The th taxicab number is the smallest number representable in ways as a sum of Positive
Cubes. The numbers derive their name from the Hardy-Ramanujan Number
(1) |
However, this property was also known as early as 1657 by F. de Bessy (Berndt and Bhargava 1993, Guy 1994).
Leech (1957) found
(2) |
(3) |
(4) |
Hardy and Wright (Theorem 412, 1979) show that the number of such sums can be made arbitrarily large but, updating Guy (1994) with Wilson's result, the least example is not known for six or more equal sums.
Sloane defines a slightly different type of taxicab numbers, namely numbers which are sums of two cubes in two or more ways, the first few of which are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ... (Sloane's A001235).
See also Diophantine Equation--Cubic, Hardy-Ramanujan Number
References
Berndt, B. C. and Bhargava, S. ``Ramanujan--For Lowbrows.'' Am. Math. Monthly 100, 645-656, 1993.
Guy, R. K. ``Sums of Like Powers. Euler's Conjecture.'' §D1 in
Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 68,
1959.
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed.
Oxford, England: Clarendon Press, 1979.
Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 564, 1989.
Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 312, 1991.
Leech, J. ``Some Solutions of Diophantine Equations.'' Proc. Cambridge Phil. Soc. 53, 778-780, 1957.
Plouffe, S. ``Taxicab Numbers.'' http://www.lacim.uqam.ca/pi/problem.html.
Rosenstiel, E.; Dardis, J. A.; and Rosenstiel, C. R. ``The Four Least Solutions in Distinct Positive Integers of the
Diophantine Equation
.'' Bull. Inst. Math. Appl. 27, 155-157, 1991.
Silverman, J. H. ``Taxicabs and Sums of Two Cubes.'' Amer. Math. Monthly 100, 331-340, 1993.
Sloane, N. J. A.
A001235 and
A011541
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.
Snow, C. P. Foreword to A Mathematician's Apology, reprinted with a foreword by C. P. Snow
(by G. H. Hardy). New York: Cambridge University Press, p. 37, 1993.
Wooley, T. D. ``Sums of Two Cubes.'' Internat. Math. Res. Not., 181-184, 1995.
© 1996-9 Eric W. Weisstein