Classification P5
| 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
P
5(1.1)
A = 0
5 = 0
P
5(1. >1)
A = nihil
P
5(2.1)
A = 2
5 = 32 =
P5(2.1)Z
P
5(2. >1)
A = nihil
P
5(3.1)
A = 3
5 = 243 =
P5(3.1)Z
P
5(3. >1)
A = nihil
P
5(4.1)
A = 4
5 = 1024
P
5(4. >1)
A = nihil
P
5(5.1)
A = 7
5 = 16807
P
5(5. >1)
A = nihil
P
5(6.1)
A = 14
5 = 537824 =
P5(6.1)Z
P
5(6. >1)
A = nihil
P
5(7.1)
A = 16
5 = 1048576 =
P5(7.1)Z
P
5(7. >1)
A = nihil
P
5(8.1)
A = 38
5 = 79235168 =
P5(8.1)Z
P
5(8.2)
A = nihil
P
5(8.3)
A = 49995
5 = 312343781246875156246875
P
5(8.4)
A = nihil
P
5(8.5)
A = 75418384
5 = 2439979134100773706931016420916722663424 =
P5(8.5)Z
P
5(8.6)
A = 2592877410
5 = 117195225794292252449115584887987847895470100000
P
5(8.7)
A = ?
5 = ?
P
5(9.1)
A = nihil
P
5(9.2)
A = 2955
5 = 225313610074846875 =
P5(9.2)Z
P
5(9.3)
A = 193125
5 = 268653488211536407470703125 =
P5(9.3)Z
P
5(9.4)
A = 10365589
5 = 119665765800843104737370354851986949
P
5(9.5)
A = 631564185
5 = 100481814610246818738257752346024507337365625
P
5(9.6)
A = ?
5 = ?
P
5(10.1)
A = nihil
P
5(10.2)
A = nihil
P
5(10.3)
A = 643905
5 = 110690152879433875483274690625
P
5(10.4)
A = 63174039
5 = 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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E-mail address : pdg@worldofnumbers.com