Taxicab number

Taxicab number

In mathematics, the "n"th 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"). So far, only the following six taxicab numbers are known OEIS|id=A011541:

:operatorname{Ta}(1) = 2 = 1^3 + 1^3

:egin{matrix}operatorname{Ta}(2)&=&1729&=&1^3 + 12^3 \&&&=&9^3 + 10^3end{matrix}

:egin{matrix}operatorname{Ta}(3)&=&87539319&=&167^3 + 436^3 \&&&=&228^3 + 423^3 \&&&=&255^3 + 414^3end{matrix}

:egin{matrix}operatorname{Ta}(4)&=&6963472309248&=&2421^3 + 19083^3 \&&&=&5436^3 + 18948^3 \&&&=&10200^3 + 18072^3 \&&&=&13322^3 + 16630^3end{matrix}

:egin{matrix}operatorname{Ta}(5)&=&48988659276962496&=&38787^3 + 365757^3 \&&&=&107839^3 + 362753^3 \&&&=&205292^3 + 342952^3 \&&&=&221424^3 + 336588^3 \&&&=&231518^3 + 331954^3end{matrix}

:egin{matrix}operatorname{Ta}(6)&=&24153319581254312065344&=&582162^3 + 28906206^3 \&&&=&3064173^3 + 28894803^3 \&&&=&8519281^3 + 28657487^3 \&&&=&16218068^3 + 27093208^3 \&&&=&17492496^3 + 26590452^3 \&&&=&18289922^3 + 26224366^3end{matrix}

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 [http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html] :

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) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9 2008. [ [http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0803&L=nmbrthry&T=0&P=1059 NMBRTHRY Archives - March 2008 (#10) ] ]

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" = "x"3+"y"3, 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.

See also

* Cabtaxi number
* Generalized taxicab number

External links

* [http://listserv.nodak.edu/scripts/wa.exe?A2=ind0207&L=nmbrthry&F=&S=&P=1278 A 2002 post to the Number Theory mailing list by Randall L. Rathbun]
* [http://euler.free.fr/ Taxicab and other maths at Euler]

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, [http://www.cix.co.uk/%7Erosenstiel/cubes/welcome.htm 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), [http://www.math.uwaterloo.ca/JIS/wilson10.html#RDR91 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


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