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 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 |