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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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'' integer cubes in ''n'' distinct ways. The most famous taxicab number is 1729 = Ta(2) = 13 + 123 = 93 + 103, also known as the Hardy–Ramanujan number. The name is derived from a conversation involving
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
. As told by Hardy:


History and definition

The pairs of summands of the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned by
Bernard Frénicle de Bessy Bernard ('' Bernhard'') is a French and West Germanic masculine given name. It has West Germanic origin and is also a surname. The name is attested from at least the 9th century. West Germanic ''Bernhard'' is composed from the two elements ''ber ...
, who published his observation in 1657. 1729 was made famous as the first taxicab number in the early 20th century by a story involving
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
in claiming it to be the smallest for his particular example of two summands. In 1938,
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and E. M. Wright proved that such numbers exist for all positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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. John Leech obtained Ta(3) in 1957. E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel found Ta(4) in 1989. J. A. Dardis found Ta(5) in 1994 and it was confirmed by David W. Wilson in 1999. Ta(6) was announced by Uwe Hollerbach on the NMBRTHRY mailing list on March 9, 2008, following a 2003 paper by Calude et al. that gave a 99% probability that the number was actually Ta(6). Upper bounds for Ta(7) to Ta(12) were found by Christian Boyer in 2006."New Upper Bounds for Taxicab and Cabtaxi Numbers" Christian Boyer, France, 2006–2008
/ref> The restriction of the 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.


Known taxicab numbers

So far, the following 6 taxicab numbers are known: \begin \operatorname(1) =& \ 2 \\ &= 1^3 + 1^3 \\ pt \operatorname(2) =& \ 1729 \\ &= 1^3 + 12^3 \\ &= 9^3 + 10^3 \\ pt \operatorname(3) =& \ 87539319 \\ &= 167^3 + 436^3 \\ &= 228^3 + 423^3 \\ &= 255^3 + 414^3 \\ pt \operatorname(4) =& \ 6963472309248 \\ &= 2421^3 + 19083^3 \\ &= 5436^3 + 18948^3 \\ &= 10200^3 + 18072^3 \\ &= 13322^3 + 16630^3 \\ pt \operatorname(5) =& \ 48988659276962496 \\ &= 38787^3 + 365757^3 \\ &= 107839^3 + 362753^3 \\ &= 205292^3 + 342952^3 \\ &= 221424^3 + 336588^3 \\ &= 231518^3 + 331954^3 \\ pt \operatorname(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


Upper bounds for taxicab numbers

For the following taxicab numbers upper bounds are known: \begin \operatorname(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\\ pt \operatorname(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 \\ pt \operatorname(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 \\ pt \operatorname(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 \\ pt \operatorname(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 \\ pt \operatorname(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


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 13. When a cubefree taxicab number is written as , the numbers and must be
relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
. Among the taxicab numbers listed above, only and 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: \begin 15170835645 &= 517^3 + 2468^3 \\ &= 709^3 + 2456^3 \\ &= 1733^3 + 2152^3 \end The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003: \begin 1801049058342701083 &= 92227^3 + 1216500^3 \\ &= 136635^3 + 1216102^3 \\ &= 341995^3 + 1207602^3 \\ &= 600259^3 + 1165884^3 \end .


See also

* * * * * * * * * * , a list of related conjectures and theorems


Notes


References

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


External links


A 2002 post to the Number Theory mailing list by Randall L. Rathbun
*
Taxicab and other maths at Euler
* {{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 Number theory Srinivasa Ramanujan