In mathematics , the -th taxicab number is defined as the smallest (natural) number that can be represented in various ways as the sum of two cubic numbers . Godfrey Harold Hardy and EM Wright have shown that for every natural number there is a taxicab number. However, the evidence says nothing about the occurrence of these numbers, so they can only be found with great (computer-aided) effort.
n
{\ displaystyle n}
n
{\ displaystyle n}
n
{\ displaystyle n}
It owes its name to a famous anecdote by Hardy. He visited Ramanujan at the bedside and mentioned that he had come in a taxi number 1729, which Hardy thought was an uninteresting number . Ramanujan did not find this by showing Hardy the properties mentioned above.
Well-known taxicab numbers
The following six taxicab numbers are known (sequence A011541 in OEIS ):
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
(
4th
)
=
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
(
6th
)
=
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}}}
Upper bounds on taxicab numbers
Upper bounds are known for the following six taxicab numbers:
Ta
(
7th
)
≤
24885189317885898975235988544
=
2648660966
3
+
1847282122
3
=
2685635652
3
+
1766742096
3
=
2736414008
3
+
1638024868
3
=
2894406187
3
+
860447381
3
=
2915734948
3
+
459531128
3
=
2918375103
3
+
309481473
3
=
2919526806
3
+
58798362
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (7) & \ leq & 24885189317885898975235988544 & = & 2648660966 ^ {3} + 1847282122 ^ {3} \\ &&& = & 2685635652 ^ {3} + 1766742096 ^ {3} \\ &&& = & 2736414008 ^ {3} + 1638024868 ^ {3} \\ &&& = & 2894406187 ^ {3} + 860447381 ^ {3} \\ &&& = & 2915734948 ^ {3} + 459531128 ^ {3} \\ &&& = & 2918375103 ^ { 3} + 309481473 ^ {3} \\ &&& = & 2919526806 ^ {3} + 58798362 ^ {3} \ end {matrix}}}
Ta
(
8th
)
≤
50974398750539071400590819921724352
=
299512063576
3
+
288873662876
3
=
336379942682
3
+
234604829494
3
=
341075727804
3
+
224376246192
3
=
347524579016
3
+
208029158236
3
=
367589585749
3
+
109276817387
3
=
370298338396
3
+
58360453256
3
=
370633638081
3
+
39304147071
3
=
370779904362
3
+
7467391974
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (8) & \ leq & 50974398750539071400590819921724352 & = & 299512063576 ^ {3} + 288873662876 ^ {3} \\ &&& = & 336379942682 ^ {3} + 234604829494 ^ {3} + 234604829494 &&& = & 341075727804 ^ {3} + 224376246192 ^ {3} \\ &&& = & 347524579016 ^ {3} + 208029158236 ^ {3} \\ &&& = & 367589585749 ^ {3} + 109276817387 ^ {3} \\ &&& = & 370298338396 3} + 58360453256 ^ {3} \\ &&& = & 370633638081 ^ {3} + 39304147071 ^ {3} \\ &&& = & 370779904362 ^ {3} + 7467391974 ^ {3} \ end {matrix}}}
Ta
(
9
)
≤
136897813798023990395783317207361432493888
=
41632176837064
3
+
40153439139764
3
=
46756812032798
3
+
32610071299666
3
=
47409526164756
3
+
31188298220688
3
=
48305916483224
3
+
28916052994804
3
=
51094952419111
3
+
15189477616793
3
=
51471469037044
3
+
8112103002584
3
=
51518075693259
3
+
5463276442869
3
=
51530042142656
3
+
4076877805588
3
=
51538406706318
3
+
1037967484386
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (9) & \ leq & 136897813798023990395783317207361432493888 & = & 41632176837064 ^ {3} + 40153439139764 ^ {3} \\ &&& = & 46756812032798 ^ {312} + \ 3263812032798 ^ {312} + 326 &&& = & 47409526164756 ^ {3} + 31188298220688 ^ {3} \\ &&& = & 48305916483224 ^ {3} + 28916052994804 ^ {3} \\ &&& = & 51094952419111 ^ {3} + 15189477616793 ^ {390} \\47 & ^& { 3} + 8112103002584 ^ {3} \\ &&& = & 51518075693259 ^ {3} + 5463276442869 ^ {3} \\ &&& = & 51530042142656 ^ {3} + 4076877805588 ^ {3} \\ &&& = & 51538406706318 ^ {3 }4 10484067063189 ^ {3} \ end {matrix}}}
Ta
(
10
)
≤
7335345315241855602572782233444632535674275447104
=
15695330667573128
3
+
15137846555691028
3
=
17627318136364846
3
+
12293996879974082
3
=
17873391364113012
3
+
11757988429199376
3
=
18211330514175448
3
+
10901351979041108
3
=
19262797062004847
3
+
5726433061530961
3
=
19404743826965588
3
+
3058262831974168
3
=
19422314536358643
3
+
2059655218961613
3
=
19426825887781312
3
+
1536982932706676
3
=
19429379778270560
3
+
904069333568884
3
=
19429979328281886
3
+
391313741613522
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (10) & \ leq & 7335345315241855602572782233444632535674275447104 & = & 15695330667573128 ^ {3} + 15137846555691028 ^ {3} \\ &&& {}} + 124631899636 ^\}}} + 12463188213636 &&& = & 17873391364113012 ^ {3} + 11757988429199376 ^ {3} \\ &&& = & 18211330514175448 ^ {3} + 10901351979041108 ^ {3} \\ &&& = & 19262797062004847 ^ {3} + 5726433061530961 & {3} + 5726433061530961 ^&38440941 ^&38 3} + 3058262831974168 ^ {3} \\ &&& = & 19422314536358643 ^ {3} + 2059655218961613 ^ {3} \\ &&& = & 19426825887781312 ^ {3} + 1536982932706676 ^ {3} \\ &&& {38270529409 ^ {3} {3} \\ &&& = & 19429979328281886 ^ {3} + 391313741613522 ^ {3} \ end {matrix}}}
Ta
(
11
)
≤
2818537360434849382734382145310807703728251895897826621632
=
11410505395325664056
3
+
11005214445987377356
3
=
12815060285137243042
3
+
8937735731741157614
3
=
12993955521710159724
3
+
8548057588027946352
3
=
13239637283805550696
3
+
7925282888762885516
3
=
13600192974314732786
3
+
6716379921779399326
3
=
14004053464077523769
3
+
4163116835733008647
3
=
14107248762203982476
3
+
2223357078845220136
3
=
14120022667932733461
3
+
1497369344185092651
3
=
14123302420417013824
3
+
1117386592077753452
3
=
14125159098802697120
3
+
657258405504578668
3
=
14125594971660931122
3
+
284485090153030494
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (11) & \ leq & 2818537360434849382734382145310807703728251895897826621632 & = & 11410505395325664056 ^ {3} + 11005214445987377356 ^ {3} \\ 24506028377356 ^ {3} \\245030428 = ^ {3} \\ 2450304277356 ^ {3} \\ 245030428 = ^ {3} \\ 2450304277356 ^ {3} \\ 24504 037 356} {3} \\ 24504502 8356 ^ {3} \\ 245060283 &&& = & 12993955521710159724 ^ {3} + 8548057588027946352 ^ {3} \\ &&& = & 13239637283805550696 ^ {3} + 7925282888762885516 ^ {3} \\ &&& = & 13600192974314732799863200 {3} + {3} \\ &&& = & 13600192974314732799863200 {3\ {3} + {3} 3} + 4163116835733008647 ^ {3} \\ &&& = & 14107248762203982476 ^ {3} + 2223357078845220136 ^ {3} \\ &&& = & 14120022667932733461 ^ {3} + 1497369344185092651 ^ {320} \\ &&& = ^ {3} + 1497369344185092651 ^ {320} \ 1470220 && 114532417} \\\ 14753247} {3} \\ &&& = & 14125159098802697120 ^ {3} + 657258405504578668 ^ {3} \\ &&& = & 14125594971660931122 ^ {3} + 284485090153030494 ^ {3} \ end {matrix}}}
Ta
(
12
)
≤
73914858746493893996583617733225161086864012865017882136931801625152
=
33900611529512547910376
3
+
32696492119028498124676
3
=
38073544107142749077782
3
+
26554012859002979271194
3
=
38605041855000884540004
3
+
25396279094031028611792
3
=
39334962370186291117816
3
+
23546015462514532868036
3
=
40406173326689071107206
3
+
19954364747606595397546
3
=
41606042841774323117699
3
+
12368620118962768690237
3
=
41912636072508031936196
3
+
6605593881249149024056
3
=
41950587346428151112631
3
+
4448684321573910266121
3
=
41960331491058948071104
3
+
3319755565063005505892
3
=
41965847682542813143520
3
+
1952714722754103222628
3
=
41965889731136229476526
3
+
1933097542618122241026
3
=
41967142660804626363462
3
+
845205202844653597674
3
{\ displaystyle {\ begin {matrix} \ operatorname {Ta} (12) & \ leq & 73914858746493893996583617733225161086864012865017882136931801625152 & = & 33900611529512547910376 ^ {3 }7 37627910376 ^ {3 }7 & # 39; & # 39; 39649211902849812} {\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\!!! &&& = & 38605041855000884540004 ^ {3} + 25396279094031028611792 ^ {3} \\ &&& = & 39334962370186291117816 ^ {3} + 23546015462514532868036 ^ {3} \\ &&&& = {3} \\ &&&&& = & 40406173326474} = 4146015462514532868037 3} + 12368620118962768690237 ^ {3} \\ &&& = & 41912636072508031936196 ^ {3} + 6605593881249149024056 ^ {3} \\ &&& = & 41950587346428151112631 ^ {3} \\ &&& = &&&&&&&&&&&&&&&&&&&&&&&&& = {3} \\ &&& = & 41965847682542813143520 ^ {3} + 1952714722754103222628 ^ {3} \\ &&& = & 41965889731136229476526 ^ {3} + 1933097542618122241026 ^ {3} \\ &&4460 \ 8356752} + {3} \\ &&4460 8359720264} {3} {3} \\ &&4460 \ 835465736} {matrix}}}
Discovery story
Ta (2) = 1,729 is also known as the Hardy-Ramanujan number due to the anecdote above ; it was published in 1657 by Bernard Frénicle de Bessy .
Ta (3) = 87,539,319 was discovered by John Leech in 1957 .
Ta (4) was found in 1991 by the amateur number theorist E. Rosenstiel
Ta (5) has been owed to David W. Wilson since 1999. Independently of that, she found Daniel Bernstein a few months later .
Ta (6) was discovered in 2003. Daniel Bernstein had previously specified an upper bound in 1998.
Generalized taxicab number
As a generalized Taxicab numbers is called a modification of the ordinary Taxicab numbers. The definition is:
Taxicab
(
k
,
j
,
n
)
{\ displaystyle \ operatorname {Taxicab} (k, y, n)}
is the smallest natural number that can be expressed in various ways using terms of -th powers .
n
{\ displaystyle n}
j
{\ displaystyle j}
k
{\ displaystyle k}
For and these are the "normal" taxicab numbers.
k
=
3
{\ displaystyle k = 3}
j
=
2
{\ displaystyle j = 2}
Leonhard Euler showed that:
T
a
x
i
c
a
b
(
4th
,
2
,
2
)
=
635.318.657
=
59
4th
+
158
4th
=
133
4th
+
134
4th
{\ displaystyle \ mathrm {Taxicab} (4,2,2) = 635,318,657 = 59 ^ {4} + 158 ^ {4} = 133 ^ {4} + 134 ^ {4}}
.
An unsolved problem in mathematics is an existence theorem for values other than and . No solutions were found for these values even with computer assistance. This problem is related to Euler's Conjecture , a generalization of Fermat's Great Theorem .
k
=
3
{\ displaystyle k = 3}
j
=
2
{\ displaystyle j = 2}
literature
Web links
Individual evidence
^ Godfrey Harold Hardy, Edward Maitland Wright: An introduction to the theory of numbers. Oxford UP, 4th edition 1975, p. 333, Theorem 412, with annotations p. 338f. The first edition is from 1938.
↑
Hardy: Ramanujan , London 1940. Hardy wrote literally:
“I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways. '”
Quotations by GH Hardy ( Memento of the original from July 16, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1 @ 2 Template: Webachiv / IABot / www-gap.dcs.st-and.ac.uk
^ Christian Boyer: New Upper Bounds for Taxicab and Cabtaxi Numbers
↑ Bruce Berndt , S. Bhargava: Ramanujan - For Lowbrows . In: American Mathematical Monthly , Volume 100, 1993, pp. 645-656.
^ J. Leech: Some Solutions of Diophantine Equations. In: Proc. Cambridge Phil. Soc. , 531957, pp. 778-780.
↑ E. Rosenstiel, JA Dardis, CR Rosenstiel: The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation In: Bull. Inst. Math. Appl. , 271991, pp. 155-157.
s
=
x
3
+
y
3
=
z
3
+
w
3
=
u
3
+
v
3
=
m
3
+
n
3
{\ displaystyle s = x ^ {3} + y ^ {3} = z ^ {3} + w ^ {3} = u ^ {3} + v ^ {3} = m ^ {3} + n ^ { 3}}
^ DW Wilson: The Fifth Taxicab Number is 48988659276962496 . In: J. Integer Sequences 2, # 99.1.9, 1999.
↑ CS Calude, E. Calude, MJ Dinneen: What Is the Value of Taxicab (6)? (PDF) In: J. Uni. Comp. Sci. , 9, 2003, pp. 1196-1203
^ Richard K. Guy : Unsolved problems in number theory (third edition) . Springer Science and Business Media, New York 2004, ISBN 0-387-20860-7 , p. 437.
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