숫자 5
" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful." Don Zagier (Mathematicians: An Outer View of the Inner World)
\(x_{i}+1=x_{i-1}x_{i+1}\), \(x_0=a\), \(x_1=b\)로 정의된 점화식이 있다고 하자.
계산을 해보면, 다음과 같은 수열을 얻게 된다.
\(a,b,\frac{1+b}{a},\frac{1+a+b}{a b},\frac{1+a}{b},a,b,\frac{1+b}{a},\frac{1+a+b}{a b},\frac{1+a}{b}\cdots\)
주기가 5인 수열이 됨을 확인할 수 있다.
\(a=3,b=4\) 인 경우라면, 위에서처럼 3,4,5/3,2/3,1,3,4,5/3 … 을 얻게 된다.
\(\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\varphi\sqrt{5}}-\varphi}\right) = 0.9981360\dots\)
\(\int_{0}^{\frac{-1+\sqrt{5}}{2}}\frac{\log(1-t)}{t}+\frac{\log(t)}{1-t}dt=-\frac{\pi^2}{5}\)
\(x\in (0,1)\) 에서의 그래프
[/pages/4855791/attachments/3056365 Roger_dilogarithm.jpg]
\(L(x)=-\frac{1}{2}\int_{0}^{x}\frac{\log(1-y)}{y}+\frac{\log(y)}{1-y}dy\)
\(0\leq x,y\leq 1\) 일 때,
\(L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\Left( \frac{1-x}{1-xy} )\right)=\frac{\pi^2}{2}\) 이 성립한다.
Andrei Zelevinsky,What is... a Cluster Algebra,Notices of the AMS,54 (2007),no .11,494-1495. http://www.ams.org/notices/200711/tx071101494p.pdf