New upper bounds of taxicab and cabtaxi numbers

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 give