Editing Taxicab number (section)

{{Short description|Class of integer}}<!-- WP:SDNOTDEF -->
{{for|number plates assigned to vehicles for hire|Taxi medallion}}
[[File:Srinivasa Ramanujan - OPC - 2.jpg|thumb|right|200px|[[Srinivasa Ramanujan]] (''picture'') was bedridden when he developed the idea of taxicab numbers, according to an anecdote from [[G. H. Hardy]].]]

In [[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 |url=https://www.numberphile.com/videos/1729-and-taxi-cabs |title=1729: Taxi Cab Number or Hardy–Ramanujan Number |last1=Grime |first1=James |last2=Bowley |first2=Roger |series=Numberphile |editor-last=Haran |editor-first=Brady |editor-link=Brady Haran}}</ref>

The name is derived from a conversation {{nowrap|''ca.''{{tsp}}1919}} involving [[mathematician]]s [[G. H. Hardy]] and [[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 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.|authorlink=Joseph H. Silverman|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>}}