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{{Short description|Smallest integer expressable as a sum of two positive integer cubes in n distinct ways}}
[[File:Srinivasa Ramanujan - OPC - 2.jpg|thumb|right|200px|Srinivasa Ramanujan developed the idea of taxicab numbers.]]

In [[Mathematics|mathematics]], the ''n''th '''taxicab number''', typically denoted Ta(''n'') or Taxicab(''n''), is defined as the smallest integer that can be expressed as a sum of two ''positive'' [[Cube (algebra)|integer cubes]] in ''n'' distinct ways.<ref>{{cite web |title=Taxicab Number |url=https://mathworld.wolfram.com/TaxicabNumber.html |website=Wolfram Mathworld}}</ref> The most famous taxicab number is [[1729 (number)|1729]] = Ta(2) = 1<sup>3</sup> + 12<sup>3</sup> = 9<sup>3</sup> + 10<sup>3</sup>, also known as the Hardy-Ramanujan number.<ref>{{cite web |title=Hardy-Ramanujan Number |url=https://mathworld.wolfram.com/Hardy-RamanujanNumber.html |website=Wolfram Mathworld}}</ref><ref>{{cite video |editor-last=Haran |editor-first=Brady |last1=Grime |first1=James |last2=Bowley |first2=Roger |title=1729: Taxi Cab Number or Hardy-Ramanujan Number |series=Numberphile |url=https://www.numberphile.com/videos/1729-and-taxi-cabs}}</ref>

The name is derived from a conversation in about 1919 involving [[Biography:Mathematician|mathematician]]s [[Biography:G. H. Hardy|G. H. Hardy]] and [[Biography:Srinivasa Ramanujan|Srinivasa Ramanujan]]. As told by Hardy:

{{quote|I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. [[1729 (number)|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."<ref>[http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html Quotations by G. H. Hardy, MacTutor History of Mathematics] {{webarchive|url=https://web.archive.org/web/20120716185939/http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html |date=2012-07-16 }}</ref><ref>{{cite journal|author=Silverman, Joseph H.|title=Taxicabs and sums of two cubes|journal=Amer. Math. Monthly|volume=100|year=1993|issue=4|pages=331–340|url=http://www.maa.org/programs/maa-awards/writing-awards/taxicabs-and-sums-of-two-cubes|doi=10.2307/2324954|jstor=2324954}}</ref>}}

==History and definition==
The pairs of summands of the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned by [[Biography:Bernard Frénicle de Bessy|Bernard Frénicle de Bessy]], who published his observation in 1657. 1729 was made famous as the first taxicab number in the early 20th century by a story involving [[Biography:Srinivasa Ramanujan|Srinivasa Ramanujan]] in claiming it to be the smallest for his particular example of two summands. In 1938, [[Biography:G. H. Hardy|G. H. Hardy]] and [[Biography:E. M. Wright|E. M. Wright]] proved that such numbers exist for all positive [[Integer|integer]]s ''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 so it cannot be used to find the actual value of Ta(''n'').

The taxicab numbers subsequent to 1729 were found with the help of computers. [[Biography:John Leech (mathematician)|John Leech]] obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1989.<ref>Numbers Count column, Personal Computer World, page 234, November 1989</ref> J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999.<ref>Numbers Count column of Personal Computer World, page 610, Feb 1995</ref><ref>[https://web.archive.org/web/20050215201136/http://www.cs.uwaterloo.ca/journals/JIS/wilson10.html "The Fifth Taxicab Number is 48988659276962496" by David W. Wilson]</ref> Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008,<ref>[https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0803&L=NMBRTHRY&F=&S=&P=5454 NMBRTHRY Archives – March 2008 (#10) "The sixth taxicab number is 24153319581254312065344" by Uwe Hollerbach]</ref> following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6).<ref>C. S. Calude, E. Calude and M. J. Dinneen: What is the value of Taxicab(6)?, [[Journal of Universal Computer Science]], Vol. 9 (2003), pp. 1196–1203</ref> Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006.<ref>[http://www.christianboyer.com/taxicab/ "'New Upper Bounds for Taxicab and Cabtaxi Numbers" Christian Boyer, France, 2006–2008]</ref>

The restriction of the [[Addition#Notation and terminology|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|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|generalized taxicab number]] allows for these values to be other than two and three, respectively.

==Known taxicab numbers==
So far, the following 6 taxicab numbers are known:

<math display=block>\begin{align}
  \operatorname{Ta}(1) =& \ 2 \\
  &= 1^3 + 1^3 \\[6pt]
  \operatorname{Ta}(2) =& \ 1729 \\
  &= 1^3 + 12^3 \\
  &= 9^3 + 10^3 \\[6pt]
  \operatorname{Ta}(3) =& \ 87539319 \\
  &= 167^3 + 436^3 \\
  &= 228^3 + 423^3 \\
  &= 255^3 + 414^3 \\[6pt]
  
  \operatorname{Ta}(4) =& \ 6963472309248 \\
  &= 2421^3 + 19083^3 \\
  &= 5436^3 + 18948^3 \\
  &= 10200^3 + 18072^3 \\
  &= 13322^3 + 16630^3 \\[6pt]
 
  \operatorname{Ta}(5) =& \ 48988659276962496 \\
  &= 38787^3 + 365757^3 \\
  &= 107839^3 + 362753^3 \\
  &= 205292^3 + 342952^3 \\
  &= 221424^3 + 336588^3 \\
  &= 231518^3 + 331954^3 \\[6pt]

  \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 + 26224366^3
\end{align}</math>

==Upper bounds for taxicab numbers==
For the following taxicab numbers upper bounds are known:

<math display=block>\begin{align}
\operatorname{Ta}(7) \le & \ 24885189317885898975235988544 \\
  &=2648660966^3 + 1847282122^3 \\ 
  &= 2685635652^3 + 1766742096^3 \\ 
  &= 2736414008^3 + 1638024868^3 \\ 
  &= 2894406187^3 + 860447381^3 \\ 
  &= 2915734948^3 + 459531128^3 \\ 
  &= 2918375103^3 + 309481473^3\\ 
  &= 2919526806^3 + 58798362^3\\[6pt]

  \operatorname{Ta}(8) \le & \ 50974398750539071400590819921724352 \\
  &= 299512063576^3 + 288873662876^3 \\ 
  &= 336379942682^3 + 234604829494^3 \\ 
  &= 341075727804^3 + 224376246192^3 \\ 
  &= 347524579016^3 + 208029158236^3 \\
  &= 367589585749^3 + 109276817387^3 \\
  &= 370298338396^3 + 58360453256^3\\ 
  &= 370633638081^3 + 39304147071^3\\ 
  &= 370779904362^3 + 7467391974^3 \\[6pt]

\operatorname{Ta}(9) \le & \ 136897813798023990395783317207361432493888 \\
  &= 41632176837064^3 + 40153439139764^3 \\ 
  &= 46756812032798^3 + 32610071299666^3 \\ 
  &= 47409526164756^3 + 31188298220688^3 \\ 
  &= 48305916483224^3 + 28916052994804^3 \\
  &= 51094952419111^3 + 15189477616793^3 \\ 
  &= 51471469037044^3 + 8112103002584^3\\ 
  &= 51518075693259^3 + 5463276442869^3\\ 
  &= 51530042142656^3 + 4076877805588^3\\ 
  &= 51538406706318^3 + 1037967484386^3 \\[6pt]

  \operatorname{Ta}(10) \le & \ 7335345315241855602572782233444632535674275447104 \\
  &= 15695330667573128^3 + 15137846555691028^3 \\ 
  &= 17627318136364846^3 + 12293996879974082^3 \\ 
  &= 17873391364113012^3 + 11757988429199376^3 \\ 
  &= 18211330514175448^3 + 10901351979041108^3 \\ 
  &= 19262797062004847^3 + 5726433061530961^3 \\ 
  &= 19404743826965588^3 + 3058262831974168^3\\ 
  &= 19422314536358643^3 + 2059655218961613^3\\ 
  &= 19426825887781312^3 + 1536982932706676^3\\ 
  &= 19429379778270560^3 + 904069333568884^3\\ 
  &= 19429979328281886^3 + 391313741613522^3 \\[6pt]

  \operatorname{Ta}(11) \le & \ 2818537360434849382734382145310807703728251895897826621632 \\
  &= 11410505395325664056^3 + 11005214445987377356^3 \\ 
  &= 12815060285137243042^3 + 8937735731741157614^3 \\ 
  &= 12993955521710159724^3 + 8548057588027946352^3 \\ 
  &= 13239637283805550696^3 + 7925282888762885516^3 \\ 
  &= 13600192974314732786^3 + 6716379921779399326^3 \\ 
  &= 14004053464077523769^3 + 4163116835733008647^3\\ 
  &= 14107248762203982476^3 + 2223357078845220136^3\\ 
  &= 14120022667932733461^3 + 1497369344185092651^3\\ 
  &= 14123302420417013824^3 + 1117386592077753452^3\\ 
  &= 14125159098802697120^3 + 657258405504578668^3\\ 
  &= 14125594971660931122^3 + 284485090153030494^3 \\[6pt]

  \operatorname{Ta}(12) \le & \ 73914858746493893996583617733225161086864012865017882136931801625152 \\
  &= 33900611529512547910376^3 + 32696492119028498124676^3 \\
  &= 38073544107142749077782^3 + 26554012859002979271194^3 \\
  &= 38605041855000884540004^3 + 25396279094031028611792^3 \\
  &= 39334962370186291117816^3 + 23546015462514532868036^3 \\
  &= 40406173326689071107206^3 + 19954364747606595397546^3 \\
  &= 41606042841774323117699^3 + 12368620118962768690237^3 \\
  &= 41912636072508031936196^3 + 6605593881249149024056^3 \\
  &= 41950587346428151112631^3 + 4448684321573910266121^3 \\
  &= 41960331491058948071104^3 + 3319755565063005505892^3 \\
  &= 41965847682542813143520^3 + 1952714722754103222628^3 \\
  &= 41965889731136229476526^3 + 1933097542618122241026^3 \\
  &= 41967142660804626363462^3 + 845205202844653597674^3 
  \end{align}</math>

==Cubefree taxicab numbers==
A more restrictive taxicab problem requires that the taxicab number be cubefree, which means that it is not divisible by any cube other than 1<sup>3</sup>. When a cubefree taxicab number {{mvar|T}} is written as {{math|1=''T'' = ''x''<sup>3</sup> + ''y''<sup>3</sup>}}, the numbers {{mvar|x}} and {{mvar|y}} must be relatively prime. Among the taxicab numbers {{math|Ta(''n'')}} listed above, only {{math|Ta(1)}} and {{math|Ta(2)}} are cubefree taxicab numbers. The smallest cubefree taxicab number with three representations was discovered by [[Biography:Paul Vojta|Paul Vojta]] (unpublished) in 1981 while he was a graduate student:

<math display=block>\begin{align}
  15170835645 &= 517^3 + 2468^3 \\
  &= 709^3 + 2456^3 \\
  &= 1733^3 + 2152^3
\end{align}</math>

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

<math display=block> \begin{align}
  1801049058342701083 &= 92227^3 + 1216500^3 \\
  &= 136635^3 + 1216102^3 \\
  &= 341995^3 + 1207602^3 \\
  &= 600259^3 + 1165884^3
\end{align}</math>
{{OEIS|id=A080642}}.

==See also==
{{div col}}
* {{annotated link|1729 (number)}}
* {{annotated link|Diophantine equation}}
* {{annotated link|Euler's sum of powers conjecture}}
* {{annotated link|Generalized taxicab number}}
* {{annotated link|Jacobi–Madden equation}}
* {{annotated link|Prouhet–Tarry–Escott problem}}
* {{annotated link|Pythagorean quadruple}}
* {{annotated link|Sums of three cubes}}
* {{annotated link|Sums of powers}}, a list of related conjectures and theorems
{{div col end}}

==Notes==
{{Reflist|2}}

==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. Camb. 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 equations'' = ''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>3</sup> + ''w''<sup>3</sup> = ''u''<sup>3</sup> + ''v''<sup>3</sup> = ''m''<sup>3</sup> + ''n''<sup>3</sup>,  ''Bull. Inst. Math. Appl.'', 27(1991) 155–157; {{MR|1125858}}, [http://www.rosenstiel.co.uk/cubes/index.htm online].
* 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]. (Wilson was unaware of J. A. Dardis' prior discovery of Ta(5) in 1994 when he wrote this.)
* D. J. Bernstein, ''Enumerating solutions to {{tmath|1=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.&nbsp;1196–1203

==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]
* {{cite video |editor-last=Haran |editor-first=Brady |last1=Grime |first1=James |last2=Bowley |first2=Roger |title=1729: Taxi Cab Number or Hardy-Ramanujan Number |series=Numberphile |url=https://www.numberphile.com/videos/1729-and-taxi-cabs}}
* [http://euler.free.fr/ Taxicab and other maths at Euler]
* {{cite web |editor-last=Haran |editor-first=Brady |last=Singh |first=Simon |title=Taxicab Numbers in Futurama |series=Numberphile |url=http://www.numberphile.com/videos/futurama.html}}

[[Category:Number theory]]

{{Sourceattribution|Taxicab number}}