Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.3

Seeds for Generalized Taxicab Numbers

Jeffrey H. Dinitz
University of Vermont
Burlington, VT 05405

Richard A. Games
Mitre Corporation
Bedford, MA 01730

Robert L. Roth
Department of Mathematics
Emory University
Atlanta, GA 30322


The generalized taxicab number T(n,m,t) is equal to the smallest number that is the sum of n positive mth powers in t ways. This definition is inspired by Ramanujan's observation that 1729 = 13+123 = 93+103 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 that T(s+k,m,t) = T(s,m,t)+k for every k ≥ 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 ≥ 4 ways.

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(Concerned with sequence A011541.)

Received January 25 2019; revised versions received May 3 2019; May 6 2019. Published in Journal of Integer Sequences, May 17 2019.

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