整值函数

\[\{ x \} = x - \lfloor x \rfloor\] \[x = \lfloor x \rfloor + \{ x \}\]
\[Spec(α) = \{ \lfloor α \rfloor, \lfloor 2α \rfloor, \lfloor 3α \rfloor, ... \}\]

\(x \mod y = x - y \lfloor x / y \rfloor, y \ne 0\).

\[c(x \mod y) = (cx) \mod (cy)\]

数论

\[m \setminus n \Leftrightarrow m > 0 且对某个整数 k 有 n = mk\] \[gcd(m, n) = max \{ k | k \setminus m 且 k \setminus n \}\]

二项式系数

\[\begin{Bmatrix} n \\ k \end{Bmatrix} = \frac{n(n-1)...(n-k+1)}{k(k-1)...(1)}\]
\[k \setminus m 和 k \setminus n \Leftrightarrow k \setminus gcd(m, n)\]
\[\sum_{m \setminus n} a_m = \sum_{m \setminus n} a_{n / m}, 整数 n > 0\] \[\sum_{m \setminus n} a_m = \sum_{k} \sum_{m > 0} a_m[n = mk]\]
\[n! \sim \sqrt{2 \pi n} \left ( \frac{n}{e} \right ) ^ n\]
\[m \perp n \Leftrightarrow m, n 是整数, 且 gcd(m, n) = 1\] \[m / gcd(m, n) \perp n / gcd(m, n)\] \[k \perp m 且 k \perp n \Leftrightarrow k \perp mn\]