广义出租车数的种子
Seeds for Generalized Taxicab Numbers
摘要：
广义出租车数$T(n,m,t)$等于以$t$方式表示的$n$的正$m$次幂之和的最小数。这个定义的灵感来自于Ramanujan的观察，$1729 = 1^3+ 12^3 =9^3 + 10^3 $是两个立方以两种方式和因此$1729= T(2,3,2)$的最小数。本文证明了对任意给定的正整数$m$和$t$，存在一个数$s$，使得对每个$k \geq 0$都有$T(s+k,m,t) =T(s,m,t) +k$.最小的这种$s$被称为广义出租车数的种子。此外，我们找到了明确的表达式，这个种子数时的方式$t$是2或3，并提出了一个猜想$t \geq 4$方式。
英文摘要：
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 $1729= T(2,3,2)$. In this paper we prove that for any given positive integers $m$ and $t$, there exists a number $s$ such $T(s+k,m,t) =T(s,m,t) +k$ for every $k \geq 0$. The smallest such $s$ 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 $t \geq 4$ ways.
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