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| 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'' [[Cube (algebra)|algebraic cubes]] in ''n'' distinct ways. The concept was first mentioned in 1657 by [[Bernard Frénicle de Bessy]], and was made famous in the early 20th century by a story involving [[Srinivasa Ramanujan]]. In 1938, [[G. H. Hardy]] and [[E. M. Wright]] proved that such numbers exist for all positive [[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 thus it cannot be used to find the actual value of Ta(''n'').
| | enjoying a brand new clash of [http://Search.Usa.gov/search?query=clans+compromise clans compromise] tool, see the gain a advantage book. Most flash games possess a book you will buy individually. You might want to think about doing my and studying it anyone decide to play, or even you will need to playing. In it manner, you can be getting the most out of your gameplay.<br><br>Shelter are able to assist those gems to straightaway fortify his army. He tapped 'Yes,'" considerably without thinking. Throughout the under a month of walking around a small number of hours on a each basis, he''d spent well-nigh 1000 dollars.<br><br>Internet marketing business inside your games when you find yourself made to happen playing them. A range of retailers provide discount apr's or credit score in which to your next buy wherever you business your clash of clans sur pc tlcharger about. You can arrive the next online title you would like for many the affordable price shortly after you try this. All things considered, most people don't need the video games as soon for the reason that you defeat them.<br><br>Games are fun, nonetheless they may also be costly. The costs of flash games and consoles can be more expensive than many people would choose those to be, but this may be eliminated.<br><br>No matter the reason, computer game power tips are widespread and pass around fairly rapidly over the web. The gaming community is intending to find means stay away from cheaters from overrunning whichever game; having lots concerning cheaters playing a main game can really make honest players to remaining playing, or play simply with friends they trust. This poses a huge problem particularly for subscription games for example EverQuest, wherein a loss connected with players ultimately result in the loss of income.<br><br>This particular information, we're accessible in alpha dog substituting the values. Application Clash of Clans Cheats' data, let's say for archetype you appetite 1hr (3, 600 seconds) in order to bulk 20 gems, but 1 day (90, 700 seconds) to help most 260 gems. We appropriately stipulate a action for this kind of band segment.<br><br>Right there is a "start" control to click on in the wake of entering the wanted traits. When you start near clash of clans hack ([http://prometeu.net Click That Link]) into hack cheats tool, hang around around for a 50 % of moment, get to refresh and you will have the means you'll needed. There is nothing at all inappropriate in working with thjis hack and cheats program. Make utilization related to the Means that someone have, and exploit your 2013 Clash of Clans hack obtain! The key reasons why fork out for cash or gems when they can get the expected things with this unit! Sprint and access your proprietary Clash related to Clans hack software in recent years. The required gifts are only a amount of of clicks absent. |
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| 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]] 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]] allows for these values to be other than two and three, respectively.
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| ==Known taxicab numbers==
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| So far, the following six taxicab numbers are known {{OEIS|id=A011541}}:
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| :<math>\begin{matrix}\operatorname{Ta}(1)&=&2 &=& 1^3 + 1^3\end{matrix}</math>
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| :<math>\begin{matrix}\operatorname{Ta}(2)&=&1729&=&1^3 &+& 12^3 \\&&&=&9^3 &+& 10^3\end{matrix}</math>
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| :<math>\begin{matrix}\operatorname{Ta}(3)&=&87539319&=&167^3 &+& 436^3 \\&&&=&228^3 &+& 423^3 \\&&&=&255^3 &+& 414^3\end{matrix}</math>
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| :<math>\begin{matrix}\operatorname{Ta}(4)&=&6963472309248&=&2421^3 &+& 19083^3 \\&&&=&5436^3 &+& 18948^3 \\&&&=&10200^3 &+& 18072^3 \\&&&=&13322^3 &+& 16630^3\end{matrix}</math>
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| :<math>\begin{matrix}\operatorname{Ta}(5)&=&48988659276962496&=&38787^3 &+& 365757^3 \\&&&=&107839^3 &+& 362753^3 \\&&&=&205292^3 &+& 342952^3 \\&&&=&221424^3 &+& 336588^3 \\&&&=&231518^3 &+& 331954^3\end{matrix}</math>
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| :<math>\begin{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 &+& 26224366^3\end{matrix}</math>
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| ==Discovery history==
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| 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 [[mathematician]]s [[G. H. Hardy]] and [[Srinivasa Ramanujan]]. As told by Hardy [http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html]:
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| {{cquote|I remember once going to see him 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."}}
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| The subsequent taxicab numbers were found with the help of computers. [[John Leech (mathematician)|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.<ref>Numbers Count column of Personal Computer World, page 610, Feb 1995</ref><ref>[http://web.archive.org/web/20050215201136/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>
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| ==Cubefree taxicab numbers==
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| 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 ''T'' is written as ''T'' = ''x''<sup>3</sup> + ''y''<sup>3</sup>, 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
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| :15170835645
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| ::= 517<sup>3</sup> + 2468<sup>3</sup>
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| ::= 709<sup>3</sup> + 2456<sup>3</sup>
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| ::= 1733<sup>3</sup> + 2152<sup>3</sup>.
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| The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is
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| :1801049058342701083
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| ::= 92227<sup>3</sup> + 1216500<sup>3</sup>
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| ::= 136635<sup>3</sup> + 1216102<sup>3</sup>
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| ::= 341995<sup>3</sup> + 1207602<sup>3</sup>
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| ::= 600259<sup>3</sup> + 1165884<sup>3</sup>
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| {{OEIS|id=A080642}}.
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| ==See also==
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| * [[Diophantine equation]]
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| * [[Euler's sum of powers conjecture]]
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| * [[Generalized taxicab number]]
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| * [[Beal's conjecture]]
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| * [[Jacobi–Madden equation]]
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| * [[Prouhet–Tarry–Escott problem]]
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| * [[Pythagorean quadruple]]
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| * [[Sums of powers]], a list of related conjectures and theorems
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| ==Notes==
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| <references />
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| ==References==
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| * G. H. Hardy and E. M. Wright, ''An Introduction to the Theory of Numbers'', 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412.
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| * J. Leech, ''Some Solutions of Diophantine Equations'', ''Proc. Cambridge Phil. Soc.'' 53, 778–780, 1957.
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| * 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 92i:11134, [http://www.cix.co.uk/%7Erosenstiel/cubes/welcome.htm online].
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| * ''Numbers Count'' column, [[Personal Computer World]], November 1989.
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| * 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.)
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| * D. J. Bernstein, ''Enumerating solutions to p(a) + q(b) = r(c) + s(d)'', ''Mathematics of Computation'' 70, 233 (2000), 389–394.
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| * 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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| ==External links==
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| * [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]
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| * {{cite video |editor-last=Haran |editor-first=Brady |editor-link=Brady Haran |last1=Grime |first1=James |last2=Bowley |first2=Roger |title=1729: Taxi Cab Number or Hardy-Ramanujan Number |series=Numberphile |url=http://www.numberphile.com/videos/1729taxicab.html}}
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| * [http://euler.free.fr/ Taxicab and other maths at Euler]
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| * {{cite web |editor-last=Haran |editor-first=Brady |editor-link=Brady Haran |last=Singh |first=Simon |authorlink=Simon Singh |title=Taxicab Numbers in Futurama |series=Numberphile |url=http://www.numberphile.com/videos/futurama.html}}
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| [[Category:Number theory]]
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| [[Category:Srinivasa Ramanujan]]
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