 
Summary: ON A GENERALIZATION OF THE FROBENIUS NUMBER
ALEXANDER BROWN, ELEANOR DANNENBERG, JENNIFER FOX, JOSHUA HANNA,
KATHERINE KECK, ALEXANDER MOORE, ZACHARY ROBBINS, BRANDON SAMPLES,
AND JAMES STANKEWICZ
Abstract. We consider a generalization of the Frobenius Problem where the object of
interest is the greatest integer which has exactly j representations by a collection of relatively
prime integers. We prove an analogue of a theorem of Brauer and Shockley and show how
it can be used for computation.
The linear diophantine problem of Frobenius has long been a celebrated problem in number
theory. Most simply put, the problem is to find the Frobenius Number of k positive relatively
prime integers (a1, . . . , ak), i.e. the greatest integer M for which there is no way to express
M as the nonnegative integral linear combination of the given integers.
A generalization, which has drawn interest both from classical study of the Frobenius
Problem ([Al05, Problem A.2.6]) and from the perspective of partition functions and integer
points on polytopes (as in [BR04]), is to ask for the greatest integer M which can be expressed
in exactly j different ways. We make this precise with the following definitions:
A representation of M by a ktuple (a1, . . . , ak) of positive, relatively prime integers is a
solution (x1, . . . , xk) Zk
0 to the equation M = k
i=1 aixi.
