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Smallest positive integer written as the sum of two integer cubes in n ways
In number theory , the n -th cabtaxi number , typically denoted Cabtaxi(n ) , is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways.[ 1] n , which follows from the analogous result for taxicab numbers .
Known cabtaxi numbers [ edit ] Only 10 cabtaxi numbers are known (sequence A047696 OEIS ):
C
a
b
t
a
x
i
(
1
)
=
1
=
1
3
+
0
3
C
a
b
t
a
x
i
(
2
)
=
91
=
3
3
+
4
3
=
6
3
−
5
3
C
a
b
t
a
x
i
(
3
)
=
728
=
6
3
+
8
3
=
9
3
−
1
3
=
12
3
−
10
3
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
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
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
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
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
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
C
a
b
t
a
x
i
(
10
)
=
933528127886302221000
=
8387730
3
+
7002840
3
=
8444345
3
+
6920095
3
=
9773330
3
−
84560
3
=
9781317
3
−
1318317
3
=
9877140
3
−
3109470
3
=
10060050
3
−
4389840
3
=
10852660
3
−
7011550
3
=
18421650
3
−
17454840
3
=
41337660
3
−
41154750
3
=
77480130
3
−
77428260
3
{\displaystyle {\begin{aligned}\mathrm {Cabtaxi} (1)=&\ 1\\&=1^{3}+0^{3}\\[6pt]\mathrm {Cabtaxi} (2)=&\ 91\\&=3^{3}+4^{3}\\&=6^{3}-5^{3}\\[6pt]\mathrm {Cabtaxi} (3)=&\ 728\\&=6^{3}+8^{3}\\&=9^{3}-1^{3}\\&=12^{3}-10^{3}\\[6pt]\mathrm {Cabtaxi} (4)=&\ 2741256\\&=108^{3}+114^{3}\\&=140^{3}-14^{3}\\&=168^{3}-126^{3}\\&=207^{3}-183^{3}\\[6pt]\mathrm {Cabtaxi} (5)=&\ 6017193\\&=166^{3}+113^{3}\\&=180^{3}+57^{3}\\&=185^{3}-68^{3}\\&=209^{3}-146^{3}\\&=246^{3}-207^{3}\\[6pt]\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}\\[6pt]\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}\\[6pt]\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}\\[6pt]\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}\\[6pt]\mathrm {Cabtaxi} (10)=&\ 933528127886302221000\\&=8387730^{3}+7002840^{3}\\&=8444345^{3}+6920095^{3}\\&=9773330^{3}-84560^{3}\\&=9781317^{3}-1318317^{3}\\&=9877140^{3}-3109470^{3}\\&=10060050^{3}-4389840^{3}\\&=10852660^{3}-7011550^{3}\\&=18421650^{3}-17454840^{3}\\&=41337660^{3}-41154750^{3}\\&=77480130^{3}-77428260^{3}\end{aligned}}}
Cabtaxi(2) was known to François Viète and Pietro Bongo in the late 16th century in the equivalent form
3
3
+
4
3
+
5
3
=
6
3
{\displaystyle 3^{3}+4^{3}+5^{3}=6^{3}}
Leonhard Euler , but its actual solution was not found until later, by Edward B. Escott in 1902.[ 1]
Cabtaxi(4) through and Cabtaxi(7) were found by Randall L. Rathbun in 1992; Cabtaxi(8) was found by Daniel J. Bernstein in 1998. Cabtaxi(9) was found by Duncan Moore in 2005, using Bernstein's method.[ 1] Christian Boyer in 2006 and verified as Cabtaxi(10) by Uwe Hollerbach and reported on the NMBRTHRY mailing list on May 16, 2008.
![{\displaystyle {\begin{aligned}\mathrm {Cabtaxi} (1)=&\ 1\\&=1^{3}+0^{3}\\[6pt]\mathrm {Cabtaxi} (2)=&\ 91\\&=3^{3}+4^{3}\\&=6^{3}-5^{3}\\[6pt]\mathrm {Cabtaxi} (3)=&\ 728\\&=6^{3}+8^{3}\\&=9^{3}-1^{3}\\&=12^{3}-10^{3}\\[6pt]\mathrm {Cabtaxi} (4)=&\ 2741256\\&=108^{3}+114^{3}\\&=140^{3}-14^{3}\\&=168^{3}-126^{3}\\&=207^{3}-183^{3}\\[6pt]\mathrm {Cabtaxi} (5)=&\ 6017193\\&=166^{3}+113^{3}\\&=180^{3}+57^{3}\\&=185^{3}-68^{3}\\&=209^{3}-146^{3}\\&=246^{3}-207^{3}\\[6pt]\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}\\[6pt]\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}\\[6pt]\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}\\[6pt]\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}\\[6pt]\mathrm {Cabtaxi} (10)=&\ 933528127886302221000\\&=8387730^{3}+7002840^{3}\\&=8444345^{3}+6920095^{3}\\&=9773330^{3}-84560^{3}\\&=9781317^{3}-1318317^{3}\\&=9877140^{3}-3109470^{3}\\&=10060050^{3}-4389840^{3}\\&=10852660^{3}-7011550^{3}\\&=18421650^{3}-17454840^{3}\\&=41337660^{3}-41154750^{3}\\&=77480130^{3}-77428260^{3}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/756f9fc5ecc3c7142fd736b9f7f8b34afc24b1ac)
