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In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the least number which can be expressed as a sum of two positive cubes in n distinct ways, up to order of summands. G. H. Hardy and E. M. Wright proved in 1954 that such numbers exist for all positive integers n; however, their proof does not help in constructing them, and so far, only the following five taxicab numbers are known ( OEIS A011541):-
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Ta(2) was first published by Bernard Frénicle de Bessy in 1657 and later immortalized by an incident involving mathematicians G. H. Hardy and Srinivasa Ramanujan:
- I [G. H. Hardy] remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number (7·13·19) 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."
The subsequent taxicab numbers 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 1991, and David W. Wilson found Ta(5) in November 19971997 was a common year starting on Wednesday (see link for calendar), and was designated the International Year of the Reef''. Events January January 3 NBC's Today Show Bryant Gumbel signs off for the last time January 8 Mister Rogers receives a star on t.
Ta(6) has not been found so far; however, Wilson also found a 6-way sum showing that the 6th taxicab number Ta(6) is ≤ 8230545258248091551205888. In 1998, Daniel J. BernsteinDaniel J. Bernstein (known among users of his software and members of his mailing lists as simply "djb") is a professor at the University of Illinois at Chicago, a mathematician, a cryptologist, and a programmer, noted as the author of the computer softwa showed that 391909274215699968 ≥ Ta(6) ≥ 1018, and in 2002, Randall L. Rathbun gave proof that Ta(6) ≤ 24153319581254312065344. Recently, in MayThis article is about the month of May. For other uses, see May (disambiguation). May is the fifth month of the year in the Gregorian Calendar, with 31 days. It may have been named for the Roman goddess Maia or more likely for the Roman goddess of fertili 20032003 is a common year starting on Wednesday (link will take you to calendar), and also: The International Year of Freshwater The European Disability Year Summary Perhaps the defining global event of the year 2003 was the Invasion of Iraq launched by the U, Stuart Gascoigne verified that Ta(6) > 6.8 · 1019, and Cristian S. Calude , Elena Calude and Michael J. Dinneen showed that with a high probability (> 99%), Ta(6) = 24153319581254312065344.
1 Also see
- Cabtaxi numberIn mathematics, the n''th cabtaxi number typically denoted Cabtaxi ''n , is defined as the smallest positive integer that can be written as the sum of two positive or negative cubes in n ways. Such numbers exist for all n (since taxicab numbers exist for
- Generalized taxicab numberIn mathematics, the generalized taxicab number ''Taxicab ''k j n is the smallest number which can be expressed as the sum of j ''k''th positive powers in n different ways. For k 3 and j 2, they coincide with Taxicab numbers. It has been shown by Euler tha
2 External links
