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

University of Vermont

Burlington, VT 05405

USA

Richard A. Games

Mitre Corporation

Bedford, MA 01730

USA

Robert L. Roth

Department of Mathematics

Emory University

Atlanta, GA 30322

USA

**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 = 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.

(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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