查看“︁的士數”︁的源代码

{{noteTA
|T=zh-hans:的士数;zh-hk:的士數;zh-tw:計程車數;
|1=zh-hans:的士;zh-hk:的士;zh-tw:計程車;
}}
{{Not|士的數}}
第<math>n</math>個'''的士數'''（{{lang|en|Taxicab number}}），一般寫作<math>\operatorname{Ta}(n)</math>或<math>\operatorname{Taxicab}(n)</math>，定義為最小的數能以<math>n</math>個不同的方法表示成兩個[[負數|正]][[立方數]]之和。1938年，[[G·H·哈代]]與[[愛德華·梅特蘭·賴特]]證明對於所有[[正整數]]<math>n</math>這樣的數也存在。可是他們的證明對找尋的士數毫無幫助，截止現時，只找到6個的士數（{{oeis|A011541}}）：

:<math>
\begin{align}
\operatorname{Ta}(1) = 2 & = 1^3 + 1^3
\end{align}
</math>

:<math>
\begin{align}
\operatorname{Ta}(2) = 1729 & = 1^3 + 12^3 \\
& = 9^3 + 10^3
\end{align}
</math>

:<math>
\begin{align}
\operatorname{Ta}(3) = 87539319 & = 167^3 + 436^3 \\
& = 228^3 + 423^3 \\
& = 255^3 + 414^3
\end{align}</math>

:<math>
\begin{align}
\operatorname{Ta}(4) = 6963472309248 & = 2421^3 + 19083^3 \\
& = 5436^3 + 18948^3 \\
& = 10200^3 + 18072^3 \\
& = 13322^3 + 16630^3
\end{align}
</math>

:<math>
\begin{align}
\operatorname{Ta}(5) = 48988659276962496 & = 38787^3 + 365757^3 \\
& = 107839^3 + 362753^3 \\
& = 205292^3 + 342952^3 \\
& = 221424^3 + 336588^3 \\
& = 231518^3 + 331954^3
\end{align}
</math>

:<math>
\begin{align}
\operatorname{Ta}(6) = 24153319581254312065344 & = 582162^3 + 28906206^3 \\
& = 3064173^3 + 28894803^3 \\
& = 8519281^3 + 28657487^3 \\
& = 16218068^3 + 27093208^3 \\
& = 17492496^3 + 26590452^3 \\
& = 18289922^3 + 26224366^3
\end{align}
</math>

<math>\operatorname{Ta}(2)</math>因為哈代和[[拉馬努金]]的故事而為人所知：
{{cquote|拉馬努金病重，哈代前往探望。哈代說：「我乘計程車來，車牌號碼是<math>1729</math>，這數真沒趣，希望不是不祥之兆。」拉馬努金答道：「不，那是個有趣得很的數。可以用兩個立方之和來表達而且有兩種表達方式的數之中，[[1729|<math>\color{blue}{1729}</math>]]是最小的。」（即<math>1729 = 1^3 + 12^3 = 9^3 + 10^3</math>，後來這類數稱為[[的士數]]。）利特爾伍德回應這宗軼聞說：「每個整數都是拉馬努金的朋友。」}}
在<math>\operatorname{Ta}(2)</math>之後，所有的的士數均用[[電腦]]來尋找。

== Ta(6)的找尋 ==
* {{lang|en|David W. Wilson}}證明了<math>\operatorname{Ta}(6) \le 8230545258248091551205888</math>。
* 1998年{{tsl|en|Daniel J. Bernstein|丹尼爾·朱利阿斯·伯恩斯坦}}證實<math>391909274215699968 \ge \operatorname{Ta}(6) \ge 10^{18}</math>
* 2002年{{lang|en|Randall L. Rathbun}}證明<math>\operatorname{Ta}(6) \le 24153319581254312065344</math>
* 2003年5月，{{lang|en|Stuart Gascoigne}}確定<math>\operatorname{Ta}(6) > 6.8 \times 10^{19}</math>，且{{lang|en|Cristian S. Calude}}、{{lang|en|Elena Calude}}及{{lang|en|Michael J. Dinneen}}顯示<math>\operatorname{Ta}(6) = 24153319581254312065344</math>的機會大於99%。

==參考文獻==
{{refbegin|2}}
* G. H. Hardy和E. M. Wright, ''An Introduction to the Theory of Numbers'', 3rd ed., Oxford University Press, London & NY, 1954, Thm. 412.
* J. Leech, ''Some Solutions of Diophantine Equations'', Proc. Cambridge Phil. Soc. 53, 778-780, 1957.
* E. Rosenstiel, J. A. Dardis and C. R. Rosenstiel, ''The four least solutions in distinct positive integers of the Diophantine equation s = x<sup>3</sup> + y<sup>3</sup> = z<sup>3</sup> + w<sup>3</sup> = u<sup>3</sup> + v<sup>3</sup> = m<sup>3</sup> + n<sup>3</sup>'', Bull. Inst. Math. Appl., 27(1991) 155-157; MR 92i:11134, [http://www.cix.co.uk/%7Erosenstiel/cubes/welcome.htm online] {{Wayback|url=http://www.cix.co.uk/%7Erosenstiel/cubes/welcome.htm |date=20050302090451 }}
* David W. Wilson, ''The Fifth Taxicab Number is 48988659276962496'', Journal of Integer Sequences, Vol. 2 (1999), [http://www.math.uwaterloo.ca/JIS/wilson10.html#RDR91 online] {{Wayback|url=http://www.math.uwaterloo.ca/JIS/wilson10.html#RDR91 |date=20040422034546 }}
* D. J. Bernstein, ''Enumerating solutions to p(a) + q(b) = r(c) + s(d)'', Mathematics of Computation 70, 233 (2000), 389—394.
* C. S. Calude, E. Calude and M. J. Dinneen: ''What is the value of Taxicab(6)?'', Journal of Universal Computer Science, Vol. 9 (2003), p. 1196-1203
{{refend}}

== 參看 ==
* [[一般化的士數]]：多個多次冪之和
* [[士的數]]：兩個不論正負的立方數之和
* [[三立方数和]]

==外部連結==
* [http://listserv.nodak.edu/scripts/wa.exe?A2=ind0207&L=nmbrthry&F=&S=&P=1278 A 2002 post to the Number Theory mailing list by Randall L. Rathbun] {{Wayback|url=http://listserv.nodak.edu/scripts/wa.exe?A2=ind0207&L=nmbrthry&F=&S=&P=1278 |date=20041010215634 }}
* {{cite video |editor-last=Haran |editor-first=Brady |editor-link=Brady Haran |last1=Grime |first1=James |last2=Bowley |first2=Roger |title=1729: Taxi Cab Number or Hardy-Ramanujan Number |series=Numberphile |url=http://www.numberphile.com/videos/1729taxicab.html |access-date=2020-10-25 |archive-date=2017-03-06 |archive-url=https://web.archive.org/web/20170306141337/http://numberphile.com/videos/1729taxicab.html |dead-url=yes }}
* [http://euler.free.fr/ Taxicab and other maths at Euler] {{Wayback|url=http://euler.free.fr/ |date=20131209163223 }}
* {{cite web |editor-last=Haran |editor-first=Brady |editor-link=Brady Haran |last=Singh |first=Simon |authorlink=Simon Singh |title=Taxicab Numbers in Futurama |series=Numberphile |url=http://www.numberphile.com/videos/futurama.html |access-date=2020-10-25 |archive-date=2015-12-03 |archive-url=https://web.archive.org/web/20151203175119/http://www.numberphile.com/videos/futurama.html |dead-url=yes }}

[[Category:整数数列|D]]
[[Category:斯里尼瓦瑟·拉马努金]]