的士數

Template:NoteTA Template:Not 第$n$個的士數（Template:Lang），一般寫作$\mathrm{Ta}(n)$或$\mathrm{Taxicab}(n)$，定義為最小的數能以$n$個不同的方法表示成兩個正立方數之和。1938年，G·H·哈代與愛德華·梅特蘭·賴特證明對於所有正整數$n$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（Template:Oeis）：

$\begin{array}{cc}\mathrm{Ta}(1)=2& =1^3+1^3\end{array}$

$\begin{array}{cc}\mathrm{Ta}(2)=1729& =1^3+12^3\\ & =9^3+10^3\end{array}$

$\begin{array}{cc}\mathrm{Ta}(3)=87539319& =167^3+436^3\\ & =228^3+423^3\\ & =255^3+414^3\end{array}$

$\begin{array}{cc}\mathrm{Ta}(4)=6963472309248& =2421^3+19083^3\\ & =5436^3+18948^3\\ & =10200^3+18072^3\\ & =13322^3+16630^3\end{array}$

$\begin{array}{cc}\mathrm{Ta}(5)=48988659276962496& =38787^3+365757^3\\ & =107839^3+362753^3\\ & =205292^3+342952^3\\ & =221424^3+336588^3\\ & =231518^3+331954^3\end{array}$

$\begin{array}{cc}\mathrm{Ta}(6)=24153319581254312065344& =582162^3+28906206^3\\ & =3064173^3+28894803^3\\ & =8519281^3+28657487^3\\ & =16218068^3+27093208^3\\ & =17492496^3+26590452^3\\ & =18289922^3+26224366^3\end{array}$

$\mathrm{Ta}(2)$因為哈代和拉馬努金的故事而為人所知： Template:Cquote 在$\mathrm{Ta}(2)$之後，所有的的士數均用電腦來尋找。

Ta(6)的找尋

• Template:Lang證明了$\mathrm{Ta}(6)\le 8230545258248091551205888$。
• 1998年Template:Tsl證實$391909274215699968\ge \mathrm{Ta}(6)\ge 10^{18}$
• 2002年Template:Lang證明$\mathrm{Ta}(6)\le 24153319581254312065344$
• 2003年5月，Template:Lang確定$\mathrm{Ta}(6)>6.8\times 10^{19}$，且Template:Lang、Template:Lang及Template:Lang顯示$\mathrm{Ta}(6)=24153319581254312065344$的機會大於99%。

參考文獻

• 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 = x³ + y³ = z³ + w³ = u³ + v³ = m³ + n³, Bull. Inst. Math. Appl., 27(1991) 155-157; MR 92i:11134, online Template:Wayback
• David W. Wilson, The Fifth Taxicab Number is 48988659276962496, Journal of Integer Sequences, Vol. 2 (1999), online Template:Wayback
• 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

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

• A 2002 post to the Number Theory mailing list by Randall L. Rathbun Template:Wayback
• Template:Cite video
• Taxicab and other maths at Euler Template:Wayback
• Template:Cite web