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Número cabtaxi , tipicamente denotado por Cabtaxi(n) Números cabtaxi existem para todo n ≥ 1.[ 1]
Difere-se do Número taxicab já que este compreende apenas a soma de dois cubos positivos, já o Número cabtaxi engloba ainda os cubos nulos ou negativos.
Até abril de 2014, eram 10 os Números cabtaxi conhecidos (sequência A047696 OEIS ), a saber:
C
a
b
t
a
x
i
(
1
)
=
1
=
1
3
+
0
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (1)&=&1&=&1^{3}+0^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
2
)
=
91
=
3
3
+
4
3
=
6
3
−
5
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (2)&=&91&=&3^{3}+4^{3}\\&&&=&6^{3}-5^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
3
)
=
728
=
6
3
+
8
3
=
9
3
−
1
3
=
12
3
−
10
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (3)&=&728&=&6^{3}+8^{3}\\&&&=&9^{3}-1^{3}\\&&&=&12^{3}-10^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
4
)
=
2741256
=
108
3
+
114
3
=
140
3
−
14
3
=
168
3
−
126
3
=
207
3
−
183
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (4)&=&2741256&=&108^{3}+114^{3}\\&&&=&140^{3}-14^{3}\\&&&=&168^{3}-126^{3}\\&&&=&207^{3}-183^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
5
)
=
6017193
=
166
3
+
113
3
=
180
3
+
57
3
=
185
3
−
68
3
=
209
3
−
146
3
=
246
3
−
207
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (5)&=&6017193&=&166^{3}+113^{3}\\&&&=&180^{3}+57^{3}\\&&&=&185^{3}-68^{3}\\&&&=&209^{3}-146^{3}\\&&&=&246^{3}-207^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
6
)
=
1412774811
=
963
3
+
804
3
=
1134
3
−
357
3
=
1155
3
−
504
3
=
1246
3
−
805
3
=
2115
3
−
2004
3
=
4746
3
−
4725
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (6)&=&1412774811&=&963^{3}+804^{3}\\&&&=&1134^{3}-357^{3}\\&&&=&1155^{3}-504^{3}\\&&&=&1246^{3}-805^{3}\\&&&=&2115^{3}-2004^{3}\\&&&=&4746^{3}-4725^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
7
)
=
11302198488
=
1926
3
+
1608
3
=
1939
3
+
1589
3
=
2268
3
−
714
3
=
2310
3
−
1008
3
=
2492
3
−
1610
3
=
4230
3
−
4008
3
=
9492
3
−
9450
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (7)&=&11302198488&=&1926^{3}+1608^{3}\\&&&=&1939^{3}+1589^{3}\\&&&=&2268^{3}-714^{3}\\&&&=&2310^{3}-1008^{3}\\&&&=&2492^{3}-1610^{3}\\&&&=&4230^{3}-4008^{3}\\&&&=&9492^{3}-9450^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
8
)
=
137513849003496
=
22944
3
+
50058
3
=
36547
3
+
44597
3
=
36984
3
+
44298
3
=
52164
3
−
16422
3
=
53130
3
−
23184
3
=
57316
3
−
37030
3
=
97290
3
−
92184
3
=
218316
3
−
217350
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (8)&=&137513849003496&=&22944^{3}+50058^{3}\\&&&=&36547^{3}+44597^{3}\\&&&=&36984^{3}+44298^{3}\\&&&=&52164^{3}-16422^{3}\\&&&=&53130^{3}-23184^{3}\\&&&=&57316^{3}-37030^{3}\\&&&=&97290^{3}-92184^{3}\\&&&=&218316^{3}-217350^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
9
)
=
424910390480793000
=
645210
3
+
538680
3
=
649565
3
+
532315
3
=
752409
3
−
101409
3
=
759780
3
−
239190
3
=
773850
3
−
337680
3
=
834820
3
−
539350
3
=
1417050
3
−
1342680
3
=
3179820
3
−
3165750
3
=
5960010
3
−
5956020
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (9)&=&424910390480793000&=&645210^{3}+538680^{3}\\&&&=&649565^{3}+532315^{3}\\&&&=&752409^{3}-101409^{3}\\&&&=&759780^{3}-239190^{3}\\&&&=&773850^{3}-337680^{3}\\&&&=&834820^{3}-539350^{3}\\&&&=&1417050^{3}-1342680^{3}\\&&&=&3179820^{3}-3165750^{3}\\&&&=&5960010^{3}-5956020^{3}\end{matrix}}}
C
a
b
t
a
x
i
(
10
)
=
933528127886302221000
=
77480130
3
−
77428260
3
=
41337660
3
−
41154750
3
=
18421650
3
−
17454840
3
=
10852660
3
−
7011550
3
=
10060050
3
−
4389840
3
=
9877140
3
−
3109470
3
=
9781317
3
−
1318317
3
=
9773330
3
−
84560
3
=
8444345
3
+
6920095
3
=
8387730
3
+
7002840
3
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (10)&=&933528127886302221000&=&77480130^{3}-77428260^{3}\\&&&=&41337660^{3}-41154750^{3}\\&&&=&18421650^{3}-17454840^{3}\\&&&=&10852660^{3}-7011550^{3}\\&&&=&10060050^{3}-4389840^{3}\\&&&=&9877140^{3}-3109470^{3}\\&&&=&9781317^{3}-1318317^{3}\\&&&=&9773330^{3}-84560^{3}\\&&&=&8444345^{3}+6920095^{3}\\&&&=&8387730^{3}+7002840^{3}\end{matrix}}}
Graficamente
n
Ca(n )
a^3+b^3
Descobridor
1
1
1,0
2
91
3,4
3
728
6,8
4
2741256
108,114
5
6017193
166,113
Randall L. Rathbun
6
1412774811
963,804
Randall L. Rathbun
7
11302198488
1926,1608
Randall L. Rathbun
8
137513849003496
22944,50058
Daniel J. Bernstein
9
424910390480793000
645210,538680
Duncan Moore
Referências 
