数论整理（upd:2021.10.7）
数论
by AmanoKumiko
1.扩欧
\[ax+by=1
\]
\[ax+by=gcd(a,b)
\]
\[ax+by=gcd(b,a\ mod\ b)
\]
\[ax+by=bx1+(a-b\lfloor\frac{a}{b}\rfloor)y1
\]
\[ax+by=bx1+ay1-b\lfloor\frac{a}{b}\rfloor·y1
\]
\[ax+by=ay1+b(x1-\lfloor\frac{a}{b}\rfloor·y1)
\]
\[x=y1,y=x1-\lfloor\frac{a}{b}\rfloor·y1
\]
2.逆元
(1)
\[ax≡1(mod\ p)
\]
(2)
\[ax+by=1
\]
(3)
\[k=\lfloor\frac{p}{i}\rfloor,r=p\ mod\ i
\]
\[ki+r≡0(mod\ p)
\]
\[kr^{-1}+i^{-1}≡0(mod\ p)
\]
\[i^{-1}≡-kr^{-1}(mod\ p)
\]
\[i^{-1}≡-\lfloor\frac{p}{i}\rfloor·(p\ mod\ i)^{-1}(mod\ p)
\]
3.φ
(1)
\[n'=\frac{n}{p1}
\]
\[p1\mid n'
\]
\[φ(n)=p1·φ(n')
\]
\[p1\nmid n'
\]
\[φ(n)=(p1-1)·φ(n')
\]
(2)
\[n=\sum_{d|n}φ(d)
\]
(3)
\[\sum_{i=1}^{n}[gcd(i,n)=1]i=\sum_{i=1}^{n}\sum_{d|gcd(i,n)}μ(d)i
\]
\[=\sum_{d|n}μ(d)\sum_{i=1}^{\frac{n}{d}}di
\]
\[=\sum_{d|n}μ(d)d\frac{(1+\frac{n}{d})\frac{n}{d}}{2}
\]
\[=\frac{n}{2}\sum_{d|n}μ(d)+μ(d)·\frac{n}{d}
\]
\[=\frac{1}{2}(nφ(n)+ε(n))
\]
(4)
\[d=gcd(i,j)
\]
\[φ(ij)=\frac{φ(i)φ(j)d}{φ(d)}
\]
\[∀n=p^a,m=p^b,a<=b,若φ(nm)=\frac{φ(n)φ(m)gcd(n,m)}{φ(gcd(n,m))},则上式成立
\]
\[n'=n*px
\]
\[φ(n'm)=\frac{φ(nm)φ(px)gcd(nm,px)}{φ(gcd(nm,px))}=\frac{φ(nm)φ(px)}{φ(1)}=φ(nm)φ(px)
\]
\[φ(p^x)=p^x-\lfloor\frac{p^x}{p}\rfloor=p^x-p^{x-1}
\]
\[右边=\frac{φ(p^a)φ(p^b)gcd(p^a,p^b)}{φ(gcd(p^a,p^b))}
\]
\[\frac{(p^a-p^{a-1})(p^b-p^{b-1})p^a}{(p^a-p^{a-1})}
\]
\[p^{a+b}-p^{a+b-1}
\]
\[φ(p^{a+b})=左边
\]
4.数论分块
\[\sum_{i=2}^{n}\lfloor\frac{n}{i}\rfloor
\]
\[l=2,r=n/(n/l)
\]
\[l=r+1
\]
\[ans=\sum(r-l+1)·(n/l)
\]
5.积性函数
(1)
\[∀x∈N,y∈N,gcd(x,y)=1,f(1)=1,都有f(xy)=f(x)f(y),则f(x)是积性函数
\]
\[∀x∈N,y∈N,f(1)=1,都有f(xy)=f(x)f(y),则f(x)是完全积性函数
\]
(2)
\[ε(n)=[n=1]
\]
\[id(n)=n
\]
\[σ(n)=\sum_{d|n}d
\]
\[I(n)=1
\]
\[φ(n)=\sum_{i=1}^{n}[gcd(i,n)=1]
\]
\[μ(n)=
\begin{cases}
(-1)^k,n=p1p2...pk\\
0,n=p^2q\\
1,n=1
\end{cases}
\]
6.狄利克雷卷积
(1)
\[(f*g)(n)=\sum_{d|n}f(d)g(\frac{n}{d})
\]
(2)
\[(f*g=g*f)
\]
\[(f*g)*h=f*(g*h)
\]
\[f*(g+h)=f*g+f*h
\]
\[f*ε=f
\]
7.莫比乌斯反演
(1)
\[\sum_{d|n}μ(d)=
\begin{cases}
1,n=1\\
0,n≠1
\end{cases}
=ε(n),μ*I=ε
\]
\[n=Π_{i=1}^kpi^{ci},n'=Π_{i=1}^kpi
\]
\[∴\sum_{d|n}μ(d)=\sum_{d|n'}μ(d)=\sum_{i=0}^kC_{i}^k·(-1)^k=(1+(-1))^k=0^k
\]
\[k=0<=>n=0,k≠0<=>n≠0
\]
\[\sum_{d|n}μ(d)=ε(n)
\]
(2)
\[f(n)=\sum_{d|n}g(\frac{n}{d})
\]
\[g(n)=\sum_{d|n}μ(d)f(\frac{n}{d})
\]
\[f=I*g
\]
\[μ*f=μ*I*g
\]
\[μ*f=ε*g=g
\]
(3)
\[\sum_{i=1}^n\sum_{j=1}^mai·bj=\sum_{i=1}^nai\sum_{j=1}^mbj
\]
\[\sum_{i=1}^nai\sum_{j=1}^mbj=\sum_{j=1}^mbj\sum_{i=1}^nai
\]
\[\sum_{i=1}^na_i\sum_{d|i}b_d=\sum_{d=1}^nb_d\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}a_{id}
\]
(4)
\[\sum_{i=1}^n\sum_{j=1}^m[gcd(i,j)=k]
\]
\[\sum_{i=1}^{\lfloor\frac{n}{k}\rfloor}\sum_{j=1}^{\lfloor\frac{m}{k}\rfloor}[gcd(i,j)=1]
\]
\[\sum_{i=1}^{\lfloor\frac{n}{k}\rfloor}\sum_{j=1}^{\lfloor\frac{m}{k}\rfloor}ε(gcd(i,j))
\]
\[\sum_{i=1}^{\lfloor\frac{n}{k}\rfloor}\sum_{j=1}^{\lfloor\frac{m}{k}\rfloor}\sum_{d|gcd(i,j)}μ(d)
\]
\[\sum_{d=1}μ(d)\sum_{i=1}^{\lfloor\frac{n}{k}\rfloor}[d\mid i]\sum_{j=1}^{\lfloor\frac{m}{k}\rfloor}[d\mid j]
\]
\[\sum_{d=1}μ(d)\lfloor\frac{n}{kd}\rfloor\lfloor\frac{m}{kd}\rfloor
\]
8.杜教筛
(1)
\[\sum_{i=1}^nf(i)
\]
\[S(n)=\sum_{i=1}^nf(i)
\]
\[h=g*f
\]
\[\sum_{i=1}^nh(i)=\sum_{i=1}^n\sum_{d|i}g(d)f(\frac{n}{d})
\]
\[\sum_{i=1}^nh(i)=\sum_{d=1}^ng(d)\sum_{i=1}^{\lfloor\frac{n}{d}\rfloor}f(i)
\]
\[\sum_{i=1}^nh(i)=\sum_{d=1}^ng(d)S(\lfloor\frac{n}{d}\rfloor)
\]
\[\sum_{i=1}^nh(i)=g(1)S(n)+\sum_{d=2}^ng(d)S(\lfloor\frac{n}{d}\rfloor)
\]
\[g(1)S(n)=\sum_{i=1}^nh(i)-\sum_{d=2}^ng(d)S(\lfloor\frac{n}{d}\rfloor)
\]
(2)
\[\sum_{i=1}^nμ(i)
\]
\[μ*I=ε
\]
\[S(n)=\sum_{i=1}^nε(i)-\sum_{d=2}^nS(\lfloor\frac{n}{d}\rfloor)
\]
\[S(n)=1-\sum_{d=2}^nS(\lfloor\frac{n}{d}\rfloor)
\]
(3)
\[\sum_{i=1}^nφ(i)
\]
\[id=φ*I
\]
\[S(n)=\sum_{i=1}^nid(i)-\sum_{d=2}^nS(\lfloor\frac{n}{d}\rfloor)
\]
\[S(n)=\sum_{i=1}^ni-\sum_{d=2}^nS(\lfloor\frac{n}{d}\rfloor)
\]
9.CRT
\[\begin{cases}
x≡a1(mod\ n1)\\
x≡a2(mod\ n2)\\
...\\
x≡ak(mod\ nk)\\
\end{cases}
\]
\[N=Π_{i=1}^kni
\]
\[mi=\frac{N}{ni}
\]
\[mixi≡1(mod\ ni)
\]
\[ci=mimi^{-1}
\]
\[x=\sum_{i=1}^kaici(mod\ N)
\]
10.Lucas
\[C_n^m=C_{n\ mod \ p}^{m\ mod \ p}·C_{n/p}^{m/p}
\]
11.Burnside&Pólya
\[A,B：有限集合，X:A到B的所有映射
\]
\[G：置换群，X/G：所有作用在X上的置换群的等价类的集合
\]
\[|X/G|=\frac{1}{|G|}∑_{i=1}^{|G|}|f(G_i)|
\]
\[f(G_i)即为经过一轮置换后仍等价于它本身的点集合
\]
\[|X/G|=\frac{1}{|G|}∑_{i=1}^{|G|}B^{c(G_i)}
\]
\[c(G_i)即为将C_i拆成不交的循环置换的个数
\]
12.Min_25
\[F_k(n)=\sum_{i>=k,p_i^2<=n}\sum_{c>=1,p_i^c<=n}f(p_i^c)([c>1]+F_{i+1}(\lfloor\frac{n}{p_i^c}\rfloor))+\sum_{i>=k}f(p_i)
\]
\[F_k(n)=\sum_{i>=k,p_i^2<=n}\sum_{c>=1,p_i^{c+1}<=n}(f(p_i^c)F_{i+1}(\lfloor\frac{n}{p_i^c}\rfloor)+f(p_i^{c+1}))+F_{prime}(n)-F_{prime}(p_{k-1})
\]
\[G_k(n)=G_{k-1}(n)-g(\lfloor\frac{n}{p_k}\rfloor)(G_{k-1}(\lfloor\frac{n}{p_k}\rfloor)-G_{k-1}(p_{k-1}))
\]
