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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 {Link without Title} : :''I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 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 {Link without Title} cubes in two different ways. The subsequent taxicab numbers were found with the help of Computer s; 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 1997 . Ta(6) has not been found so far; however, Wilson found a 6-way sum showing that the 6th taxicab number Ta(6) ≤ 8230545258248091551205888. In 1998, Daniel J. Bernstein showed that a lower bound was Ta(6)≥391909274215699968. In 2002, Randall L. Rathbun improved the upper bound to Ta(6) ≤ 24153319581254312065344. More recently, in May 2003 , Stuart Gascoigne verified that Ta(6) > 68000000000000000000. Cristian S. Calude , Elena Calude and Michael J. Dinneen have shown that with a high "probability" (> 99%), Ta(6) = 24153319581254312065344. course this "probability" is merely evidential: the real probability is either 1 or 0, but as yet unknown :Ta(6) ≤ 24153319581254312065344 = 2890620633 + 58216233 ::= 2889480333 + 306417333 ::= 2865748733 + 851928133 ::= 2709320833 + 1621806833 ::= 2659045233 + 1749249633 ::= 2622436633 + 1828992233. SEE ALSO EXTERNAL LINKS REFERENCES
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