# 接近整数

Ed Pegg jr.先生發現上圖中的線段d長度為$\frac{1}{2}\sqrt{\frac{1}{30}(61421-23\sqrt{5831385})}$，非常接近7（數值為7.0000000857）[1]

## 有關黃金比例及其他皮索特-维贡伊拉卡文数

• $\phi^{17}=\frac{3571+1597\sqrt5}{2}\approx 3571.00028\,$
• $\phi^{18}=2889+1292\sqrt5 \approx 5777.999827\,$
• $\phi^{19}=\frac{9349+4181\sqrt5}{2}\approx 9349.000107\,$

$\phi \overline{\phi} = -1$

$\phi+\overline{\phi} = 1$

$\phi^n + \overline{\phi}^n$可以用$\phi \overline{\phi}$$\phi+\overline{\phi}$來表示，由於二根之和及二根之積均為整數，計算所得的結果也是一個正整數，假設為一正整數K，則$\phi^n$可以用下式表示

$\phi^n = K - \overline{\phi}^n$

$\phi^n \approx K$

## 有關黑格納數

• $e^{\pi\sqrt{43}}\approx 884736743.999777466\,$
• $e^{\pi\sqrt{67}}\approx 147197952743.999998662454\,$
• $e^{\pi\sqrt{163}}\approx 262537412640768743.99999999999925007\,$

$e^{\pi\sqrt{43}}=12^3(9^2-1)^3+744-2.225\cdots\times 10^{-4}\,$
$e^{\pi\sqrt{67}}=12^3(21^2-1)^3+744-1.337\cdots\times 10^{-6}\,$
$e^{\pi\sqrt{163}}=12^3(231^2-1)^3+744-7.499\cdots\times 10^{-13}\,$

## 有關π及e

$e^{\pi}-\pi=19.999099979189\cdots\,$

• $22{\pi}^4=2143.0000027480\cdots\,$
• ${\pi}^3=31.006276\cdots\,$
• ${\pi}^3-\frac{\pi}{500}=30.999993494\cdots\,$
• ${\pi}^2+\frac{\pi}{24}=10.000504\cdots\,$

## 其他例子

 ${}_{\cos\left\{\pi\cos\left[\pi\cos\ln\left(\pi+20\right)\right]\right\}\approx -0.9999999999999999999999999999999999606783 }$ ${}_{\sin2017\sqrt[5]2\approx -0.9999999999999999785}$ ${}_{\sum_{k=1}^{\infty}\frac{\lfloor n\tanh \pi \rfloor}{10^n}-\frac{1}{81}\approx 1.11\times10^{-269}}$ ${}_{\sqrt{29}\left(\cos\frac{2\pi}{59}-\cos\frac{24\pi}{59}\right)-\frac{19}{5}\approx 3.057684294154\times10^{-6}}$ ${}_{1+\frac{103378831900730205293632}{e^{3\pi\sqrt{163}}}-\frac{196884}{e^{2\pi\sqrt{163}}}-\frac{262537412640768744}{e^{\pi\sqrt{163}}}\approx 1.161367900476\times10^{-59}}$ ${}_{\frac{\ln^2262537412640768744}{\pi^2}-163\approx 2.32167\times10^{-29}}$ ${}_{10\tanh\frac{28}{15}\pi-\frac{\pi^9}{e^8}\approx 3.661398\times10^{-8}}$ ${}_{ \sqrt[4]{\frac{91}{10}}-\frac{33}{19}\approx 3.661398\times10^{-8}}$ ${}_{ \gamma-{10\over81}\left(11-2\sqrt{10}\right)=\int_0^{\infty}\left(\frac{1}{e^x-1}-\frac{1}{xe^x}\right){\rm{d}}x-{10\over81}\left(11-2\sqrt{10}\right)\approx 2.72\times10^{-7}}$ ${}_{\frac{\left(5+\sqrt5\right)\Gamma\left({3\over4}\right)}{e^{\frac{5}{6}\pi}}\approx1.000000000000045422}$ ${}_{{1\over4}\left(\cos{1\over10}+\cosh{1\over10}+2\cos{\sqrt2\over20}\cosh{\sqrt2\over20}\right)\approx 1.000000000000248 }$ ${}_{e^6-\pi^5-\pi^4\approx1.7673\times10^{-5}}$ ${}_{\sqrt{29}\left(\cos\frac{2\pi}{59}-\cos\frac{24\pi}{59}\right)\approx 3.0576842941540143382\times 10^{-6}}$ ${}_{ \left(3\sqrt5\right)^{\gamma}=\left(3\sqrt5\right)^{\int_0^{\infty}\left(\frac{1}{e^x-1}-\frac{1}{xe^x}\right){\rm{d}}x}\approx 3.000060964}$ ${}_{ e^{\phi_0\left(\frac{2+\sqrt3}{4}\right)}=e^{\int_0^{\infty}\left(\frac{1}{te^t}-\frac{e^{\frac{2-\sqrt3}{4}t}}{e^t-1}\right){\rm{d}}t}\approx 1.99999969}$ ${}_{ \frac{\sqrt[3]9}{3\ln 2}\approx 1.00030887}$ ${}_{\sum_{k=-\infty}^{\infty}10^{-\frac{k^2}{10000}}-100\sqrt{\frac{\pi}{\ln10}}=\theta_3\left(0,\frac{1}{\sqrt[10000]{10}}\right)-100\sqrt{\frac{\pi}{\ln10}}\approx1.3809\times10^{-18613}}$ ${}_{ {\pi^9\over e^8}\approx 9.998387}$ ${}_{ e^{\pi}-\pi\approx 19.999099979}$ ${}_{ \frac{e^{\pi}-\ln3}{\ln2}-\frac{4}{5}\approx 31.0000000033}$ ${}_{\frac{\pi^{11}}{e^3}-\Gamma\left[\Gamma\left(\pi+1\right)+1\right]=\frac{\pi^{11}}{e^3}-\int_0^{\infty}\frac{t^{\int_0^{\infty}\frac{u^{\pi}}{e^u}{\rm{d}}u}}{e^t} {\rm{d}}t\approx 7266.9999993632596}$ ${}_{ 163\left(\pi-e\right)\approx 68.999664}$ ${}_{ \left(\frac{23}{9}\right)^5=\frac{6436343}{59049}\approx 109.00003387}$ ${}_{ 88\ln 89\approx 395.00000053}$ ${}_{ 510\lg 7\approx 431.00000040727098}$ ${}_{ 272\log_{\pi}97\approx 1087.000000204}$ ${}_{ \frac{53453}{\ln 53453}\approx 4910.00000122}$ ${}_{ \frac{53453}{\ln 53453}+\frac{163}{\ln 163}\approx 4941.99999995925082 }$ ${}_{\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}+4\pi e^{\pi}\right)}-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}-4\pi e^{\pi}\right)}\approx 2.570287024592328869357\times 10^{-6}}$ ${}_{10-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}-4\pi e^{\pi}\right)}\approx 2.57055302118\times 10^{-6}}$ ${}_{10-\sqrt[8]{\frac{\sqrt2}{4}\left(\pi^{17}-4e^{2\pi}+4\pi e^{\pi}\right)}\approx 2.65996596963\times 10^{-10}}$ ${}_{ \frac{163}{\ln 163}\approx 31.9999987343}$
${}_{ \ln K_0-\ln\ln K_0\approx 1.0000744}$ ，其中$K_0$辛钦常数
${}_{\frac{10}{81}-\sum_{n=1}^\infty\frac{\sum_{k=10^{n-1}}^{10^n-1}10^{-n\left[k-(10^{n-1}-1)\right]}k}{10^{\sum_{k=0}^{n-1}9\times 10^{k-1}k}}=\frac{10}{81}-\sum_{n=1}^\infty\sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{kn-9\sum_{k=0}^{n-1}10^k(n-k)}}\approx 1.022344\times10^{-9}}$

${}_{-\frac{1}{5} +e^{\frac{6}{5}} {}_4F_3\left(-\frac{1}{5},\frac{1}{20},\frac{3}{10},\frac{11}{20};\frac{1}{5},\frac{2}{5},\frac{3}{5};\frac{256}{3125e^6}\right)+\frac{2}{25e^{\frac{6}{5}}}{}_4F_3\left(\frac{1}{5},\frac{9}{20},\frac{7}{10},\frac{19}{20};\frac{3}{5},\frac{4}{5},\frac{7}{5};\frac{256}{3125e^6}\right)-\frac{4}{125e^{\frac{12}{5}}}{}_4F_3\left(\frac{2}{5},\frac{13}{20},\frac{9}{10},\frac{23}{20};\frac{4}{5},\frac{6}{5},\frac{8}{5};\frac{256}{3125e^6}\right)+\frac{7}{625e^{\frac{18}{5}}}{}_4F_3\left(\frac{3}{5},\frac{17}{20},\frac{11}{10},\frac{27}{20};\frac{6}{5},\frac{7}{5},\frac{9}{5};\frac{256}{3125e^6}\right)-\pi\approx 2.89221114964408683\times10^{-8}}$

${}_{\qquad\mbox{Root of } x^6-615x^5+151290x^4-18608670x^3+1144433205x^2-28153057165x+39605=0} \,$
${}_{\frac{615-55\sqrt5-\sqrt[3]{7451370+3332354\sqrt5+6\sqrt{8890710030+3976046490\sqrt5}}-\sqrt[3]{7451370+3332354\sqrt5-6\sqrt{8890710030+3976046490\sqrt5}}}{6}\approx 1.40677447684\times10^{-6}}$

${}_{\qquad\mbox{Root of } 312500000x^5-6843750000x^4+6826250000x^3+10476025000x^2-7886869750x-72099=0} \,$
${}_{\tan\left(\frac{\arctan 4}{5}+\frac{4\pi}{5}\right)+\frac{19}{50}=\frac{219}{50}+\frac{-1-\sqrt5+\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884+799{\rm{i}}}+\frac{-1-\sqrt5-\sqrt{10-2\sqrt5}{\rm{i}}}{4}\sqrt[5]{884-799{\rm{i}}}+\frac{-1+\sqrt5-\sqrt{10+2\sqrt5}{\rm{i}}}{4}\sqrt[5]{1156+289{\rm{i}}}+\frac{-1+\sqrt5+\sqrt{10+2\sqrt5}{\rm{i}}}{4}\sqrt[5]{1156-289{\rm{i}}}\approx -9.141538637378949398666277\times 10^{-6}}$

${}_{\rm{erfi}\left(\rm{erfi}\frac{\sqrt3}{3}\right)=\frac{2}{\sqrt\pi}\int_0^{\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t} e^{u^2} \rm{d} u =\frac{2}{\sqrt\pi}e^{\left(\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right)^2}\int_0^{\infty}\frac{\sin\left[\frac{4u\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right]}{e^{u^2}}{\rm{d}}u =\frac{2}{\sqrt\pi}\int_0^{{}_{\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t}} e^{u^2} {\rm{d}} u =\frac{2}{\sqrt\pi}e^{\left(\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)^2}\int_0^{\infty}\frac{\sin\left(\frac{4u}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)}{e^{u^2}} {\rm{d}} u\approx 1.00002087363809430195879}$