公式测试
纯手打
\(\begin{bmatrix}a_{11} & a_{12} & a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\a_{41}&a_{42}&a_{43}&a_{44}\\\end{bmatrix} \cdot\begin{bmatrix}x\\y\\z\\1\end{bmatrix}=\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}\)
\(a\cdot b=\begin{cases}0&\text{if }a=0\\\\2\cdot\frac{a}{2}\cdot b&\text{if }a>0\text{ and }a\text{ even}\\\\2\cdot\frac{a-1}{2}\cdot b+b&\text{if }a>0\text{ and }a\text{ odd}\end{cases}\)
\(\left\lfloor\dfrac bs\right\rfloor\cdot s+b\bmod s\)
\(a^b=a^{\lfloor\frac bs\rfloor\cdot s}\times a^{b\bmod s}\)
\(a\times b\bmod m=a\times b-\left\lfloor\dfrac{ab}m\right\rfloor\times m=\left(a\times b-\left\lfloor\dfrac{ab}m\right\rfloor\times m\right)\bmod 2^{64}\)
\(\begin{aligned}a^n&=(a^{n_t 2^t+\cdots+n_0 2^0})\\\\&=a^{n_0 2^0}\times a^{n_1 2^1}\times\cdots\times a^{n_t2^t}\end{aligned}\)
\(a^{n}=\underbrace{a\times a\cdots\times a}_{n\text{ 个 a}}\)
\(\dfrac{a_4 base+a_3}{b_3+b_2 base^{-1}+(b_1+1)base^{-2}}\)
\(\begin{aligned}x&=x_1\cdot 10^m+x_0, \\y&=y_1\cdot 10^m+y_0, \\x\cdot y&=z_2\cdot 10^{2m}+z_1\cdot 10^m + z_0\end{aligned}\)
\(\begin{aligned}\frac{a+b\text{i}}{c+d\text{i}}&=\frac{(a+b\text{i})(c-d\text{i})}{(c+d\text{i})(c-d\text{i})}\\&=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}\text{i} &(c+d\text{i}\not=0)\end{aligned}\)
\(\tan\theta=\frac{y}{x}\)
\(\theta=\operatorname{Arg}z\)
\(-\pi<\operatorname{arg}z\le\pi\)
\(f(z)=e^x(\cos y+i\sin y)\)
\(\operatorname{exp}z=e^x(\cos y+i\sin y)\)
\(\begin{aligned}\omega_n^n&=1\\\omega_n^k&=\omega_{2n}^{2k}\\\omega_{2n}^{k+n}&=-\omega_{2n}^k\\\end{aligned}\)
\(\{\omega_n^k\mid 0\le k<n,\gcd(n,k)=1\}\)
\(\left(g^eg^{-(e\bmod 2^k)})\right)^{2^{n-1-k}}\equiv g^{2^{n-1}\cdot e_k}\equiv\begin{cases}1\pmod p&\text{if }e_k=0\text{,}\\-1\pmod p&\text{if }e_k=1\text{.}\end{cases}\)
\(\begin{aligned}(a_0+a_1b,a_0-a_1b)&=\phi(a_0+a_1x)\\&=\phi(r-x)^{(p-1)/2}\\&=((r-b)^{(p-1)/2},(r+b)^{(p-1)/2})\\&=(\pm 1,\mp 1)\end{aligned}\)
\(\begin{aligned}g^{2^{n-1}}&\equiv r^{2^{n-1}\cdot m}\pmod p\\&\equiv r^{(p-1)/2}\pmod p\\&\equiv -1\pmod p\end{aligned}\)
\(\begin{aligned}\phi:\mathbb{F}_p\lbrack x\rbrack/(x^2-a)&\to\mathbb{F}_p\times\mathbb{F}_p\\x&\mapsto(b,-b)\end{aligned}\)
\(\begin{aligned}(a_0+a_1x)^2&=a_0^2+2a_0a_1x+a_1^2x^2\\&\equiv a_0^2+2a_0a_1x+a_1^2(r^2-a)\pmod{f(x)}\end{aligned}\)
\(\begin{aligned}(a_0+a_1x)^2&=a_0^2+2a_0a_1x+a_1^2x^2\\&\equiv(r-v)^{p+1}\pmod{f(x)}\\&\equiv(r-x)^p(r-x)\pmod{f(x)}\\&\equiv(r+x)(r-x)\pmod{f(x)}\\&\equiv r^2-x^2\pmod{f(x)}\\&\equiv a\pmod{f(x)}\end{aligned}\)
\(\begin{aligned}(a+b)^p&=\sum_{i=0}^p\binom{p}{i}a^ib^{p-i}\\&=\sum_{i=0}^p\frac{p!}{i!(p-i)!}a^ib^{p-i}\\&\equiv a^p+b^p\pmod p\end{aligned}\)
令 \(f(x)=x^2-(r^2-a)\in\mathbb{F}_p\lbrack x\rbrack\) 和 \(a_0+a_1x=(r-x)^{(p+1)/2}\bmod (x^2-(r^2-a))\) 那么有 \(a_0^2\equiv a\pmod p\) 且 \(a_1\equiv 0\pmod p\).
\(\begin{aligned}x^p&\equiv x(x^2)^{(p-1)/2}\pmod{f(x)}\\&\equiv x(r^2-a)^{(p-1)/2}\pmod{f(x)}&\quad (\because{x^2\equiv r^2-a\pmod{f(x)}})\\&\equiv -x\pmod{f(x)}&\quad (\because{r^2-a}\text{ 为二次非剩余})\end{aligned}\)
\(x^2-(r^2-a)\in\mathbb{F}_p\lbrack x\rbrack\)
\(\mathbb{F}_p\lbrack x\rbrack /(x^2-(r^2-a))\)
\(\begin{aligned}&{}g^{p-1}\equiv 1\pmod p\\\iff &g^{p-1}-1\equiv 0\pmod p\\\iff &\left(g^{(p-1)/2}-1\right)\cdot\left(g^{(p-1)/2}+1\right)\equiv 0\pmod p\\\implies &g^{(p-1)/2}\equiv -1\pmod p\end{aligned}\)
\(\left(\frac{a}{p}\right)=\begin{cases}1,\,&p\nmid a \text{ 且 }a\text{ 是模 }p\text{ 的二次剩余}\\-1,\,&p\nmid a \text{ 且 }a\text{ 不是模 }p\text{ 的二次剩余}\\0,\,&p\mid a\end{cases}\)
一段牛逼的代码(大概意思是指中国剩余定理)
\(\begin{array}{ll}&\textbf{Chinese Remainder Algorithm }\operatorname{cra}(\mathbf{v}, \mathbf{m})\text{:} \\&\textbf{Input}\text{: }\mathbf{m}=(m_0,m_1,\dots ,m_{n-1})\text{, }m_i\in\mathbb{Z}^+\land\gcd(m_i,m_j)=1\text{ for all } i\neq j\text{,} \\&\qquad \mathbf{v}=(v_0,\dots ,v_{n-1}) \text{ where }v_i=x\bmod m_i\text{.} \\&\textbf{Output}\text{: }x\bmod{\prod_{i=0}^{n-1} m_i}\text{.} \\1&\qquad \textbf{for }i\text{ from }1\text{ to }(n-1)\textbf{ do} \\2&\qquad \qquad C_i\gets\left(\prod_{j=0}^{i-1}m_j\right)^{-1}\bmod{m_i} \\3&\qquad x\gets v_0 \\4&\qquad \textbf{for }i\text{ from }1\text{ to }(n-1)\textbf{ do} \\5&\qquad \qquad u\gets (v_i-x)\cdot C_i\bmod{m_i} \\6&\qquad \qquad x\gets x+u\prod_{j=0}^{i-1}m_j \\7&\qquad \textbf{return }(x)\end{array}\)
\(\varphi(n)=n\times\prod_{i=1}^s{\dfrac{p_i-1}{p_i}}\)
\(\begin{aligned}\varphi(n)&=\prod_{i=1}^{s} \varphi(p_i^{k_i})\\&=\prod_{i=1}^{s}(p_i-1)\times{p_i}^{k_i-1}\\&=\prod_{i=1}^{s}{p_i}^{k_i}\times(1-\frac{1}{p_i})\\&=n~ \prod_{i=1}^{s}(1-\frac{1}{p_i})&\square\end{aligned}\)
扩展欧拉定理
\(a^b\equiv\begin{cases}a^{b\bmod\varphi(p)},\,&\gcd(a,\,p)=1\\a^b,&\gcd(a,\,p)\ne1,\,b<\varphi(p)\\a^{b\bmod\varphi(p)+\varphi(p)},&\gcd(a,\,p)\ne1,\,b\ge\varphi(p)\end{cases}\pmod p\)
\(\sum\limits_{i=0}^{\left[\frac{n}{a}\right]}\left[\frac{n-ia}{b}\right]\)
\(\phi\left(x,a\right)=\#\big\{n\le x\mid n\bmod p=0\Rightarrow p>p_a\big\}\)
\(P_k\left(x,a\right)=\#\big\{n\le x\mid n=q_1q_2\cdots q_k\Rightarrow\forall i,q_i>p_a\big\}\)
\(P_2\left(x,a\right)=\sum\limits_{y<p\le \sqrt{x}}{\left(\pi\left(\dfrac{x}{p}\right)-\pi\left(p\right)+1\right)}\)
一个神奇的格式
\(\phi\left(x,a\right)=\sum\limits_{1\le n\le x\ P^+\left(n\right)\le y}{\mu\left(n\right)\left[x/n\right]}\)
\(\begin{matrix}&&\phi\left(x,a\right)&&\\&\swarrow&&\searrow&\\&\phi\left(x,a-1\right)&&-\phi\left(\frac{x}{p_a},a-1\right)&\\\swarrow&\downarrow&&\downarrow&\searrow\\\phi\left(x,a-2\right)&\phi\left(\frac{x}{p_{a-1}},a-2\right)&&-\phi\left(\frac{x}{p_a},a-2\right)&\phi\left(\frac{x}{p_ap_{a-1}},a-2\right)\end{matrix}\)
\(\vdots\)
\(S=\sum\limits_{n/\delta\left(n\right)\le y\le n}{\mu\left(n\right)\phi\left(\dfrac{x}{n},\pi\left(\delta\left(n\right)\right)-1 \right)}\)
\(S=-\sum\limits_{p\le y}{\ \sum\limits_{\delta\left(m\right)>p\ m\le y<mp}{\mu\left(m\right)\phi\left(\dfrac{x}{mp},\pi\left(p\right)-1\right)}}\)
\(S_1=-\sum\limits_{x^{1/3}<p\le y}{\ \sum\limits_{\delta\left(m\right)>p\ m\le y<mp}{\mu\left(m\right)\phi\left(\dfrac{x}{mp},\pi\left(p\right)-1\right)}}\)
\(S_2=-\sum\limits_{x^{1/4}<p\le x^{1/3}}{\ \sum\limits_{\delta\left(m\right)>p\ m\le y<mp}{\mu\left(m\right)\phi\left(\dfrac{x}{mp},\pi\left(p\right)-1\right)}}\)
\(S_3=-\sum\limits_{p\le x^{1/4}}{\ \sum\limits_{\delta\left(m\right)>p\ m\le y<mp}{\mu\left(m\right)\phi\left(\dfrac{x}{mp},\pi\left(p\right)-1\right)}}\)
\(S_1=\dfrac{\left(\pi\left(y\right)-\pi\left(x^{1/3}\right)\right)\left(\pi\left(y\right)-\pi\left(x^{1/3}\right)-1\right)}{2}\)
\(\sum_{d\mid n}\mu(b)f(\frac{n}{d})=\sum_{d\mid n}\mu(d)\sum_{k\mid \frac{n}{d}}g(k)=\sum_{k\mid n}g(k)\sum_{d\mid \frac{n}{k}}\mu(d)=g(n)\)
\(qwq\)
\(\begin{aligned}&\sum_{n|d}{\mu(\frac{d}{n})f(d)}\\=& \sum_{k=1}^{+\infty}{\mu(k)f(kn)}= \sum_{k=1}^{+\infty}{\mu(k)\sum_{kn|d}{g(d)}}\\=& \sum_{n|d}{g(d)\sum_{k|\frac{d}{n}}{\mu(k)}}= \sum_{n|d}{g(d)\epsilon(\frac{d}{n})}\\=& g(n)\end{aligned}\)
\(\begin{aligned}T(n)&= \sum_{i^{2} \le n} O\left(\pi\left(\sqrt{i}\right)\right) + \sum_{i^{2} \le n} O\left(\pi\left(\sqrt{\frac{n}{i}}\right)\right) \\&= \sum_{i^{2} \le n} O\left(\frac{\sqrt{i}}{\ln{\sqrt{i}}}\right) + \sum_{i^{2} \le n} O\left(\frac{\sqrt{\frac{n}{i}}}{\ln{\sqrt{\frac{n}{i}}}}\right) \\&= O\left(\int_{1}^{\sqrt{n}} \frac{\sqrt{\frac{n}{x}}}{\log{\sqrt{\frac{n}{x}}}} \mathrm{d} x\right) \\&= O\left(\frac{n^{\frac{3}{4}}}{\log{n}}\right)\end{aligned}\)
\(s_{k} := F_{\mathrm{prime}}(p_{k})\)
\(F_{\mathrm{prime}}(p_{k-1})\)
\(f(n) = \begin{cases}1 & n = 1 \\ p \operatorname{xor} c & n = p ^ {c} \\ f(a)f(b) & n = ab \land a \perp b\end{cases}\)
\(\sum_{i=1}^nf(i)=\sum_{i=1}^n\left[\exists d\in(\sqrt n,n]\cap\mathbb P,d\mid i\right]f(i)+\sum_{i=1}^n\left[\forall d\in(\sqrt n, n]\cap\mathbb P,d\nmid i\right]f(i)\)
\(\sum_{i=1}^nf(i)=\sum_{i=1}^{\sqrt n}f(i)\cdot\left(\sum_{d=\lfloor\sqrt n\rfloor + 1}^{\lfloor\frac ni\rfloor}[d\in\mathbb P]\right)+\sum_{i=1}^n\left[\forall d\in(\sqrt n,n]\cap\mathbb P,d\nmid i\right]f(i)\)
\(\displaystyle\sum_{i=1}^{\sqrt n}f(i)\cdot\left(\sum_{d=\lfloor\sqrt n\rfloor+1}^{\lfloor\frac ni\rfloor}[d\in\mathbb P]f(d)\right)\)
\(\displaystyle\sum_{i=1}^n\left[\forall d\in(\sqrt n,n]\cap\mathbb P,d\nmid i\right]f(i)\)
\(\displaystyle h(t,l)=\sum_{i=1}^l\left[i=\prod_{j=t}^mp_j^{c_j},c_j\in\mathbb N\right]f(i)\)
\(\displaystyle h(t,l)=h(t+1,l)+\sum_{c\in\mathbb N^*}f(p_t^c)\cdot h\left(t+1,\left\lfloor\frac l{p_t^c}\right\rfloor\right)\)
\(\displaystyle\sum_{i=p_{t_l}}^{\min(l,\sqrt n)}[i\in\mathbb P]f(i)\)
\(\mathcal O\left(\dfrac{n^{\frac34}}{\log n}\right)\)
又是一个神奇的格式
\(\begin{array}{lllllllllllll}F_1=\{&\frac{0}{1},&&&&&&&&&&\frac{1}{1}&\}\\F_2=\{&\frac{0}{1},&&&&&\frac12,&&&&&\frac11&\}\\F_3=\{&\frac01,&&&\frac13&&\frac12,&&\frac23,&&&\frac11&\}\\F_4=\{&\frac01,&&\frac14,&\frac13,&&\frac12,&&\frac23,&\frac34,&&\frac11&\}\\F_5=\{&\frac01,&\frac15,&\frac14,&\frac13,&\frac25,&\frac12,&\frac35,&\frac23,&\frac34,&\frac45,&\frac11&\}\end{array}\)
\(H_c=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}\sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot\Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].\)
各种数学公式
\(n^2\) \(2_a\) \(b_{a-2}\) \(\frac{a}{3}\) \(\frac{y}{\frac{3}{x}+b}\) \(\sqrt{y^2}\) \(\sqrt[x]{y^2}\) \(\sum_{x=1}^5 y^z\) \(\int_a^b f(x)\) \(\alpha\) \(\beta\) \(\delta, \Delta\) \(\pi, \Pi\) \(\sigma, \Sigma\) \(\phi, \Phi, \varphi\) \(\psi, \Psi\) \(\omega, \Omega\) \(e=mc^2\) \(\pi=\frac{c}{d}\) \(\frac{d}{dx}e^x=e^x\) \(\frac{d}{dx}\int_{0}^{\infty} f(s)ds=f(x)\) \(f(x)=\sum_i 0^{\infty}\frac{f^{(i)}(0)}{i!}x^i\) \(x=\sqrt(\frac{x_i}{z}y)\)
\(\hat{a}\) \(\check{a}\) \(\tilde{a}\) \(\acute{a}\) \(\grave{a}\) \(\dot{a}\) \(\ddot{a}\) \(\breve{a}\) \(\bar{a}\) \(\vec{a}\) \(\widehat{a}\) \(\widetilde{a}\) \(\alpha\) \(\beta\) \(\gamma\) \(\delta\) \(\epsilon\) \(\varepsilon\) \(\zeta\) \(\eta\) \(\theta\) \(\vartheta\) \(\iota\) \(\kappa\) \(\lambda\) \(\mu\) \(\nu\) \(\xi\) \(\pi\) \(\varpi\) \(\rho\) \(\varrho\) \(\sigma\) \(\varsigma\) \(\tau\) \(\upsilon\) \(\phi\) \(\varphi\) \(\chi\) \(\psi\) \(\omega\) \(\Gamma\) \(\Delta\) \(\Theta\) \(\Lambda\) \(\Xi\) \(\Pi\) \(\Sigma\) \(\Upsilon\) \(\Phi\) \(\Psi\) \(\Omega\)
\(<\) \(>\) \(=\) \(\leq or \le\) \(\geq or \ge\) \(\equiv\) \(\ll\) \(\gg\) \(\doteq\) \(\prec\) \(\succ\) \(\sim\) \(\preceq\) \(\simeq\) \(\subset\) \(\supset\) \(\approx\) \(\subseteq\) \(\supseteq\) \(\cong\) \(\sqsubset\) \(\sqsupset\) \(\Join\) \(\sqsubseteq\) \(\sqsupseteq\) \(\bowtie\) \(\in\) \(\ni or \owns\) \(\propto\) \(\vdash\) \(\dashv\) \(\models\) \(\mid\) \(\parallel\) \(\perp\) \(\smile\) \(\frown\) \(\asymp\) \(:\) \(\notin\) \(\neq or \ne\) \(+\) \(-\) \(\pm\) \(\mp\) \(\triangleleft\) \(\cdot\) \(\div\) \(\triangleright\) \(\times\) \(\setminus\) \(\star\) \(\cup\) \(\cap\) \(\ast\) \(\sqcup\) \(\sqcap\) \(\circ\) \(\vee or \lor\) \(\wedge or \land\) \(\bullet\) \(\oplus\) \(\ominus\) \(\diamond\) \(\odot\) \(\oslash\) \(\uplus\) \(\otimes\) \(\bigcirc\) \(\amalg\) \(\bigtriangleup\) \(\bigtriangledown\) \(\dagger\) \(\lhd\) \(\rhd\) \(\ddagger\) \(\unlhd\) \(\unrhd\) \(\wr\) \(\sum\) \(\prod\) \(\coprod\) \(\int\) \(\bigcup\) \(\bigcap\) \(\bigsqcup\) \(\oint\) \(\bigvee\) \(\bigwedge\) \(\bigoplus\) \(\bigotimes\) \(\bigodot\) \(\biguplus\)
\(\leftarrow or \gets\) \(\rightarrow or \to\) \(\leftrightarrow\) \(\Leftarrow\) \(\Rightarrow\) \(\Leftrightarrow\) \(\mapsto\) \(\hookleftarrow\) \(\leftharpoonup\) \(\leftharpoondown\) \(\longleftarrow\) \(\longrightarrow\) \(\longleftrightarrow\) \(\Longleftarrow\) \(\Longrightarrow\) \(\Longleftrightarrow\) \(\longmapsto\) \(\hookrightarrow\) \(\rightharpoonup\) \(\rightharpoondown\) \(\iff (bigger spaces)\) \(\uparrow\) \(\downarrow\) \(\updownarrow\) \(\Uparrow\) \(\Downarrow\) \(\Updownarrow\) \(\nearrow\) \(\searrow\) \(\swarrow\) \(\nwarrow\) \(\leadsto\) \((\) \()\) \([ or \lbrack\) \(] or \rbrack\) \(\{ or \lbrace\) \(\} or \rbrace\) \(\langle\) \(\rangle\) \(\lfloor\) \(\rfloor\) \(/\) \(\backslash\) \(\uparrow\) \(\downarrow\) \(\updownarrow\) \(| or \vert\) \(\lceil\) \(\Uparrow\) \(\Downarrow\) \(\Updownarrow\) \(\| or \Vert\) \(\rceil\) \(\lgroup\) \(\rgroup\) \(\arrowvert\) \(\Arrowvert\) \(\lmoustache\) \(\bracevert\) \(\rmoustache\) \(\dots\) \(\cdots\) \(\vdots\) \(\ddots\) \(\hbar\) \(\imath\) \(\jmath\) \(\ell\) \(\Re\) \(\Im\) \(\aleph\) \(\wp\) \(\forall\) \(\exists\) \(\mho\) \(\partial\) \('\) \(\prime\) \(\emptyset\) \(\infty\) \(\nabla\) \(\triangle\) \(\Box\) \(\Diamond\) \(\bot\) \(\top\) \(\angle\) \(\surd\) \(\diamondsuit\) \(\heartsuit\) \(\clubsuit\) \(\spadesuit\) \(\neg or \lnot\) \(\flat\) \(\natural\) \(\sharp\) \(\dag\) \(\ddag\) \(\S\) \(\P\) \(\copyright\) \(\pounds\) \(\ulcorner\) \(\urcorner\) \(\llcorner\) \(\lrcorner\) \(\lvert\) \(\rvert\) \(\lVert\) \(\rVert\) \(\digamma\) \(\varkappa\) \(\beth\) \(\daleth\) \(\gimel\)
还有我就不写了，太费肝了
