Taxicab Number

The $n$th taxicab number ${\rm Ta}(n)$ is the smallest number representable in $n$ ways as a sum of Positive Cubes. The numbers derive their name from the Hardy-Ramanujan Number

$${\rm Ta}(2) = 1729$$

$$= 1^3+12^3$$

$$= 9^3+10^3,$$ (1)

which is associated with the following story told about Ramanujan by G. H. Hardy. ``Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, `rather a dull number,' adding that he hoped that wasn't a bad omen. `No, Hardy,' said Ramanujan, `it is a very interesting number. It is the smallest number expressible as the sum of two [Positive] cubes in two different ways''' (Hofstadter 1989, Kanigel 1991, Snow 1993).

However, this property was also known as early as 1657 by F. de Bessy (Berndt and Bhargava 1993, Guy 1994). Leech (1957) found

$${\rm Ta}(3) = 87539319$$

$$= 167^3+436^3$$

$$= 228^3+423^3$$

$$= 255^3+414^3.$$ (2)

Rosenstiel et al. (1991) recently found

$${\rm Ta}(4) = 6963472309248$$

$$= 2421^3+19083^3$$

$$= 5436^3+18948^3$$

$$= 10200^3+18072^3$$

$$= 13322^3+16630^3.$$ (3)

D. Wilson found

$${\rm Ta}(5) = 48988659276962496$$

$$= 38787^3 + 365757^3$$

$$= 107839^3 + 362753^3$$

$$= 205292^3 + 342952^3$$

$$= 221424^3 + 336588^3$$

$$= 231518^3 + 331954^3.$$ (4)

The first few taxicab numbers are therefore 2, 1729, 87539319, 6963472309248, ... (Sloane's A011541).

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 $s=x^3+y^3=z^3+w^3=u^3+v^3=m^3+n^3$.'' 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.