Taxicab numbers
In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Hardy–Ramanujan number, is defined as the smallest number that can be expressed as a sum of two positive cube numbers in n distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103.
The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:
The concept was first mentioned in 1657 by Bernard Frénicle de Bessy, and was made famous in the early 20th century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers n, and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are the smallest possible and thus it cannot be used to find the actual value of Ta(n).
The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in n distinct ways. The concept of a cabtaxi number has been introduced to allow for alternative, less restrictive definitions of this nature. In a sense, the specification of two summands and powers of three is also restrictive; a generalized taxicab number allows for these values to be other than two and three, respectively.
So far, the following six taxicab numbers are known (sequence in the OEIS):
Ta(2), also known as the Hardy–Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657.
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