Classification P5

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Numbers whose digits occur with same frequency ~ Classification P5
Class_P2 Class_P3 Class_P4

	Frequency with which the digits occur in P₅
'd' differ. digits	1	2	3	4	5	6	7	...
1	2	0	0	0	0	0	0	...
2	1	0	0	0	0	0	0	...
3	1	0	0	0	0	0	0	...
4	2	0	0	0	0	0	0	...
5	2	0	0	0	0	0	0	...
6	1	0	0	0	0	0	0	...
7	1	0	0	0	0	0	?	...
8	1	0	3	0	1	11	?	...
9	0	1	1	8	84	?	?	...
10	0	0	7	133	?	?	?	...

The Making of an Exhaustive List

The smallest solutions

P₅(1.1)^A = 0⁵ = 0
P₅(1. >1)^A = nihil

P₅(2.1)^A = 2⁵ = 32 = P₅(2.1)^Z
P₅(2. >1)^A = nihil

P₅(3.1)^A = 3⁵ = 243 = P₅(3.1)^Z
P₅(3. >1)^A = nihil

P₅(4.1)^A = 4⁵ = 1024
P₅(4. >1)^A = nihil

P₅(5.1)^A = 7⁵ = 16807
P₅(5. >1)^A = nihil

P₅(6.1)^A = 14⁵ = 537824 = P₅(6.1)^Z
P₅(6. >1)^A = nihil

P₅(7.1)^A = 16⁵ = 1048576 = P₅(7.1)^Z
P₅(7. >1)^A = nihil

P₅(8.1)^A = 38⁵ = 79235168 = P₅(8.1)^Z
P₅(8.2)^A = nihil
P₅(8.3)^A = 49995⁵ = 312343781246875156246875
P₅(8.4)^A = nihil
P₅(8.5)^A = 75418384⁵ = 2439979134100773706931016420916722663424 = P₅(8.5)^Z
P₅(8.6)^A = 2592877410⁵ = 117195225794292252449115584887987847895470100000
P₅(8.7)^A = ?⁵ = ?

P₅(9.1)^A = nihil
P₅(9.2)^A = 2955⁵ = 225313610074846875 = P₅(9.2)^Z
P₅(9.3)^A = 193125⁵ = 268653488211536407470703125 = P₅(9.3)^Z
P₅(9.4)^A = 10365589⁵ = 119665765800843104737370354851986949
P₅(9.5)^A = 631564185⁵ = 100481814610246818738257752346024507337365625
P₅(9.6)^A = ?⁵ = ?

P₅(10.1)^A = nihil
P₅(10.2)^A = nihil
P₅(10.3)^A = 643905⁵ = 110690152879433875483274690625
P₅(10.4)^A = 63174039⁵ = 1006220638584953725761454302767138894199

The largest solutions
P₅(1.1)^Z = 1⁵ = 1
P₅(1. >1)^Z = nihil

P₅(2.1)^Z = 2⁵ = 32 = P₅(2.1)^A
P₅(2. >1)^Z = nihil

P₅(3.1)^Z = 3⁵ = 243 = P₅(3.1)^A
P₅(3. >1)^Z = nihil

P₅(4.1)^Z = 5⁵ = 3125
P₅(4. >1)^Z = nihil

P₅(5.1)^Z = 8⁵ = 32768
P₅(5. >1)^Z = nihil

P₅(6.1)^Z = 14⁵ = 537824 = P₅(6.1)^A
P₅(6. >1)^Z = nihil

P₅(7.1)^Z = 16⁵ = 1048576 = P₅(7.1)^A
P₅(7. >1)^Z = nihil

P₅(8.1)^Z = 38⁵ = 79235168 = P₅(8.1)^A
P₅(8.2)^Z = nihil
P₅(8.3)^Z = 61557⁵ = 883867634090375064499557
P₅(8.4)^Z = nihil
P₅(8.5)^Z = 75418384⁵ = 2439979134100773706931016420916722663424 = P₅(8.5)^A
P₅(8.6)^Z = 3835020303⁵ = 829541919891852538715325897428343419274377521743
P₅(8.7)^Z = ?⁵ = ?

P₅(9.1)^Z = nihil
P₅(9.2)^Z = 2955⁵ = 225313610074846875 = P₅(9.2)^A
P₅(9.3)^Z = 193125⁵ = 268653488211536407470703125 = P₅(9.3)^A
P₅(9.4)^Z = 14768868⁵ = 702645136480123183310487625524077568
P₅(9.5)^Z = 992758715⁵ = 964314153757084568107154987089060693334196875
P₅(9.6)^Z = ?⁵ = ?

P₅(10.1)^Z = nihil
P₅(10.2)^Z = nihil
P₅(10.3)^Z = 984927⁵ = 926872965448613570921013853407
P₅(10.4)^Z = 99900828⁵ = 9950512253367684107778194042198520634368

The total number of solutions

P₅(1.1)^T = 2

0⁵ = 0

1⁵ = 1

P₅(1. >1)^T = 0

P₅(2.1)^T = 1    a unique solution !

2⁵ = 32

P₅(2. >1)^T = 0    all nonexistant ?

P₅(3.1)^T = 1    a unique solution !

3⁵ = 243

P₅(3. >1)^T = 0    all nonexistant ?

P₅(4.1)^T = 2

4⁵ = 1024

5⁵ = 3125

P₅(4. >1)^T = 0    all nonexistant ?

P₅(5.1)^T = 2

7⁵ = 16807

8⁵ = 32768

P₅(5. >1)^T = 0    all nonexistant ?

P₅(6.1)^T = 1    a unique solution !

14⁵ = 537824

P₅(6. >1)^T = 0    all nonexistant ?

P₅(7.1)^T = 1    a unique solution !

16⁵ = 1048576

P₅(7. >1)^T = 0    all nonexistant ?

P₅(8.1)^T = 1    a unique solution !

38⁵ = 79235168

P₅(8.2)^T = 0
P₅(8.3)^T = 3

49995⁵ = 312343781246875156246875

57615⁵ = 634859804603764980759375

61557⁵ = 883867634090375064499557

P₅(8.4)^T = 0
P₅(8.5)^T = 1    a unique solution !

75418384⁵ = 2439979134100773706931016420916722663424

P₅(8.6)^T = 11

2592877410⁵ = 117195225794292252449115584887987847895470100000

2937411918⁵ = 218687652680475264071477052082441458720015161568

3054351006⁵ = 265824327313481538631122845681762715546354487776

3145504479⁵ = 307929672182062108611086171702308879332696738399

3225852489⁵ = 349319251551263912652396312024040050366594601449

3383231856⁵ = 443260270395509692467230060455427333476529995776

3405362379⁵ = 457948529798626755266418971251122416415746887899

3574381257⁵ = 583450975970633003445766344389490869567868798057

3658841442⁵ = 655719271130757967396019103309526657120920563232

3823554789⁵ = 817215487524126241979815788947692466565186572949

3835020303⁵ = 829541919891852538715325897428343419274377521743

P₅(8.7)^T = ?

P₅(9.1)^T = 0
P₅(9.2)^T = 1    a unique solution !

2955⁵ = 225313610074846875

P₅(9.3)^T = 1    a unique solution !

193125⁵ = 268653488211536407470703125

P₅(9.4)^T = 8

10365589⁵ = 119665765800843104737370354851986949

11228103⁵ = 178456322746061358045561821732080743

12036828⁵ = 252673835969632125494758974118714368

12343788⁵ = 286577619853439827259931426441735168

12608302⁵ = 318627322049731980106674841799864032

12736024⁵ = 335096077304295591976171102543642624

14175405⁵ = 572370680423381730540165646748128125

14768868⁵ = 702645136480123183310487625524077568

P₅(9.5)^T = 84
P₅(9.6)^T = ?

P₅(10.1)^T = 0
P₅(10.2)^T = 0
P₅(10.3)^T = 7

643905⁵ = 110690152879433875483274690625

680061⁵ = 145458581697864327220703996301

720558⁵ = 194242706843325709850513196768

775113⁵ = 279785436151445000683198226793

840501⁵ = 419460598737334268928156702501

878613⁵ = 523586118778694132765090044293

984927⁵ = 926872965448613570921013853407

P₅(10.4)^T = 133
P₅(10.5)^T = ?

Here is my collection of such numbers with palindromic fifth roots.
0⁵ = 0
1⁵ = 1

Contributions

Jeff Heleen (email) from New Hampshire, USA.

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Patrick De Geest - Belgium

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