Classification P3

The Making of an Exhaustive List

The smallest solutions

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P₃(1.1)^A = 0³ = 0
P₃(1. >1)^A = nihil

P₃(2.1)^A = 3³ = 27
P₃(2.2)^A = 11³ = 1331 = P₃(2.2)^Z
P₃(2. >2)^A = nonexistant ?

P₃(3.1)^A = 5³ = 125
P₃(3.2)^A = 62³ = 238328 = P₃(3.2)^Z
P₃(3.3)^A = 888³ = 700227072 = P₃(3.3)^Z
P₃(3. >3)^A = nonexistant ?

P₃(4.1)^A = 12³ = 1728
P₃(4.2)^A = 303³ = 27818127 = P₃(4.2)^Z
P₃(4. >2)^A = nonexistant ?

P₃(5.1)^A = 22³ = 10648
P₃(5.2)^A = 1412³ = 2815166528
P₃(5.3)^A = 56592³ = 181244621426688
P₃(5.4)^A = 2830479³ = 22676697737363992239 = P₃(5.4)^Z
P₃(5.5)^A = 120023142³ = 1728999927211172788179288
P₃(5.6)^A = 6351783105³ = 256263633328535368685258882625
P₃(5.7)^Z = 267745815817³ = 19194114355415391344355399945943513 (GR)

P₃(6.1)^A = 59³ = 205379
P₃(6.2)^A = 4755³ = 107510668875
P₃(6.3)^A = 496941³ = 122719757596965621
P₃(6.4)^A = 47802109³ = 109229808820515515981029
P₃(6.5)^A = 4849795571³ = 114069699564596150060045954411

P₃(7.1)^A = 135³ = 2460375
P₃(7.2)^A = 22111³ = 10809986553631
P₃(7.3)^A = 4654191³ = 100816729790622689871
P₃(7.4)^A = 1001802781³ = 1005418098917074987897545541

P₃(8.1)^A = 289³ = 24137569
P₃(8.2)^A = 110185³ = 1337726800581625
P₃(8.3)^A = 46440546³ = 100159454656890483891336
P₃(8.4)^A = 21546921816³ = 10003585938508415641831399466496

P₃(9.1)^A = nihil
P₃(9.2)^A = 470095³ = 103885969226107375
P₃(9.3)^A = 464209617³ = 100032794277593492545888113
P₃(9.4)^A = 464165579022³ = 100004327671547355628148253326718648

P₃(10.1)^A = nihil
P₃(10.2)^A = 2158479³ = 10056421854778936239
P₃(10.3)^A = 4642110594³ = 100033726751278963952981464584

The largest solutions

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P₃(1.1)^Z = 2³ = 8
P₃(1. >1)^Z = nihil

P₃(2.1)^Z = 4³ = 64
P₃(2.2)^Z = 11³ = 1331 = P₃(2.2)^A
P₃(2. >2)^Z = nihil ?

P₃(3.1)^Z = 9³ = 729
P₃(3.2)^Z = 62³ = 238328 = P₃(3.2)^A
P₃(3.3)^Z = 888³ = 700227072 = P₃(3.3)^A
P₃(3. >3)^Z = nonexistant ?

P₃(4.1)^Z = 21³ = 9261
P₃(4.2)^Z = 303³ = 27818127 = P₃(4.2)^A
P₃(4. >2;2)^Z = nonexistant ?

P₃(5.1)^Z = 41³ = 68921
P₃(5.2)^Z = 2078³ = 8972978552
P₃(5.3)^Z = 78183³ = 477899958554487
P₃(5.4)^Z = 2830479³ = 22676697737363992239 = P₃(5.4)^A
P₃(5.5)^Z = 131158825³ = 2256277705056006702765625
P₃(5.6)^Z = 8439649482³ = 601136681133830360316800808168

P₃(6.1)^Z = 97³ = 912673
P₃(6.2)^Z = 9709³ = 915215787829
P₃(6.3)^Z = 988458³ = 965772115692567912
P₃(6.4)^Z = 99259996³ = 977963756545433564479936
P₃(6.5)^Z = 9946706804³ = 984097094780740676997868806464

P₃(7.1)^Z = 203³ = 8365427
P₃(7.2)^Z = 45675³ = 95287441921875
P₃(7.3)^Z = 9964932³ = 989516449813334165568
P₃(7.4)^Z = 2152735824³ = 9976362336324497457957556224

P₃(8.1)^Z = 319³ = 32461759
P₃(8.2)^Z = 210916³ = 9382716173855296
P₃(8.3)^Z = 99785364³ = 993574730695766088308544
P₃(8.4)^Z = 46415089682³ = 99994838144343656268511251322568

P₃(9.1)^Z = nihil
P₃(9.2)^Z = 982617³ = 948752253468679113
P₃(9.3)^Z = 999774573³ = 999323871440541388225070517
P₃(9.4)^Z = 999989603912³ = 999968812060234813516120343055446528

P₃(10.1)^Z = nihil
P₃(10.2)^Z = 4631793³ = 99368200745116834257
P₃(10.3)^Z = 9999257781³ = 999777350826262438514680310541

The total number of solutions

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P₃(1.1)^T = 3

0³ = 0

1³ = 1

2³ = 8

P₃(1. >1)^T = 0    all nonexistant ?

P₃(2.1)^T = 2

3³ = 27

4³ = 64

P₃(2.2)^T = 1    a unique solution !

11³ = 1331

P₃(2. >2)^T = 0    all nonexistant ?

P₃(3.1)^T = 4

5³ = 125

6³ = 216

8³ = 512

9³ = 729

P₃(3.2)^T = 1    a unique solution !

62³ = 238328

P₃(3.3)^T = 1    a unique solution !

888³ = 700227072

P₃(3. >3)^T = 0    all nonexistant ?

P₃(4.1)^T = 7

12³ = 1728  |  18³ = 5832

13³ = 2197  |  19³ = 6859

16³ = 4096  |  21³ = 9261

17³ = 4913

P₃(4.2)^T = 1    a unique solution !

303³ = 27818127

P₃(4. >2)^T = 0    all nonexistant ?

P₃(5.1)^T = 8

22³ = 10648  |  32³ = 32768

24³ = 13824  |  35³ = 42875

27³ = 19683  |  38³ = 54872

29³ = 24389  |  41³ = 68921

P₃(5.2)^T = 3

1412³ = 2815166528

1694³ = 4861163384

2078³ = 8972978552

P₃(5.3)^T = 5

56592³ = 181244621426688

58524³ = 200448128101824

65577³ = 282003588255033

70869³ = 355933540044909

78183³ = 477899958554487

P₃(5.4)^T = 1    a unique solution !

2830479³ = 22676697737363992239

P₃(5.5)^T = 2

120023142³ = 1728999927211172788179288

131158825³ = 2256277705056006702765625

P₃(5.6)^T = 4

6351783105³ = 256263633328535368685258882625

7364180055³ = 399367937595765965775933666375

8001100080³ = 512211244405555444021120512000

8439649482³ = 601136681133830360316800808168

P₃(6.1)^T = 10

59³ = 205379  |  76³ = 438976

66³ = 287496  |  84³ = 592704

69³ = 328509  |  88³ = 681472

73³ = 389017  |  93³ = 804357

75³ = 421875  |  97³ = 912673

P₃(6.2)^T = 8

4755³ = 107510668875  |  6443³ = 267463420307

5399³ = 157376536199  |  9078³ = 748118742552

6181³ = 236143627741  |  9413³ = 834034807997

6274³ = 246963938824  |  9709³ = 915215787829

P₃(6.3)^T = 19
P₃(6.4)^T = 45

P₃(7.1)^T = 3

135³ = 2460375

145³ = 3048625

203³ = 8365427

P₃(7.2)^T = 28
P₃(7.3)^T = 110

P₃(8.1)^T = 4

289³ = 24137569

297³ = 26198073

302³ = 27543608

319³ = 32461759

P₃(8.2)^T = 41
P₃(8.3)^T = 831

P₃(9.1)^T = 0
P₃(9.2)^T = 74
P₃(9.3)^T = 6858

P₃(10.1)^T = 0
P₃(10.2)^T = 138
P₃(10.3)^T = ?

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Here is my collection of such numbers with palindromic cuberoots.
0³ = 0
1³ = 1
2³ = 8

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