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# Taxicab number

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, 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 is thus useless in finding Ta(n).

Known taxicab numbers

So far, the following six taxicab numbers are known (sequence A011541 in OEIS):

Discovery history

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. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.[1][2] Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,[3] following a 2003 paper by Calude et al. that gave a 99% chance that the number was actually Ta(6).[4] Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.[5]

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 for all pairs (x, y). Among the taxicab numbers Ta(n) listed above, only Ta(1) and Ta(2) are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by Paul Vojta (unpublished) in 1981 while he was a graduate student. It is

15170835645

= 5173 + 24683
= 7093 + 24563
= 17333 + 21523.

The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is

1801049058342701083

= 922273 + 12165003
= 1366353 + 12161023
= 3419953 + 12076023
= 6002593 + 11658843

(sequence A080642 in OEIS)

* Cabtaxi number
* Generalized taxicab number

Notes

1. ^ Numbers Count column of Personal Computer World, page 610, Feb 1995
2. ^ "The Fifth Taxicab Number is 48988659276962496" by David W. Wilson
3. ^ NMBRTHRY Archives - March 2008 (#10) "The sixth taxicab number is 24153319581254312065344" by Uwe Hollerbach
4. ^ 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
5. ^ "'New Upper Bounds for Taxicab and Cabtaxi Numbers" Christian Boyer, France, 2006-2008

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. (Wilson was unaware of J. A. Dardis's prior discovery of Ta(5) in 1994 when he wrote this.)
* 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