gamma}$ for which
$n+{\Gamma}^{\star}$ is contained in ${\Gamma}^{\star}$. The purpose
of this article is to determine an algorithm which can be used to
obtain the set ${\cal S}^{\star}$ in the three variable case. In
particular, we show that the set ${\cal S}^{\star}(a_1,a_2,a_3)$ has
at most two elements. We also obtain a formula for $g(a_1,a_2,a_3)$,
the largest integer not representable by the form
$a_1x_1+a_2x_2+a_3x_3$ with the $x_i$'s nonnegative integers.
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