Classification P4

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Numbers whose digits occur with same frequency ~ Classification P4
Class_P2 Class_P3 Class_P5

	Frequency with which the digits occur in P₄
'd' differ. digits	1	2	3	4	5	6	7	...
1	2	0	0	0	0	0	0	...
2	2	0	0	0	0	0	0	...
3	2	0	0	0	0	0	0	...
4	3	0	0	0	0	0	0	...
5	4	1	0	0	0	0	0	...
6	3	0	0	0	0	0	?	...
7	5	1	1	5	2	?	?	...
8	0	3	5	40	?	?	?	...
9	0	2	25	759	?	?	?	...
10	0	4	172	?	?	?	?	...

The Making of an Exhaustive List

The smallest solutions

P₄(1.1)^A = 0⁴ = 0
P₄(1. >1)^A = nihil

P₄(2.1)^A = 2⁴ = 16
P₄(2. >1)^A = nihil

P₄(3.1)^A = 4⁴ = 256
P₄(3. >1)^A = nihil

P₄(4.1)^A = 6⁴ = 1296
P₄(4. >1)^A = nihil

P₄(5.1)^A = 12⁴ = 20736
P₄(5.2)^A = 207⁴ = 1836036801 = P₄(5.2)^Z
P₄(5. >2)^A = nihil

P₄(6.1)^A = 18⁴ = 104976
P₄(6. >1)^A = nihil

P₄(7.1)^A = 32⁴ = 1048576
P₄(7.2)^A = 2452⁴ = 36147799388416 = P₄(7.2)^Z
P₄(7.3)^A = 130398⁴ = 289123718973983667216 = P₄(7.3)^Z
P₄(7.4)^A = 5694207⁴ = 1051315345334684056886604801
P₄(7.5)^A = 426424828⁴ = 33065106952901022329359695121613056
P₄(7.6)^A = 18304641024⁴ = 112265125207703310302573655130362165067776

P₄(8.1)^A = nihil
P₄(8.2)^A = 8692⁴ = 5707933051146496
P₄(8.3)^A = 569058⁴ = 104863930698324110228496
P₄(8.4)^A = 57081391⁴ = 10616422434730826078387783204161
P₄(8.5)^A = 5629546339⁴ = 1004369679090466326141409237273977132241
P₄(8.6)^A = 562379596107⁴ = 100027225332362313560971766691937509957725591601

P₄(9.1)^A = nihil
P₄(9.2)^A = 24267⁴ = 346788239145769521
P₄(9.3)^A = 3182043⁴ = 102523677648570833624514801
P₄(9.4)^A = 562606291⁴ = 100188606547504649933395818497375761
P₄(9.5)^A = 100005244522⁴ = 100020979738358361519703325186685226719978256

P₄(10.1)^A = nihil
P₄(10.2)^A = 69636⁴ = 23514473895962870016
P₄(10.3)^A = 17824719⁴ = 100946384385027947782661539521
P₄(10.4)^A = 5623720662⁴ = 1000218682348975505736229176987914443536

The largest solutions
P₄(1.1)^Z = 1⁴ = 1
P₄(1. >1)^Z = nihil

P₄(2.1)^Z = 3⁴ = 81
P₄(2. >1)^Z = nihil

P₄(3.1)^Z = 5⁴ = 625
P₄(3. >1)^Z = nihil

P₄(4.1)^Z = 8⁴ = 4096
P₄(4. >1)^Z = nihil

P₄(5.1)^Z = 17⁴ = 83521
P₄(5.2)^Z = 207⁴ = 1836036801 = P₄(5.2)^A
P₄(5. >2)^Z = nihil

P₄(6.1)^Z = 25⁴ = 390625
P₄(6. >1)^Z = nihil

P₄(7.1)^Z = 49⁴ = 5764801
P₄(7.2)^Z = 2452⁴ = 36147799388416 = P₄(7.2)^A
P₄(7.3)^Z = 130398⁴ = 289123718973983667216 = P₄(7.3)^A
P₄(7.4)^Z = 9681618⁴ = 8786011595668091975085109776
P₄(7.5)^Z = 487331540⁴ = 56402464255939342395392324966560000
P₄(7.6)^Z = 31159138983⁴ = 942631163354926125294614695396339214455521

P₄(8.1)^Z = nihil
P₄(8.2)^Z = 9595⁴ = 8475784699200625
P₄(8.3)^Z = 977847⁴ = 914289286476569154757281
P₄(8.4)^Z = 99266285⁴ = 97097282557302398839867366500625
P₄(8.5)^Z = 9998070694⁴ = 9992285009045746654672422078758808756496
P₄(8.6)^Z = 999817502946⁴ = 999270211590737036231702317961557666152302935056

P₄(9.1)^Z = nihil
P₄(9.2)^Z = 30059⁴ = 816390822057597361
P₄(9.3)^Z = 5577408⁴ = 967675311854837821442359296
P₄(9.4)^Z = 999592427⁴ = 998370704423712389607141298663608241
P₄(9.5)^Z = 177822924599⁴ = 999887167503511209336573208035869782275261601

P₄(10.1)^Z = nihil
P₄(10.2)^Z = 80892⁴ = 42817597245013360896
P₄(10.3)^Z = 31607052⁴ = 998012460822049787453371535616
P₄(10.4)^Z = 9999112926⁴ = 9996452176112247953600451338048028738576

The total number of solutions

P₄(1.1)^T = 2

0⁴ = 0

1⁴ = 1

P₄(1. >1)^T = 0

P₄(2.1)^T = 2

2⁴ = 16

3⁴ = 81

P₄(2. >1)^T = 0

P₄(3.1)^T = 2

4⁴ = 256

5⁴ = 625

P₄(3. >1)^T = 0

P₄(4.1)^T = 3

6⁴ = 1296

7⁴ = 2401

8⁴ = 4096

P₄(4. >1)^T = 0

P₄(5.1)^T = 4

12⁴ = 20736

13⁴ = 28561

14⁴ = 38416

17⁴ = 83521

P₄(5.2)^T = 1    a unique solution !

207⁴ = 1836036801

P₄(5. >2)^T = 0

P₄(6.1)^T = 3

18⁴ = 104976

23⁴ = 279841

25⁴ = 390625

P₄(6. >1)^T = 0

P₄(7.1)^T = 5

32⁴ = 1048576

38⁴ = 2085136

44⁴ = 3748096

48⁴ = 5308416

49⁴ = 5764801

P₄(7.2)^T = 1    a unique solution !

2452⁴ = 36147799388416

P₄(7.3)^T = 1    a unique solution !

130398⁴ = 289123718973983667216

P₄(7.4)^T = 5

5694207⁴ = 1051315345334684056886604801

7423704⁴ = 3037264324502656425730707456

8255138⁴ = 4644054960561353915900113936

9454392⁴ = 7989772631868111973223632896

9681618⁴ = 8786011595668091975085109776

P₄(7.5)^T = 2

426424828⁴ = 33065106952901022329359695121613056

487331540⁴ = 56402464255939342395392324966560000

P₄(8.1)^T = 0
P₄(8.2)^T = 3

8692⁴ = 5707933051146496

8721⁴ = 5784490950217281

9595⁴ = 8475784699200625

P₄(8.3)^T = 5

569058⁴ = 104863930698324110228496

674778⁴ = 207321173700596599312656

770637⁴ = 352695100362907795721361

977643⁴ = 913526563244942905031601

977847⁴ = 914289286476569154757281

P₄(8.4)^T = 40
P₄(8.5)^T = ?

P₄(9.1)^T = 0
P₄(9.2)^T = 2

24267⁴ = 346788239145769521

30059⁴ = 816390822057597361

P₄(9.3)^T = 25
P₄(9.4)^T = 759
P₄(9.5)^T = ?

P₄(10.1)^T = 0
P₄(10.2)^T = 4

69636⁴ = 23514473895962870016

70215⁴ = 24306341799881750625

77058⁴ = 35259076387041812496

80892⁴ = 42817597245013360896

P₄(10.3)^T = 172
P₄(10.4)^T = ?

Here is my collection of such numbers with palindromic fourth roots.
0⁴ = 0
1⁴ = 1

Contributions

Jeff Heleen (email) from New Hampshire, USA.

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Patrick De Geest - Belgium

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E-mail address : pdg@worldofnumbers.com