In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number which can be expressed as a sum of two positive cubes in n distinct ways, up to order of summands. G. H. Hardy and E. M. Wright proved in 1954 that such numbers exist for all positive integers n; however, their proof does not help in constructing them, and so far, only the following five taxicab numbers are known :

Ta(2), also known as the Hardy-Ramanujan number, was first published by Bernard Frénicle de Bessy in 1657 and later immortalized by an incident involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy [1]:

I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.

The subsequent taxicab numbers were found with the help of computers; John Leech obtained Ta(3) in 1957, E. Rosenstiel , J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1991, and David W. Wilson found Ta(5) in November 1997.

Ta(6) has not been found so far; however, Wilson also found a 6-way sum showing that the 6th taxicab number Ta(6) is ≤ 8230545258248091551205888. In 1998, Daniel J. Bernstein showed that 391909274215699968 ≥ Ta(6) ≥ 10¹⁸, and in 2002, Randall L. Rathbun gave proof that Ta(6) ≤ 24153319581254312065344. Recently, in May 2003, Stuart Gascoigne verified that Ta(6) > 6.8 · 10¹⁹, and Cristian S. Calude , Elena Calude and Michael J. Dinneen showed that with a high probability (> 99%), Ta(6) = 24153319581254312065344.

Also see

• Cabtaxi number
• Generalized taxicab number

External links

• A 2002 post to the Number Theory mailing list by Randall L. Rathbun

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412.
• J. Leech, Some Solutions of Diophantine Equations, Proc. Cambridge Phil. Soc. 53, 778-780, 1957.
• E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equation s = x³ + y³ = z³ + w³ = u³ + v³ = m³ + n³, Bull. Inst. Math. Appl., 27(1991) 155-157; MR 92i:11134, online
• David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online
• D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d), Mathematics of Computation 70, 233 (2000), 389--394.
• C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, Journal of Universal Computer Science, Vol. 9 (2003), p. 1196-1203