New upper bounds for taxicab and cabtaxi numbers

Boyer, Christian

Journal of integer sequences, Tome 11 (2008) no. 1.

Résumé

Summary: 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 $Taxicab(2) = 1729 = 9^{3} + 10^{3} = 1^{3} + 12^{3}, Taxicab(n)$ 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, $Cabtaxi(2) = 91 = 3^{3} + 4^{3} = 6^{3} - 5^{3}$. 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 $Cabtaxi(20)$.

Détail

Classification : 11D25
Keywords: taxicab number, cabtaxi number, Hardy-Ramanujan number, bernard frenicle de bessy, fran$$###$$cois vi`ete, sum of two cubes, difference of two cubes, magic square of cubes