Taxicab number

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In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest number that 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 (sequence A011541 in OEIS):

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 found a 6-way sum showing that the 6th taxicab number Ta(6) ≤ 8230545258248091551205888. In 1998, Daniel J. Bernstein showed that a lower bound was Ta(6)≥391909274215699968. In 2002, Randall L. Rathbun improved the upper bound to Ta(6) ≤ 24153319581254312065344. More recently, in May 2003, Stuart Gascoigne verified that Ta(6) > 68000000000000000000. Cristian S. Calude, Elena Calude and Michael J. Dinneen have shown that with a high "probability" (> 99%), Ta(6) = 24153319581254312065344. [Of course this "probability" is merely evidential: the real probability is either 1 or 0, but as yet unknown]

Ta(6) ≤ 24153319581254312065344 = 289062063³ + 5821623³

= 288948033³ + 30641733³

= 286574873³ + 85192813³

= 270932083³ + 162180683³

= 265904523³ + 174924963³

= 262243663³ + 182899223³.

A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 1³. When a cubefree taxicab number T is written as T = x³+y³, the numbers x and y must be relatively prime. Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. There is only one cubefree taxicab number known with three representations. It was discovered by Paul Vojta (unpublished) in 1981 while he was a graduate student. It is

15170835645

=517³ + 2468³

= 709³ + 2456³

= 1733³ + 2152³.

[edit] See also

• Cabtaxi number
• Generalized taxicab number

[edit] External links

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

[edit] 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. See also Numbers Count Personal Computer World November 1989.
• 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