書藉網站：http://www-cs-faculty.stanford.edu/~uno/gkp.html
符號
\(\lfloor x \rfloor\) 小於等於x的最大整數
\(\lceil x \rceil\) 大於等於x的最小整數
\({x} = x - \lfloor x \rfloor\) x的分數部份
\([\alpha .. \beta]\) 即[]區間
質數：只有二個因數（1和它自己）的數合成數：有三個以上因數的數
prime number theorem
小於x的質數個數 \(\pi(x) \sim \frac{x}{\ln x}\)
算數基本定理 (Fundamental Theorem of Arithmetic)
若x為整數且x > 1，則x的因式分解唯一
$$
\binom{r}{0} = 1, \binom{r}{1} = r, \binom{r}{2} = \frac{r(r - 1)}{2}
$$
p.156 symmetry identidy
$$
\binom{n}{k} = \binom{n}{n - k}
$$
p.157 absorption identidy
$$
\binom{r}{k} = \frac{r}{k} \binom{r - 1}{k - 1} \\
=> k \binom{r}{k} = r \binom{r - 1}{k - 1} \\
=> (r - k) \binom{r}{k} = r \binom{r - 1}{k}
$$
p.158 addtion formula
$$
\binom{r}{k} = \binom{r - 1}{k} + \binom{r - 1}{k - 1} \\
(r - k) \binom{r}{k} + k \binom{r}{k} = r \binom{r - 1}{k} + r \binom{r - 1}{k - 1}
$$
p.160 summation on the upper index
$$
\sum_{0 \leq k \leq n} \binom{k}{m} = \binom{0}{m} + \binom{1}{m} + \dots + \binom{n}{m} = \binom{n+1}{m+1}
$$
p.161
$$
\sum_{k \leq n} \binom{m + k}{k} = \binom{m + n + 1}{m + 1} = \binom{m + n + 1}{n}
$$
p.164 negating the upper index (或upper negation)
$$
\binom{r}{k} = (-1)^k \binom{k - r - 1}{k}
$$
p.165
$$
(-1)^m \binom{-n - 1}{m} = (-1)^n \binom{-m - 1}{n} \\
\sum_{k \leq m} \binom{r}{k} (-1)^k = \binom{r}{0} - \binom{r}{1} + \dots + (-1)^m \binom{r}{m} = (-1)^m \binom{r - 1}{m}
$$