Editing Taxicab number (section)

==History and definition==
The pairs of summands of the Hardy–Ramanujan number Ta(2) = 1729 were first mentioned by [[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 [[Srinivasa Ramanujan]] in claiming it to be the smallest for his particular example of two summands. 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 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 (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>{{sfn|Wilson|1999}} 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).{{sfn|Calude|Calude|Dinneen|2003}} 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. {{OEIS|id=A293647}}. 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.