提示 ：此条目页的主题不是
士的數 。
第
n
{\displaystyle n}
的士數 （Taxicab number ），一般寫作
Ta
(
n
)
{\displaystyle \operatorname {Ta} (n)}
Taxicab
(
n
)
{\displaystyle \operatorname {Taxicab} (n)}
n
{\displaystyle n}
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
{\displaystyle n}
A011541
Ta
(
1
)
=
2
=
1
3
+
1
3
{\displaystyle {\begin{aligned}\operatorname {Ta} (1)=2&=1^{3}+1^{3}\end{aligned}}}
Ta
(
2
)
=
1729
=
1
3
+
12
3
=
9
3
+
10
3
{\displaystyle {\begin{aligned}\operatorname {Ta} (2)=1729&=1^{3}+12^{3}\\&=9^{3}+10^{3}\end{aligned}}}
Ta
(
3
)
=
87539319
=
167
3
+
436
3
=
228
3
+
423
3
=
255
3
+
414
3
{\displaystyle {\begin{aligned}\operatorname {Ta} (3)=87539319&=167^{3}+436^{3}\\&=228^{3}+423^{3}\\&=255^{3}+414^{3}\end{aligned}}}
Ta
(
4
)
=
6963472309248
=
2421
3
+
19083
3
=
5436
3
+
18948
3
=
10200
3
+
18072
3
=
13322
3
+
16630
3
{\displaystyle {\begin{aligned}\operatorname {Ta} (4)=6963472309248&=2421^{3}+19083^{3}\\&=5436^{3}+18948^{3}\\&=10200^{3}+18072^{3}\\&=13322^{3}+16630^{3}\end{aligned}}}
Ta
(
5
)
=
48988659276962496
=
38787
3
+
365757
3
=
107839
3
+
362753
3
=
205292
3
+
342952
3
=
221424
3
+
336588
3
=
231518
3
+
331954
3
{\displaystyle {\begin{aligned}\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{aligned}}}
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
{\displaystyle {\begin{aligned}\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{aligned}}}
Ta
(
2
)
{\displaystyle \operatorname {Ta} (2)}
拉馬努金 的故事而為人所知：
“
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
1729
{\displaystyle 1729}
1729
{\displaystyle \color {blue}{1729}}
1729
=
1
3
+
12
3
=
9
3
+
10
3
{\displaystyle 1729=1^{3}+12^{3}=9^{3}+10^{3}}
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
Ta
(
2
)
{\displaystyle \operatorname {Ta} (2)}
電腦 來尋找。
David W. Wilson 證明了
Ta
(
6
)
≤
8230545258248091551205888
{\displaystyle \operatorname {Ta} (6)\leq 8230545258248091551205888}
1998年丹尼爾·朱利阿斯·伯恩斯坦
391909274215699968
≥
Ta
(
6
)
≥
10
18
{\displaystyle 391909274215699968\geq \operatorname {Ta} (6)\geq 10^{18}}
2002年Randall L. Rathbun 證明
Ta
(
6
)
≤
24153319581254312065344
{\displaystyle \operatorname {Ta} (6)\leq 24153319581254312065344}
2003年5月，Stuart Gascoigne 確定
Ta
(
6
)
>
6.8
×
10
19
{\displaystyle \operatorname {Ta} (6)>6.8\times 10^{19}}
Cristian S. Calude 、Elena Calude 及Michael J. Dinneen 顯示
Ta
(
6
)
=
24153319581254312065344
{\displaystyle \operatorname {Ta} (6)=24153319581254312065344}
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 （页面存档备份 ，存于互联网档案馆 ）
David W. Wilson, The Fifth Taxicab Number is 48988659276962496 , Journal of Integer Sequences, Vol. 2 (1999), online （页面存档备份 ，存于互联网档案馆 ）
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
