User:Hyp cos/Numbers close to powers of 10

Powers of 2
These results can be generated using continued fraction of \(\lg2\). \begin{eqnarray*} 2^{3} &=& 0.8000\times10^{1}\\ 2^{10} &=& 1.02400\times10^{3}\\ 2^{93} &\approx& 0.990352\times10^{28}\\ 2^{196} &\approx& 1.004336\times10^{59}\\ 2^{485} &\approx& 0.998960\times10^{146}\\ 2^{2136} &\approx& 1.0001629\times10^{643}\\ 2^{13301} &\approx& 0.99993628\times10^{4004}\\ 2^{28738} &\approx& 1.00003544\times10^{8651}\\ 2^{42039} &\approx& 0.99997172\times10^{12655}\\ 2^{70777} &\approx& 1.000007160\times10^{21306}\\ 2^{254370} &\approx& 0.999993200\times10^{76573}\\ 2^{325147} &\approx& 1.0000003603\times10^{97879}\\ 2^{6107016} &\approx& 0.9999996863\times10^{1838395}\\ 2^{6432163} &\approx& 1.00000004669\times10^{1936274}\\ 2^{44699994} &\approx& 0.99999996648\times10^{13456039}\\ 2^{51132157} &\approx& 1.00000001317\times10^{15392313}\\ 2^{146964308} &\approx& 0.999999992815\times10^{44240665}\\ 2^{198096465} &\approx& 1.000000005983\times10^{59632978}\\ 2^{345060773} &\approx& 0.999999998798\times10^{103873643}\\ 2^{1578339557} &\approx& 1.000000001174\times10^{475127550}\\ 2^{1923400330} &\approx& 0.99999999997214\times10^{579001193}\\ 2^{82361153417} &\approx& 1.000000000004088\times10^{24793177656}\\ 2^{496090320832} &\approx& 0.999999999996664\times10^{149338067129}\\ 2^{578451474249} &\approx& 1.0000000000007514\times10^{174131244785}\\ 2^{2809896217828} &\approx& 0.9999999999996692\times10^{845863046269}\\ 2^{6198243909905} &\approx& 1.00000000000008983\times10^{1865857337323}\\ 2^{21404627947543} &\approx& 0.99999999999993871\times10^{6443435058238}\\ 2^{27602871857448} &\approx& 1.00000000000002854\times10^{8309292395561}\\ 2^{76610371662439} &\approx& 0.999999999999995780\times10^{23062019849360}\\ 2^{487265101832082} &\approx& 1.000000000000003216\times10^{146681411491721}\\ 2^{563875473494521} &\approx& 0.999999999999998996\times10^{169743431341081}\\ 2^{2178891522315645} &\approx& 1.0000000000000002049\times10^{655911705514964}\\ 2^{9279441562757101} &\approx& 0.9999999999999998159\times10^{2793390253400937}\\ 2^{11458333085072746} &\approx& 1.00000000000000002086\times10^{3449301958915901}\\ 2^{100946106243339069} &\approx& 0.99999999999999998279\times10^{30387805924728145}\\ 2^{112404439328411815} &\approx& 1.000000000000000003652\times10^{33837107883644046}\\ 2^{550563863556986329} &\approx& 0.999999999999999997401\times10^{165736237459304329}\\ 2^{662968302885398144} &\approx& 1.000000000000000001052\times10^{199573345342948375}\\ 2^{1876500469327782617} &\approx& 0.9999999999999999995055\times10^{564882928145201079}\\ 2^{4415969241540963378} &\approx& 1.00000000000000000006324\times10^{1329339201633350533}\\ 2^{32788285160114526263} &\approx& 0.99999999999999999994812\times10^{9870257339578654810}\\ 2^{37204254401655489641} &\approx& 1.00000000000000000001136\times10^{11199596541212005343}\\ 2^{181605302766736484827} &\approx& 0.999999999999999999993565\times10^{54668643504426676182}\\ 2^{218809557168391974468} &\approx& 1.000000000000000000004926\times10^{65868240045638681525}\end{eqnarray*}