"숫자 163"의 두 판 사이의 차이

2009년 8월 14일 (금) 11:10 판

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

셋 모두 끝 세 자리가 744

\(j(\tau) = \frac{1}[[:틀:Q]] + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots\)
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.

@@ 88번째 줄: / 88번째 줄: @@
 <h5>관련된 다른 주제들</h5>
+*
 * [[오일러의 소수생성다항식 x²+x+41|오일러의 소수생성다항식 x² +x+41]]
-* [[#]]
+* [[가우스의 class number one 문제]]
+* [[라마누잔과 파이]]
+* [[라마누잔의 class invariants]]