# Seeds for Generalized Taxicab Numbers

## TL;DR

This paper introduces the concept of seed numbers for generalized taxicab numbers, proves their existence, and provides explicit formulas for certain cases, advancing understanding of these special numbers.

## Contribution

It establishes the existence of seed numbers for generalized taxicab numbers and derives explicit formulas for cases with two or three ways, proposing a conjecture for higher cases.

## Key findings

- Existence of seed numbers for all positive integers m and t
- Explicit formulas for seed numbers when t=2 or 3
- Conjecture for seed numbers when t ≥ 4

## Abstract

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.