的關係，使得上式中出現平方項。常數e^{\pi\sqrt{163}}\,有時會稱為拉馬努金常數。
有關π及e
許多有關π及e的常數也是接近整數，例如
:e^{\pi}-\pi=19.999099979189\cdots\,
以及
:e^{5\pi}=6635623.999341134233\cdots\,
格尔丰德常数（e^{\pi}\,）接近\pi+20\,，至2011年為止還沒找到出現此特性的原因，因此只能視為一數學巧合。另一個有關格尔丰德常数的常數也是接近整數
\frac{e^\pi-\pi-1}{6\pi}=1.00793356\cdots\,
以下也是一些接近整數的例子
*22{\pi}^4=2143.0000027480\cdots\,
*{\pi}^5=306.019684\cdots\,
*{\pi}^3=31.006276\cdots\,
*{\pi}^{13/2}=1704.017978\cdots\,
*{\pi}^3-\frac{\pi}{500}=30.999993494\cdots\,
*{\pi}^2+\frac{\pi}{24}=10.000504\cdots\,
*{\pi}^5-3{\pi}^3=213.0008547\cdots\,
*e^\pi-\pi+0.0009=19.9999999791\cdots\,
*e^3=20.0855369\cdots\,
*e^{9/2}=90.0171313\cdots\,
*e^{\pi\sqrt{2}}=85.019695\cdots\,
*e^6-{\pi}^5-{\pi}^4=0.00001767\cdots\,
*9{\pi}^5-2e^3=2714.00608922\cdots\,
*e^{13/2}-\pi=662.00004039\cdots\,
*{\pi}^{13/2}-e^{9/2}=1614.00084707\cdots\,
其他例子
:
:2^{2^{2/3}}\approx 3.00507511272
:4 \ln 2.117\approx 2.99999996861
:{}_{ \ln K_0-\ln\ln K_0\approx 1.0000744} ，其中K_0是辛钦常数
:{}_{\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}
*\sin 11 = -0.999990207...，這是由於3.5 \pi \approx 10.9956 \approx 11的緣故，另一個類似的例子為\sin 355 = -0.00003014435335948844921433...
*\sqrt{1^2+2^2+3^2+4^2+.......+552057^2} \approx 236818619.0000004307
*\frac{5^6}{6^5} \approx 2
外部連結
- [http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf J.S. Markovitch Coincidence, data compression, and Mach's concept of economy of thought]
註釋
參考資料
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