User:Aadenboy/Self-equaling squares

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In base 10, you may notice that 5 and 6 to any power will always result in a number that ends in itself. This also applies to 0 and 1, however that goes for any base, and aren't interesting. I wanted to know what these numbers would be in other bases, and generalized the problem to searching for digits in base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d^2 \equiv d \ \mathrm{mod}\ n, \ \forall \ d < n} . I only need to check Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d^2} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d^x \equiv d \times d \times \ldots \times d^2} .

After making this article, User:PkmnQ pointed out that the amount of self-equaling squares in a base is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^{\omega(n)}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega{n}} is the amount of unique prime factors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} : i.e, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega{36} = 2} since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 36 = 2^2 * 3^2} , with only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3} as its unique prime factors.

For the first 100 bases...

Base Total Digits
2 2 0, 1
3 2 0, 1
4 2 0, 1
5 2 0, 1
6 4 0, 1, 3, 4
7 2 0, 1
8 2 0, 1
9 2 0, 1
10 4 0, 1, 5, 6
11 2 0, 1
12 4 0, 1, 4, 9
13 2 0, 1
14 4 0, 1, 7, 8
15 4 0, 1, 6, 10
16 2 0, 1
17 2 0, 1
18 4 0, 1, 9, 10
19 2 0, 1
20 4 0, 1, 5, 16
21 4 0, 1, 7, 15
22 4 0, 1, 11, 12
23 2 0, 1
24 4 0, 1, 9, 16
25 2 0, 1
26 4 0, 1, 13, 14
27 2 0, 1
28 4 0, 1, 8, 21
29 2 0, 1
30 8 0, 1, 6, 10, 15, 16, 21, 25
31 2 0, 1
32 2 0, 1
33 4 0, 1, 12, 22
34 4 0, 1, 17, 18
35 4 0, 1, 15, 21
36 4 0, 1, 9, 28
37 2 0, 1
38 4 0, 1, 19, 20
39 4 0, 1, 13, 27
40 4 0, 1, 16, 25
41 2 0, 1
42 8 0, 1, 7, 15, 21, 22, 28, 36
43 2 0, 1
44 4 0, 1, 12, 33
45 4 0, 1, 10, 36
46 4 0, 1, 23, 24
47 2 0, 1
48 4 0, 1, 16, 33
49 2 0, 1
50 4 0, 1, 25, 26
51 4 0, 1, 18, 34
52 4 0, 1, 13, 40
53 2 0, 1
54 4 0, 1, 27, 28
55 4 0, 1, 11, 45
56 4 0, 1, 8, 49
57 4 0, 1, 19, 39
58 4 0, 1, 29, 30
59 2 0, 1
60 8 0, 1, 16, 21, 25, 36, 40, 45
61 2 0, 1
62 4 0, 1, 31, 32
63 4 0, 1, 28, 36
64 2 0, 1
65 4 0, 1, 26, 40
66 8 0, 1, 12, 22, 33, 34, 45, 55
67 2 0, 1
68 4 0, 1, 17, 52
69 4 0, 1, 24, 46
70 8 0, 1, 15, 21, 35, 36, 50, 56
71 2 0, 1
72 4 0, 1, 9, 64
73 2 0, 1
74 4 0, 1, 37, 38
75 4 0, 1, 25, 51
76 4 0, 1, 20, 57
77 4 0, 1, 22, 56
78 8 0, 1, 13, 27, 39, 40, 52, 66
79 2 0, 1
80 4 0, 1, 16, 65
81 2 0, 1
82 4 0, 1, 41, 42
83 2 0, 1
84 8 0, 1, 21, 28, 36, 49, 57, 64
85 4 0, 1, 35, 51
86 4 0, 1, 43, 44
87 4 0, 1, 30, 58
88 4 0, 1, 33, 56
89 2 0, 1
90 8 0, 1, 10, 36, 45, 46, 55, 81
91 4 0, 1, 14, 78
92 4 0, 1, 24, 69
93 4 0, 1, 31, 63
94 4 0, 1, 47, 48
95 4 0, 1, 20, 76
96 4 0, 1, 33, 64
97 2 0, 1
98 4 0, 1, 49, 50
99 4 0, 1, 45, 55
100 4 0, 1, 25, 76

On its own, this is cool, but I then was interested about which bases have the highest amount of self-equaling squares. Filtering this so that only bases whose totals are greater than all before it...

Base Total Digits
2 2 0, 1
6 4 0, 1, 3, 4
30 8 0, 1, 6, 10, 15, 16, 21, 25
210 16 0, 1, 15, 21, 36, 70, 85, 91, 105, 106, 120, 126, 141, 175, 190, 196
2310 32 0, 1, 210, 231, 330, 385, 441, 540, 561, 595, 616, 715, 771, 826, 925, 946, 1155, 1156, 1365, 1386, 1485, 1540, 1596, 1695, 1716, 1750, 1771, 1870, 1926, 1981, 2080, 2101
30030 64 0, 1, 715, 1365, 1716, 2080, 2640, 2926, 3081, 4005, 4291, 5005, 6006, 6370, 6721, 6930, 7371, 7645, 8086, 8295, 8646, 9010, 10011, 10725, 11011, 11935, 12090, 12376, 12936, 13300, 13651, 14301, 15015, 15016, 15730, 16380, 16731, 17095, 17655, 17941, 18096, 19020, 19306, 20020, 21021, 21385, 21736, 21945, 22386, 22660, 23101, 23310, 23661, 24025, 25026, 25740, 26026, 26950, 27105, 27391, 27951, 28315, 28666, 29316
510510 128 0, 1, 715, 9010, 11935, 12376, 13651, 34035, 37401, 38676, 46410, 47125, 47685, 51051, 51766, 58345, 60061, 60775, 62986, 71995, 85086, 85800, 94095, 97020, 97461, 98176, 98736, 107185, 109396, 110110, 111826, 118405, 123046, 132210, 136851, 143430, 145146, 145860, 148071, 156520, 157080, 157795, 158236, 161161, 169456, 170170, 183261, 192270, 194481, 195195, 196911, 203490, 204205, 207571, 208131, 208846, 216580, 217855, 221221, 241605, 242880, 243321, 246246, 254541, 255255, 255256, 255970, 264265, 267190, 267631, 268906, 289290, 292656, 293931, 301665, 302380, 302940, 306306, 307021, 313600, 315316, 316030, 318241, 327250, 340341, 341055, 349350, 352275, 352716, 353431, 353991, 362440, 364651, 365365, 367081, 373660, 378301, 387465, 392106, 398685, 400401, 401115, 403326, 411775, 412335, 413050, 413491, 416416, 424711, 425425, 438516, 447525, 449736, 450450, 452166, 458745, 459460, 462826, 463386, 464101, 471835, 473110, 476476, 496860, 498135, 498576, 501501, 509796
9699690 256 0, 1, 62986, 157795, 203490, 217855, 241605, 293931, 307021, 352716, 367081, 510511, 570571, 573496, 633556, 728365, 752115, 766480, 804441, 812175, 864501, 877591, 969970, 1032955, 1081081, 1119196, 1144066, 1182181, 1263900, 1276990, 1322685, 1337050, 1375011, 1385671, 1413126, 1480480, 1540540, 1543465, 1603525, 1616616, 1629706, 1679601, 1689766, 1692691, 1752751, 1774410, 1834470, 1847560, 1896181, 1923636, 1956241, 1983696, 2051050, 2089165, 2114035, 2127126, 2152150, 2187186, 2190111, 2200276, 2250171, 2263261, 2344980, 2355640, 2383095, 2466751, 2494206, 2504866, 2586585, 2599675, 2649570, 2659735, 2662660, 2697696, 2722720, 2735811, 2760681, 2798796, 2866150, 2893605, 2926210, 2953665, 3002286, 3015376, 3075436, 3097095, 3157155, 3160080, 3170245, 3220140, 3233230, 3246321, 3306381, 3309306, 3369366, 3436720, 3464175, 3474835, 3512796, 3527161, 3572856, 3585946, 3667665, 3705780, 3730650, 3768765, 3816891, 3879876, 3972255, 3985345, 4037671, 4045405, 4083366, 4097731, 4121481, 4216290, 4276350, 4279275, 4339335, 4482765, 4497130, 4542825, 4555915, 4608241, 4631991, 4646356, 4692051, 4786860, 4849845, 4849846, 4912831, 5007640, 5053335, 5067700, 5091450, 5143776, 5156866, 5202561, 5216926, 5360356, 5420416, 5423341, 5483401, 5578210, 5601960, 5616325, 5654286, 5662020, 5714346, 5727436, 5819815, 5882800, 5930926, 5969041, 5993911, 6032026, 6113745, 6126835, 6172530, 6186895, 6224856, 6235516, 6262971, 6330325, 6390385, 6393310, 6453370, 6466461, 6479551, 6529446, 6539611, 6542536, 6602596, 6624255, 6684315, 6697405, 6746026, 6773481, 6806086, 6833541, 6900895, 6939010, 6963880, 6976971, 7001995, 7037031, 7039956, 7050121, 7100016, 7113106, 7194825, 7205485, 7232940, 7316596, 7344051, 7354711, 7436430, 7449520, 7499415, 7509580, 7512505, 7547541, 7572565, 7585656, 7610526, 7648641, 7715995, 7743450, 7776055, 7803510, 7852131, 7865221, 7925281, 7946940, 8007000, 8009925, 8020090, 8069985, 8083075, 8096166, 8156226, 8159151, 8219211, 8286565, 8314020, 8324680, 8362641, 8377006, 8422701, 8435791, 8517510, 8555625, 8580495, 8618610, 8666736, 8729721, 8822100, 8835190, 8887516, 8895250, 8933211, 8947576, 8971326, 9066135, 9126195, 9129120, 9189180, 9332610, 9346975, 9392670, 9405760, 9458086, 9481836, 9496201, 9541896, 9636705
223092870 512 0, 1, 367081, 728365, 864501, 1144066, 1322685, 1540540, 2187186, 2466751, 5616325, 6032026, 6113745, 6393310, 7039956, 7050121, 7354711, 7436430, 7715995, 7803510, 7852131, 8083075, 8219211, 8580495, 8895250, 8947576, 9541896, 9903180, 11623326, 12299365, 12946011, 13468455, 14196820, 14486550, 14612521, 14791140, 15630616, 15935206, 16302286, 16384005, 16663570, 16799706, 17475745, 18122391, 19706401, 20203821, 20518576, 20879860, 21246940, 21526506, 21551530, 21893586, 22048950, 22569625, 22705761, 22985326, 23067045, 23371635, 23738715, 24882781, 27142830, 27558531, 27955005, 28234570, 28286896, 28881216, 29099071, 29466151, 29827435, 30132025, 30421755, 30474081, 30788836, 31150120, 31286256, 31565821, 32319210, 34715395, 35362041, 35641606, 35723325, 36038080, 36139026, 36453781, 36535500, 36815065, 36902580, 37182145, 37461711, 37828791, 38046646, 38225265, 38504830, 38640966, 39002250, 39369331, 42045081, 43078035, 43295890, 43890210, 43942536, 44222101, 44618575, 44985655, 45034276, 45401356, 45483075, 45762640, 46409286, 46724041, 46805760, 47085325, 47221461, 48805471, 50128156, 50625576, 50650600, 50992656, 51297246, 51658530, 51668695, 51973285, 52315341, 52470705, 52837785, 53566150, 53981851, 54160470, 55304536, 56241900, 57385966, 57564585, 57980286, 58708651, 59075731, 59231095, 59573151, 59877741, 59887906, 60249190, 60553780, 60895836, 60920860, 61418280, 62740965, 64324975, 64461111, 64740676, 64822395, 65137150, 65783796, 66063361, 66145080, 66512160, 66560781, 66927861, 67324335, 67603900, 67656226, 68250546, 68468401, 69501355, 72177105, 72544186, 72905470, 73041606, 73321171, 73499790, 73717645, 74084725, 74364291, 74643856, 74731371, 75010936, 75092655, 75407410, 75508356, 75823111, 75904830, 76184395, 76831041, 79227226, 79980615, 80260180, 80396316, 80757600, 81072355, 81124681, 81414411, 81719001, 82080285, 82447365, 82665220, 83259540, 83311866, 83591431, 83987905, 84403606, 86663655, 87807721, 88174801, 88479391, 88561110, 88840675, 88976811, 89497486, 89652850, 89994906, 90019930, 90299496, 90666576, 91027860, 91342615, 91840035, 93424045, 94070691, 94746730, 94882866, 95244150, 95611230, 95915820, 96755296, 97059886, 97349616, 99923110, 101643256, 102004540, 102598860, 102651186, 103742926, 104110006, 104506480, 105153126, 105514410, 109359250, 110005896, 110402370, 111546436, 111913516, 112274800, 112410936, 112869120, 114013186, 117162760, 117660180, 118596556, 118901146, 119262430, 119398566, 119629510, 119765646, 120126930, 123845800, 124492446, 125014890, 126158956, 127930440, 129022180, 129668826, 131252836, 131750256, 134116060, 134252196, 134613480, 134918070, 135285150, 136429216, 139104966, 139501440, 140645506, 141012586, 141373870, 141678460, 141968190, 142020516, 143112256, 146261830, 146908476, 147269760, 148000216, 148361500, 148728580, 149008146, 149375226, 149771700, 150915766, 153591516, 154624470, 155768536, 156165010, 156532090, 157029510, 158270476, 158631760, 158767896, 160351906, 163215130, 163519720, 163861776, 164017140, 164384220, 165528286, 169111020, 170255086, 170622166, 170777530, 171119586, 171424176, 174287400, 175871410, 176007546, 176368830, 177609796, 178107216, 178474296, 178870770, 180014836, 181047790, 183723540, 184867606, 185264080, 185631160, 185910726, 186277806, 186639090, 187369546, 187730830, 188377476, 192671116, 193265436, 193626720, 193993800, 195534340, 199354156, 199721236, 204970480, 209624416, 210146860, 214512376, 214873660, 215009796, 215240740, 215376876, 215738160, 220626120

There are two very interesting patterns. Each base is the multiple of the first Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} primes, and the amount of self-equaling squares is exactly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^{n-1}} .

Base Primes Digits
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2} 2
6 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3} 4
30 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5} 8
210 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7} 16
2310 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11} 32
30030 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13} 64
510510 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17} 128
9699690 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19} 256
223092870 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23} 512
6469693230 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29} 1024
200560490130 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31} 2048
7420738134810 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37} 4096
304250263527210 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41} 8192
13082761331670030 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43} 16384
614889782588491410 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47} 32768
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