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出租车数

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第 n 个出租车数 Ta(n) 是可以用 n 种方式表示为正立方数之和的最小数。这些数字的名字来源于 哈代-拉马努金数

Ta(2)=1729
(1)
=1^3+12^3
(2)
=9^3+10^3,
(3)

这与 G. H. 哈代 (Hofstadter 1989, Kanigel 1991, Snow 1993) 讲述的关于拉马努金的故事有关。

1729 in Futurama

1729 的这个性质在 2005 年的电影证明中被安东尼·霍普金斯扮演的角色罗伯特（有时是精神失常的数学家）提及。它也是动画电视剧飞出个未来第二季 DVD 2ACV02 集（Greenwald；左图）中出现的 Nimbus BP-1729 宇宙飞船的名称的一部分，以及机器人角色本德的序列号，如圣诞贺卡在圣诞故事一集中所描绘的那样（第 2 卷 DVD，Georgoulias et al. 2004；右图）。

然而，早在 1657 年，F. de Bessy (Berndt and Bhargava 1993, Guy 1994) 就已经知道了这个性质。Leech (1957) 发现

Ta(3)=87539319
(4)
=167^3+436^3
(5)
=228^3+423^3
(6)
=255^3+414^3.
(7)

Rosenstiel et al. (1991) 最近发现

Ta(4)=6963472309248
(8)
=2421^3+19083^3
(9)
=5436^3+18948^3
(10)
=10200^3+18072^3
(11)
=13322^3+16630^3.
(12)

Wilson (1999) 发现

Ta(5)=48988659276962496
(13)
=38787^3+365757^3
(14)
=107839^3+362753^3
(15)
=205292^3+342952^3
(16)
=221424^3+336588^3
(17)
=231518^3+331954^3.
(18)

因此，前几个出租车数是 2, 1729, 87539319, 6963472309248, 48988659276962496, ... (OEIS A011541)。

第六个出租车数是

Ta(6)=24153319581254312065344
(19)
=28906206^3+582162^3
(20)
=28894803^3+3064173^3
(21)
=28657487^3+8519281^3
(22)
=27093208^3+16218068^3
(23)
=26590452^3+17492496^3
(24)
=26224366^3+18289922^3
(25)

(Calude et al. 2003, Hollerbach 2008)。

Hardy 和 Wright (1979 年定理 412) 表明，这种和的数量可以任意大，但是，根据威尔逊的结果更新 Guy (1994) 的结果，对于六个或更多相等的和，最小的例子仍然未知。

Sloane 定义了一种略有不同的出租车数类型，即可以用两种或多种方式表示为两个立方和的数字，其中前几个是 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ... (OEIS A001235)。

另请参见

Cabtaxi 数, 丢番图方程——3 次幂, 哈代-拉马努金数, 出租车度量

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参考文献

Berndt, B. C. and Bhargava, S. "Ramanujan--For Lowbrows." Am. Math. Monthly 100, 645-656, 1993.Butler, B. "Ramanujan Numbers and the Taxicab Problem." http://www.durangobill.com/Ramanujan.html.Calude,C. S.; Calude, E.; and Dinneen, M. J. "What Is the Value of Taxicab(6)?" J. Uni. Comp. Sci. 9, 1196-1203, 2003. http://www.cs.auckland.ac.nz/~cristian/taxicab.pdf.Georgoulias, T.; Greenwald, S. J.; and Wichterich, M. "Futurama pik: Mathematics in the Year 3000." Math Horizons, 12-15, Apr. 2004.Greenwald, S. "Dr. Sarah's Futurama pik--Mathematics in the Year 3000." http://www.mathsci.appstate.edu/~sjg/futurama/.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, pp. 12 and 68, 1999.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.Hollerbach, U. "The Sixth Taxicab Number Is 24153319581254312065344." Mar. 8, 2008. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0803&L=nmbrthry&T=0&F=&S=&P=1059.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.Meyrignac, J. "The Taxicab Problem." http://euler.free.fr/taxicab.htm.Plouffe, S. "Taxicab Numbers." http://pi.lacim.uqam.ca/eng/problem_en.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. Sequences A001235 and A011541 in "The On-Line Encyclopedia of Integer Sequences."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.Wilson, D. W. "The Fifth Taxicab Number is 48988659276962496." J. Integer Sequences 2, #99.1.9, 1999.Wooley, T. D. "Sums of Two Cubes." Internat. Math. Res. Not. No. 4, 181-184, 1995.

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出租车数

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Weisstein, Eric W. "出租车数。" 来自 —— 资源。 https://mathworld.net.cn/TaxicabNumber.html

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