Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.6

New Upper Bounds for Taxicab and Cabtaxi Numbers

Christian Boyer
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Hardy was surprised by Ramanujan's remark about a London taxi numbered 1729: "it is a very interesting number, it is the smallest number expressible as a sum of two cubes in two different ways". In memory of this story, this number is now called Taxicab(2) = 1729 = 93 + 103 = 13 + 123, Taxicab(n) being the smallest number expressible in n ways as a sum of two cubes. We can generalize the problem by also allowing differences of cubes: Cabtaxi(n) is the smallest number expressible in n ways as a sum or difference of two cubes. For example, Cabtaxi(2) = 91 = 33 + 43 = 63 - 53. Results were only known up to Taxicab(6) and Cabtaxi(9). This paper presents a history of the two problems since Fermat, Frenicle and Viète, and gives new upper bounds for Taxicab(7) to Taxicab(19), and for Cabtaxi(10) to Cabtaxi(30). Decompositions are explicitly given up to Taxicab(12) and Cabtaxi(20).

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(Concerned with sequences A011541 and A047696 .)

Received June 26 2007; revised version received February 22 2008. Published in Journal of Integer Sequences, March 7 2008.

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