New Upper Bounds for 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 = 9 + 10 = 1 + 12, 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 + 4 = 6 − 5. 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). 1 A Fermat problem solved by Frenicle Our story starts 350 years ago, with letters exchanged between France and England during the reign of Louis XIV and the protectorate of Oliver Cromwell. On August 15th 1657, from Castres (in the south of France), Pierre de Fermat sent to Kenelm Digby some mathematical problems. Translated into English, two of them were: 1. Find two cube numbers of which the sum is equal to two other cube numbers. 2. Find two cube numbers of which the sum is a cube.