User:Aadenboy/Self-equaling squares

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 $n$ where $d^{2}\equiv d\ \mathrm {mod} \ n,\ \forall \ d<n$. I only need to check $d^{2}$ as $d^{x}\equiv d\times d\times \ldots \times d^{2}$.

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 $n$ primes, and the amount of self-equaling squares is exactly $2^{n-1}$.

Base	Primes	Digits
2	$2$	2
6	$2\times 3$	4
30	$2\times 3\times 5$	8
210	$2\times 3\times 5\times 7$	16
2310	$2\times 3\times 5\times 7\times 11$	32
30030	$2\times 3\times 5\times 7\times 11\times 13$	64
510510	$2\times 3\times 5\times 7\times 11\times 13\times 17$	128
9699690	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19$	256
223092870	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23$	512
6469693230	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29$	1024
200560490130	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31$	2048
7420738134810	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31\times 37$	4096
304250263527210	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31\times 37\times 41$	8192
13082761331670030	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31\times 37\times 41\times 43$	16384
614889782588491410	$2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29\times 31\times 37\times 41\times 43\times 47$	32768