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Introduction
Details on wikipedia.
Refers to A011541 OEIS sequence, and A001235 OEIS sequence for non-primitives solutions.
See also Boyer's site and Boyer's 2008 PDF paper (or archive).
See also Po-Chi Su's 2016 PDF paper.

Taxicab(n) - or Ta(n) or T(n) or T(n, 1) or Taxicab(3, n, n) - is the smallest number expressible as a sum of two positive cubes in n different ways.
For n > 6, as described in Boyer's paper, there is an upper bound list, as BTa(n) described in OEIS sequence.

Ta(n) <= BTa(n) = r * s, with r < s
r = x + y = sqrt(3xy + s), with x < y
s = x² - xy + y² = x² + y * (y - x) = r² - 3xy = xy + (y - x)² = xy + delta_b
delta_a = 12s - 3r², delta_a = 9 * delta_b
delta_b = (4s - r²) / 3 = r² - 4xy = (y - x)²
x = (3r - sqrt(delta_a)) / 6 = (r - sqrt(delta_b)) / 2
y = (3r + sqrt(delta_a)) / 6 = (r + sqrt(delta_b)) / 2
sqrt(delta_a) = 3 * (y - x) = 3 * (s - x²) / y
xy = (r² - s) / 3
y - x = sqrt(delta_a) / 3 = sqrt(delta_b)

Note that s can also be written: s = (y - x)² + x * y
then for each {x,y}, Ta(n) = sum * ( diff² + product )
and Ta(n) = r³ - 3xyr then sum² - diff² = 4 * product

Let x_{(n, i)} and y_{(n, i)} denote the i th x and y for BTa(n) or Ta(n).

If x or y is odd (but not both):

w = (y + x + 1) / 2 = (r + 1) / 2
v = (y - x - 1) / 2 = y - w = (sqrt(delta_a) - 1) / 6
Ta(n) <= BTa(n) = (w - v - 1)³ + (v + w)³

else (if x and y are both even or are both odd)

w = (x + y) / 2 = r / 2
v = (y - x) / 2 = y - w = sqrt(delta_a) / 6
Ta(n) <= BTa(n) = (w - v)³ + (v + w)³

Decomposition of Ta(n) or BTa(n)

• source code contains all {x,y} values with {r,s} and delta_a components for checking matches
• text output of the program execution shows if values are correct
• taxicab_x contains x values
• taxicab_y contains y values
• taxicab_r contains r values
• taxicab_s contains s values

Relationships

• BTa(24) <= 381489286124430409319768596454988766303873589432584556125228221326120950488322054793438654594075836788861479033831645270800877473265863391069226260470316824340088274129522154177976000000000

  • z = 59 * 1543 = 91037
  • BTa(24) = z^3 * BTa(23)

• BTa(23) <= 505624992450102221744244678441689845434887715317085055675501100061664233777550644179891369792998312033624027623507268993575520089167244066787797454194149200985506392000000000

  • z = 97 * 491 = 47627
  • BTa(23) = z^3 * BTa(22)

• BTa(22) <= 4680247859298792255465896583920257161776853980641720488623945439873944558056759119053262369879441902130153472288312996410513410535472957059046536164424000000000

  • z = 11 * 31 * 103 = 35123
  • BTa(22) = z^3 * BTa(21)

• BTa(21) <= 108017480260055891349349434078007118409816211596074421874782013152781209964290521967447172971431469705047741717793222992124981940703226072000000000

  • z = 127 * 197 = 25019
  • BTa(21) = z^3 * BTa(20)
  • {x_{(21, i)}, y_{(21, i)}} = z * {x_{(20, i)}, y_{(20, i)}} (except for i=7)

• BTa(20) <= 6897380753715950027554119762858815120976857451913869723850868496922922856256366585322070924221550029921862628606999892276808000000000

  • z = 5^2 * 457 * 521 = 5952425
  • BTa(20) = z^3 * BTa(18)
  • {x_{(20, i)}, y_{(20, i)}} = z * {x_{(18, i)}, y_{(18, i)}} (except for i=14)

• BTa(19) <= 89155799373431424956779039683208074195259067508030416952016859893056008200986665364956362114476625742414288155368000000000

• BTa(18) <= 32704115261272222010108668321371490956498583891096244871994833214348230808345220932546725287574509356813791744000

  • z = 37 * 181 = 6697
  • BTa(18) = z^3 * BTa(17)
  • {x_{(18, i)}, y_{(18, i)}} = z * {x_{(17, i)}, y_{(17, i)}} (except for i=1)

• BTa(17) <= 108883358434560363503260016942467566965657808716401027019007732065428469239764403684536752685040128000

  • z = 4261
  • BTa(17) = z^3 * BTa(16)
  • {x_{(17, i)}, y_{(17, i)}} = z * {x_{(16, i)}, y_{(16, i)}} (except for i=3)

• BTa(16) <= 1407430328457240141244921568479580896498768360005757901394557724136294559835243494681088000

  • z = 2 * 607 = 1214
  • BTa(16) = z^3 * BTa(15)
  • {x_{(16, i)}, y_{(16, i)}} = z * {x_{(15, i)}, y_{(15, i)}} (except for i=12)

• BTa(15) <= 786630615595626829796137755567437844038832146564530660386728812791938555036352000

  • z = 503
  • BTa(15) = z^3 * BTa(14)
  • {x_{(15, i)}, y_{(15, i)}} = z * {x_{(14, i)}, y_{(14, i)}} (except for i=4)

• BTa(14) <= 6181115942163278484307116174514304039670628856329988877227399275142976000

  • z = 397
  • BTa(14) = z^3 * BTa(13)
  • {x_{(14, i)}, y_{(14, i)}} = z * {x_{(13, i)}, y_{(13, i)}} (except for i=6)

• BTa(13) <= 98785992977316717572070208794037178343163969803121800608526912000

  • z = 3 * 61 = 183
  • BTa(13) = z^3 * BTa(12)
  • {x_{(13, i)}, y_{(13, i)}} = z * {x_{(12, i)}, y_{(12, i)}} (except for i=4)

• BTa(12) <= 16119148654034302034428760115512552827992287460693283776000

  • z = 3 * 19 = 57
  • BTa(12) = z^3 * BTa(11)
  • {x_{(12, i)}, y_{(12, i)}} = z * {x_{(11, i)}, y_{(11, i)}} (except for i=2)
  • {x_(12,12), y_(12,12)} ​​do not contain 7
  • {x_{(12, 5)}, y_{(12, 5)}} ​​do not contain 13
  • {x_{(12, 4)}, y_{(12, 4)}} ​​do not contain 17

• BTa(11) <= 87039729655193781808322993393446581825405320183232000

  • {x_(11,11), y_(11,11)} ​​do not contain 7
  • {x_{(11, 4)}, y_{(11, 4)}} ​​do not contain 13
  • {x_{(11, 3)}, y_{(11, 3)}} ​​do not contain 17

• BTa(10) <= 7335345315241855602572782233444632535674275447104

  • z = 13 * 29 = 377
  • BTa(10) = z^3 * BTa(9)
  • {x_{(10, i)}, y_{(10, i)}} = z * {x_{(9, i)}, y_{(9, i)}} (except for i=2)
  • {x₂, y₂} ​​do not contain 13 * 29
  • {x₃, y₃} ​​do not contain 139
  • {x₅, y₅} ​​do not contain 101

• BTa(9) <= 136897813798023990395783317207361432493888

  • z = 139
  • BTa(9) = z^3 * BTa(8)
  • {x_{(9, i)}, y_{(9, i)}} = z * {x_{(8, i)}, y_{(8, i)}} (except for i=2)

• BTa(8) <= 50974398750539071400590819921724352

  • z = 127
  • BTa(8) = z^3 * BTa(7)
  • {x_{(8, i)}, y_{(8, i)}} = z * {x_{(7, i)}, y_{(7, i)}} (except for i=8)

• BTa(7) <= 24885189317885898975235988544

  • z = 101
  • BTa(7) = z^3 * BTa(6)
  • {x_{(7, i)}, y_{(7, i)}} = z * {x_{(6, i)}, y_{(6, i)}} (except for i=3)

• Ta(6) = 24153319581254312065344

  • z = 79
  • BTa(6) = z^3 * BTa(5)
  • {x_{(6, i)}, y_{(6, i)}} = z * {x_{(5, i)}, y_{(5, i)}} (except for i=1)

• Ta(5) = 48988659276962496

• Ta(4) = 6963472309248

• Ta(3) = 87539319

  • Ta(3) = (302 - 134 -1)³ + (302 + 134)³ = (326 - 97 - 1)³ + (326 + 97)³ = (335 - 79 - 1)³ + (335 + 79)³

• Ta(2) = 1729

• Ta(1) = 2

Others non-primitive solutions

T(19,?) <= 20400824749409517528805616329601248054238975120047899653306832282155305931553802798551427446267572330642814901020208524571911529017539708401834884288000 = 2^9 * 3^12 * 5^3 * 7^4 * 11^3 * 13^4 * 17^3 * 19 * 29^3 * 41^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 271^3 * 349^3 * 601^3 * 727^3 * 733^3 * 2311^3 * 2971^3 * 4327^3 * 7549^3

T(18,?) <= 1669102760262770599073633207770749663711833995754118800009115774272350540676865796358235477762975575548685960795354954969975483624898697728 = 2^9 * 3^12 * 7^4 * 11^3 * 13^4 * 19 * 29^3 * 41^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 349^3 * 601^3 * 727^3 * 733^3 * 2311^3 * 2971^3 * 4327^3 * 7549^3

T(17,?) <= 390632494425592857308683115941747067399542361030496398592266116103976300384768055853952870509972688279894938439376270144482816 = 2^9 * 3^3 * 7^4 * 11^3 * 13^4 * 19 * 29^3 * 41^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 349^3 * 727^3 * 733^3 * 2311^3 * 2971^3 * 4327^3 * 7549^3

T(16,?) <= 133333879575067044255496407342541909372921060713622256889696224426242257299346008208169434186136206003482924680704 = 2^9 * 3^3 * 7^4 * 11^3 * 13^4 * 19 * 29^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 727^3 * 733^3 * 2311^3 * 2971^3 * 4327^3 * 7549^3

T(15,?) <= 254361007450418467683691805874871224061387429621450309607817376336263717732850126485501423293412997632 = 2^9 * 3^3 * 7^4 * 13^4 * 19 * 29^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 727^3 * 2311^3 * 2971^3 * 4327^3

T(14,?) <= 591265120715306076227178149379165201865336472346517495072293984450950965960475614042157568 = 2^9 * 3^3 * 7^4 * 13^4 * 19 * 29^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 727^3 * 2311^3 * 2971^3 * 4327^3

T(13,?) <= 5988146776742829080553965820313279739849705084894534523771076163371248442670016 = 2^6 * 3^3 * 7^4 * 13^4 * 19 * 29^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157 * 727^3 * 2971^3 * 4327^3

T(7, ?) = 64866613980488624314617087936 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 79^3 * 97 * 139^3 * 157
x₁ = 80920518 = 2 * 3 * 7 * 83 * 139 * 167
x₂ = 317835644 = 2^2 * 7 * 79 * 143687
x₃ = 425920047 = 3 * 7 * 79 * 139 * 1847
x₄ = 1184180059 = 79 * 139 * 107839
x₅ = 2254311452 = 2^2 * 17 * 79 * 139 * 3019
x₆ = 2431456944 = 2^4 * 3 * 7 * 79 * 139 * 659
x₇ = 2542299158 = 2 * 7 * 23 * 79 * 139 * 719
y₇ = 3645186874 = 2 * 7 * 79 * 131 * 139 * 181
y₆ = 3696072828 = 2^2 * 3 * 7 * 79 * 139 * 4007
y₅ = 3765955912 = 2^3 * 79 * 139 * 163 * 263
y₄ = 3983390693 = 79 * 139 * 362753
y₃ = 4016377617 = 3 * 7 * 79 * 139 * 17417
y₂ = 4017310528 = 2^6 * 7 * 11 * 17 * 79 * 607
y₁ = 4017962634 = 2 * 3 * 7 * 139 * 311 * 2213

T(4, ?) = 1801049058342701083 = 7 * 31 * 37 * 43 * 163 * 193 * 9151 * 18121
x₁ = 92227
x₂ = 136635 = 3 * 5 * 9109
x₃ = 341995 = 5 * 68399
x₄ = 600259 = 11 * 197 * 277
y₄ = 1165884 = 2^2 * 3 * 97157
y₃ = 1207602 = 2 * 3^3 * 11 * 19 * 107
y₂ = 1216102 = 2 * 23 * 26437
y₁ = 1216500 = 2^2 * 3 * 5^3 * 811

T(3, ?) = 15170835645 = 3^2 * 5 * 7 * 31 * 37 * 199 * 211
x₁ = 517 = 11 * 47
x₂ = 709
x₃ = 1733
y₃ = 2152 = 2^3 * 269
y₂ = 2456 = 2^3 * 307
y₁ = 2468 = 2^2 * 617

T(3, 4) = 175959000 = 35^3 ∗ T(2, 2)
x₁ = 70 = 2 * 5 * 7
x₂ = 198 = 2 * 3^2 * 11
x₃ = 315 = 3^2 * 5 * 7
y₃ = 525 = 3 * 5^2 * 7
y₂ = 552 = 2^3 * 3 * 23
y₁ = 560 = 2^4 * 5 * 7

T(2, 2) = 4104 = (12 − 3)³ + (12 + 3)³ = 2³ * (1³ + 8³)
x₁ = 2
x₂ = 9 = 3^2
y₂ = 15 = 3 * 5
y₁ = 16 = 2^4