Seeds for Generalized Taxicab Numbers

The generalized taxicab number $T(n,m,t)$ is equal to the smallest number that is the sum of $n$ positive $m$th powers in $t$ ways. This definition is inspired by Ramanujan’s observation that $1729=1^3+{12}^3=9^3+{10}^3$ is the smallest number that is the sum of two cubes in two ways and thus $1729= T(2,3,2)$\mathrm{.}Inthispaperweprovethatforanygivenpositiveintegers$m$and$t$,thereexistsanumber$s$such$T(s+k,m,t) =T(s,m,t) +k$forevery$k \geq 0$\mathrm{.}Thesmallestsuch$s$ is termed the seed for the generalized taxicab number. Furthermore, we find explicit expressions for this seed number when the number of ways $t$ is 2 or 3 and present a conjecture for $t\ge 4$ ways.