숫자 163

\(\large e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots\approx 262537412640768744\)
\(e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785\approx 884736744\)
\(e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613\approx 147197952744\)

셋 모두 끝 세 자리가 744

\(j(\tau)= {E_4(\tau)^3\over \Delta(\tau)}= q^{-1}+744+196884q+21493760q^2+\cdots\)

\(j(\tau)=1728\frac{g_2^3}{g_2^3-27g_3^2}\)

\( E_4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots\)

\((\sigma_3(n)=\sum_{d|n}d^3)\)

\(\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots\)

\(j(\tau) = \frac{1}[[:틀:Q]] + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots\)
이 때, \({q} = e^{2\pi i\tau}\)

\( j(\tau)= {E_4(\tau)^3\over \Delta(\tau)}= q^{-1}+744+196884q+21493760q^2+\cdots\)

\( E_4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots\)

\((\sigma_3(n)=\sum_{d|n}d^3)\)

\(\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots\)

Ramanujan observed that \(\large e^{\pi \sqrt{163}}=262537412640768743.99999999999925\cdots\) is within \(10^{-12}\) of an integer and used this to obtain approximations to \(\pi\). In his Field’s Medal lecture, Richard Borcherds said that every mathematician should see once in his/her life why this should be the case, and this essay is an attempt to do just that.
\(640320^3= 262537412640768744\)

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