"숫자 163"의 두 판 사이의 차이

2009년 4월 29일 (수) 10:38 판

\(\large e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots\)

\(e^{\pi \sqrt{163}} \approx 262537412640768744\)

\(e^{\pi \sqrt{163}} - 262537412640768744 \approx 7.5 \times 10^{-13}\)

\(e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785\)
\(e^{\pi \sqrt{43}} \approx 884736744\)

\(e^{\pi \sqrt{67}} \approx 147197952744\)

\(e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613\)

셋 모두 끝 세 자리가 744

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

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

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

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

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

\( j(\frac {-1+\sqrt{-43}} {2})=884736744\)

\( j(\frac {-1+\sqrt{-67}} {2})=147197952744\)

\( j(\frac {-1+\sqrt{-163}} {2})=262537412640768744\)

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.

@@ 2번째 줄: / 2번째 줄: @@
 <math>\large e^{\pi \sqrt{163}}=262537412640768743.9999999999992500725\cdots</math>
+<math>e^{\pi \sqrt{163}} \approx 262537412640768744</math>
 <math>e^{\pi \sqrt{163}} - 262537412640768744 \approx 7.5 \times 10^{-13}</math>
@@ 7번째 줄: / 9번째 줄: @@
+<math>e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785</math><br><math>e^{\pi \sqrt{43}} \approx 884736744</math>
-<math>e^{\pi \sqrt{43}} = 884736743.9997774660349066619374620785</math><br><math>e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613</math><br><math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007259</math>
-<math>e^{\pi \sqrt{43}} \approx 884736744</math>
 <math>e^{\pi \sqrt{67}} \approx 147197952744</math>
+<math>e^{\pi \sqrt{67}} = 147197952743.9999986624542245068292613</math>
-<math>e^{\pi \sqrt{163}} \approx 262537412640768744</math>
@@ 62번째 줄: / 52번째 줄: @@
 <math> j(\frac {-1+\sqrt{-43}} {2})=884736744</math><br><br><math> j(\frac {-1+\sqrt{-67}} {2})=147197952744</math><br><br><math> j(\frac {-1+\sqrt{-163}} {2})=262537412640768744</math>
-<math>j({{\sqrt{-163}+1}\over{2}})</math>