20200920 (Sunday)
6
Raise the x to itself and the result must be an integer: x ^ x = Z
x ^ x = Z
Raise the x to itself and the result must be an integer. The upper case "Z" stands for Zahl ("number" in German language) or Zahlen ("numbers" in German language), but in the large set of numbers, the "Z" represents the integer numbers.
Now, clearing the x is a difficult task, which is achieved by some:
(A) by iterations.
(B) by applying the Newton-Raphson method, which is a root-finding algorithm that produces successively better approximations to the roots (or zeroes) of a real-valued function.
(C) by applying some empirical methods, such as paying many visits to wolframalpha and typing successively in its search box the following equations (one at a time): x ^ x = 1, x ^ x = 2, x ^ x = 3, x ^ x = 4, and so on.
(D) by paying a visit to x ^ x = Z, clearing of x, solutions for Z = from 2 to 99. In such a web page, someone (a programmer, coder, software creator, mathematician, geek, etcetera) entered a pertinent program.
(E) by other methods not considered here.
x ^ x = Z
0 ^ 0 = 1
1 ^ 1 = 1
1.5596104694623693500 ^ 1.5596104694623693500 = 2
1.8254550229248300400 ^ 1.8254550229248300400 = 3
2 ^ 2 = 4
2.1293724827601566964 ^ 2.1293724827601566964 = 5
2.2318286244090093674 ^ 2.2318286244090093674 = 6
2.3164549587856123013 ^ 2.3164549587856123013 = 7
2.3884234844993385564 ^ 2.3884234844993385564 = 8
2.4509539280155796306 ^ 2.4509539280155796306 = 9
2.5061841455887692563 ^ 2.5061841455887692563 = 10
2.5556046121008206153 ^ 2.5556046121008206153 = 11
2.6002950000539155877 ^ 2.6002950000539155877 = 12
2.6410619164843958084 ^ 2.6410619164843958084 = 13
2.6785234858912995813 ^ 2.6785234858912995813 = 14
2.7131636040042392096 ^ 2.7131636040042392096 = 15
2.7453680235674634847 ^ 2.7453680235674634847 = 16
2.7754491049442334313 ^ 2.7754491049442334313 = 17
2.8036632456580215497 ^ 2.8036632456580215497 = 18
2.8302234384970308956 ^ 2.8302234384970308956 = 19
2.8553085030012414128 ^ 2.8553085030012414128 = 20
2.8790699928935659256 ^ 2.8790699928935659256 = 21
2.9016374467499625981 ^ 2.9016374467499625981 = 22
2.9231224361312694255 ^ 2.9231224361312694255 = 23
2.9436217266289826968 ^ 2.9436217266289826968 = 24
2.9632197748934563283 ^ 2.9632197748934563283 = 25
2.9819907219821817027 ^ 2.9819907219821817027 = 26
3 ^ 3 = 27
3.0173056385312550860 ^ 3.0173056385312550860 = 28
3.0339593356304303347 ^ 3.0339593356304303347 = 29
3.0500073424230602731 ^ 3.0500073424230602731 = 30
3.0654911988615763645 ^ 3.0654911988615763645 = 31
3.0804483496557610280 ^ 3.0804483496557610280 = 32
3.0949126630131026190 ^ 3.0949126630131026190 = 33
3.1089148699947203136 ^ 3.1089148699947203136 = 34
3.1224829386046084566 ^ 3.1224829386046084566 = 35
3.1356423938890448229 ^ 3.1356423938890448229 = 36
3.1484165931165360971 ^ 3.1484165931165360971 = 37
3.1608269633816938595 ^ 3.1608269633816938595 = 38
3.1728932076149117966 ^ 3.1728932076149117966 = 39
3.1846334838989551957 ^ 3.1846334838989551957 = 40
3.1960645621301305824 ^ 3.1960645621301305824 = 41
3.2072019613676607979 ^ 3.2072019613676607979 = 42
3.2180600706537838993 ^ 3.2180600706537838993 = 43
3.2286522556309704270 ^ 3.2286522556309704270 = 44
3.2389909529099377650 ^ 3.2389909529099377650 = 45
3.2490877538360709208 ^ 3.2490877538360709208 = 46
3.2589534790493330299 ^ 3.2589534790493330299 = 47
3.2685982450234625521 ^ 3.2685982450234625521 = 48
3.2780315235960593661 ^ 3.2780315235960593661 = 49
3.2872621953555806526 ^ 3.2872621953555806526 = 50
3.2962985976291226481 ^ 3.2962985976291226481 = 51
3.3051485677119986694 ^ 3.3051485677119986694 = 52
3.3138194818931829219 ^ 3.3138194818931829219 = 53
3.3223182907569548994 ^ 3.3223182907569548994 = 54
3.3306515511783378154 ^ 3.3306515511783378154 = 55
3.3388254553763678047 ^ 3.3388254553763678047 = 56
3.3468458573433744041 ^ 3.3468458573433744041 = 57
3.3547182969290745029 ^ 3.3547182969290745029 = 58
3.3624480218243707439 ^ 3.3624480218243707439 = 59
3.3700400076604626243 ^ 3.3700400076604626243 = 60
3.3774989764135260917 ^ 3.3774989764135260917 = 61
3.3848294132832120798 ^ 3.3848294132832120798 = 62
3.3920355821940675324 ^ 3.3920355821940675324 = 63
3.3991215400522833124 ^ 3.3991215400522833124 = 64
3.4060911498755757812 ^ 3.4060911498755757812 = 65
3.4129480929012200816 ^ 3.4129480929012200816 = 66
3.4196958797660251505 ^ 3.4196958797660251505 = 67
3.4263378608421623488 ^ 3.4263378608421623488 = 68
3.4328772358040515954 ^ 3.4328772358040515954 = 69
3.4393170624938175028 ^ 3.4393170624938175028 = 70
3.4456602651460217411 ^ 3.4456602651460217411 = 71
3.4519096420263438355 ^ 3.4519096420263438355 = 72
3.4580678725335237118 ^ 3.4580678725335237118 = 73
3.4641375238091118120 ^ 3.4641375238091118120 = 74
3.4701210568953241348 ^ 3.4701210568953241348 = 75
3.4760208324775074623 ^ 3.4760208324775074623 = 76
3.4818391162443299251 ^ 3.4818391162443299251 = 77
3.4875780838957766712 ^ 3.4875780838957766712 = 78
3.4932398258263085523 ^ 3.4932398258263085523 = 79
3.4988263515080974834 ^ 3.4988263515080974834 = 80
3.5043395935970540472 ^ 3.5043395935970540472 = 81
3.5097814117823834793 ^ 3.5097814117823834793 = 82
3.5151535963986212384 ^ 3.5151535963986212384 = 83
3.5204578718174877087 ^ 3.5204578718174877087 = 84
3.5256958996354445272 ^ 3.5256958996354445272 = 85
3.5308692816715160941 ^ 3.5308692816715160941 = 86
3.5359795627887444703 ^ 3.5359795627887444703 = 87
3.5410282335515612300 ^ 3.5410282335515612300 = 88
3.5460167327303745096 ^ 3.5460167327303745096 = 89
3.5509464496637733925 ^ 3.5509464496637733925 = 90
3.5558187264879358896 ^ 3.5558187264879358896 = 91
3.5606348602420831508 ^ 3.5606348602420831508 = 92
3.5653961048581440529 ^ 3.5653961048581440529 = 93
3.5701036730421746074 ^ 3.5701036730421746074 = 94
3.5747587380545100605 ^ 3.5747587380545100605 = 95
3.5793624353951090450 ^ 3.5793624353951090450 = 96
3.5839158644000741571 ^ 3.5839158644000741571 = 97
3.5884200897548978110 ^ 3.5884200897548978110 = 98
3.5928761429295825100 ^ 3.5928761429295825100 = 99
3.5972850235404175055 ^ 3.5972850235404175055 = 100
3.6016477006428562317 ^ 3.6016477006428562317 = 101
3.6059651139596282584 ^ 3.6059651139596282584 = 102
3.6102381750479334055 ^ 3.6102381750479334055 = 103
3.6144677684093019458 ^ 3.6144677684093019458 = 104
3.6186547525454615427 ^ 3.6186547525454615427 = 105
3.6227999609633269665 ^ 3.6227999609633269665 = 106
3.6269042031320210987 ^ 3.6269042031320210987 = 107
3.6309682653946438386 ^ 3.6309682653946438386 = 108
3.6349929118373279477 ^ 3.6349929118373279477 = 109
3.6389788851179564184 ^ 3.6389788851179564184 = 110
3.6429269072567635547 ^ 3.6429269072567635547 = 111
3.6468376803909006198 ^ 3.6468376803909006198 = 112
3.6507118874949157384 ^ 3.6507118874949157384 = 113
3.6545501930689759371 ^ 3.6545501930689759371 = 114
3.6583532437965460065 ^ 3.6583532437965460065 = 115
3.6621216691731336053 ^ 3.6621216691731336053 = 116
3.6658560821076120777 ^ 3.6658560821076120777 = 117
3.6695570794975412510 ^ 3.6695570794975412510 = 118
3.6732252427798215089 ^ 3.6732252427798215089 = 119
3.6768611384579372157 ^ 3.6768611384579372157 = 120
[...]
3.9983597012366129896 ^ 3.9983597012366129896 = 255
4 ^ 4 = 256
4.0016336236357945063 ^ 4.0016336236357945063 = 257
4.5555357051951280237 ^ 4.5555357051951280237 = 1000
5 ^ 5 = 3125
5.4385826960183050548 ^ 5.4385826960183050548 = 10000
6 ^ 6 = 46656
6.1212785507000868400 ^ 6.1212785507000868400 = 65536
6.2709195555620448175 ^ 6.2709195555620448175 = 100000
7 ^ 7 = 823543
7.0657967282996206111 ^ 7.0657967282996206111 = 1 000 000
7.8313895115911773025 ^ 7.8313895115911773025 = 10 000 000
8 ^ 8 = 16 777 216
8.5731845080373830322 ^ 8.5731845080373830322 = 100 000 000
9 ^ 9 = 387 420 489
9.2950869003762184806 ^ 9.2950869003762184806 = 1 000 000 000
10 ^ 10 = 10 000 000 000
11 ^ 11 = 285 311 670 611
Etcetera.
#math #integers #unknown #raise #clearing #power #x
20200920 (Sunday)