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app1.tex
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\chapter{常用的表}
\section{梅森素数表}
Mersenne exponents: primes $p$ such that $2^p - 1$ is prime. Then $2^p - 1$ is called a Mersenne prime.
见表\ref{tab:Mersenne-prime}。
\begin{table}[!htbp]
\centering
\caption{梅森素数指数 \label{tab:Mersenne-prime}}
\begin{tabular}{|c|c|c|c|}
\hline
n & a(n) & 24 & 19937 \\ \hline
1 & 2 & 25 & 21701 \\ \hline
2 & 3 & 26 & 23209 \\ \hline
3 & 5 & 27 & 44497 \\ \hline
4 & 7 & 28 & 86243 \\ \hline
5 & 13 & 29 & 110503 \\ \hline
6 & 17 & 30 & 132049 \\ \hline
7 & 19 & 31 & 216091 \\ \hline
8 & 31 & 32 & 756839 \\ \hline
9 & 61 & 33 & 859433 \\ \hline
10 & 89 & 34 & 1257787 \\ \hline
11 & 107 & 35 & 1398269 \\ \hline
12 & 127 & 36 & 2976221 \\ \hline
13 & 521 & 37 & 3021377 \\ \hline
14 & 607 & 38 & 6972593 \\ \hline
15 & 1279 & 39 & 13466917 \\ \hline
16 & 2203 & 40 & 20996011 \\ \hline
17 & 2281 & 41 & 24036583 \\ \hline
18 & 3217 & 42 & 25964951 \\ \hline
19 & 4253 & 43 & 30402457 \\ \hline
20 & 4423 & 44 & 32582657 \\ \hline
21 & 9689 & 45 & 37156667 \\ \hline
22 & 9941 & 46 & 42643801 \\ \hline
23 & 11213 & 47 & 43112609 \\ \hline
\end{tabular}
\end{table}
\section{卡米歇尔数}
见表\ref{tab:Carmichael-numbers}。
\begin{table}[!htbp]
\centering
\caption{卡米歇尔数前面一点 \label{tab:Carmichael-numbers}}
\begin{tabular}{|c|c|}
\hline
a & a(n) \\ \hline
1 & 561 \\ \hline
2 & 1105 \\ \hline
3 & 1729 \\ \hline
4 & 2465 \\ \hline
5 & 2821 \\ \hline
6 & 6601 \\ \hline
7 & 8911 \\ \hline
8 & 10585 \\ \hline
9 & 15841 \\ \hline
10 & 29341 \\ \hline
11 & 41041 \\ \hline
12 & 46657 \\ \hline
13 & 52633 \\ \hline
14 & 62745 \\ \hline
15 & 63973 \\ \hline
16 & 75361 \\ \hline
17 & 101101 \\ \hline
18 & 115921 \\ \hline
19 & 126217 \\ \hline
20 & 162401 \\ \hline
21 & 172081 \\ \hline
22 & 188461 \\ \hline
23 & 252601 \\ \hline
24 & 278545 \\ \hline
25 & 294409 \\ \hline
\end{tabular}
\end{table}
\section{常见的素数及其原根}
见表\ref{tab:ntt-primes}。
\begin{table}[!htbp]
\centering
\caption{常见的素数及其原根 \label{tab:ntt-primes}}
\begin{tabular}{|c|c|c|c|}
\hline
$p=k*2^m + 1$ & $k$ & $m$ & $groot$ \\ \hline
3 & 1 & 1 & 2 \\ \hline
5 & 1 & 2 & 2 \\ \hline
17 & 1 & 4 & 3 \\ \hline
97 & 3 & 5 & 5 \\ \hline
193 & 3 & 6 & 5 \\ \hline
257 & 1 & 8 & 3 \\ \hline
7681 & 15 & 9 & 17 \\ \hline
12289 & 3 & 12 & 11 \\ \hline
40961 & 5 & 13 & 3 \\ \hline
65537 & 1 & 16 & 3 \\ \hline
786433 & 3 & 18 & 10 \\ \hline
5767169 & 11 & 19 & 3 \\ \hline
7340033 & 7 & 20 & 3 \\ \hline
23068673 & 11 & 21 & 3 \\ \hline
104857601 & 25 & 22 & 3 \\ \hline
167772161 & 5 & 25 & 3 \\ \hline
469762049 & 7 & 26 & 3 \\ \hline
{ \color{red}998244353} & 119 & 23 & 3 \\ \hline
{\color{red}1004535809} & 479 & 21 & 3 \\ \hline
2013265921 & 15 & 27 & 31 \\ \hline
{\color{red}2281701377} & 17 & 27 & 3 \\ \hline
3221225473 & 3 & 30 & 5 \\ \hline
75161927681 & 35 & 31 & 3 \\ \hline
77309411329 & 9 & 33 & 7 \\ \hline
206158430209 & 3 & 36 & 22 \\ \hline
2061584302081 & 15 & 37 & 7 \\ \hline
2748779069441 & 5 & 39 & 3 \\ \hline
6597069766657 & 3 & 41 & 5 \\ \hline
39582418599937 & 9 & 42 & 5 \\ \hline
79164837199873 & 9 & 43 & 5 \\ \hline
263882790666241 & 15 & 44 & 7 \\ \hline
1231453023109121 & 35 & 45 & 3 \\ \hline
1337006139375617 & 19 & 46 & 3 \\ \hline
3799912185593857 & 27 & 47 & 5 \\ \hline
4222124650659841 & 15 & 48 & 19 \\ \hline
7881299347898369 & 7 & 50 & 6 \\ \hline
31525197391593473 & 7 & 52 & 3 \\ \hline
180143985094819841 & 5 & 55 & 6 \\ \hline
1945555039024054273 & 27 & 56 & 5 \\ \hline
4179340454199820289 & 29 & 57 & 3 \\ \hline
\end{tabular}
\end{table}
\section{一些公式}
\subsection{切比雪夫多项式}
$$
\cos n x=\left\{\begin{array}{l}{\sum_{k=0}^{\frac{n}{2}}(-1)^{\frac{n-2 k}{2}} \frac{n \cdot(n+2 k-2) ! !}{(2 k) !(n-2 k) ! !} \cos ^{2 k} x} \ , \quad n\ is\ even \\
{\sum_{k=1}^{\frac{n+1}{2}}(-1)^{\frac{n+1-2 k}{2}} \frac{n \cdot(n+2 k-3) !}{(2 k-1) !(n+1-2 k) ! !} \cos ^{2 k-1} x} \ , \quad n\ is\ odd \end{array}\right.
$$
\newpage
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