  Generalized taxicab numbers are the smallest positive integers that are the
sum of exactly $j$, positive $k$-th powers in exactly $m$ distinct ways. This
paper is considers for which values of $m$ does a smallest such integer exist
as $j$ gets large. There appear to be only two possible outcomes, leading to
curious results like there is no positive integer that can be expressed as the
sum of exactly $10$ positive squares in exactly $3$ ways. This paper resolves a
number of conjectures found in the OEIS by considering generalized Taxicab
numbers in the setting of the theory of partitions.
