Number theory (Primer)

Several important things to keep in mind:

1. Euclid theorem (Euclidean law)

int gcd (intint  b) {    return b==0? a:gcd (b, a%b);}

2. Extended Euclidean (Ax+by = gcd (A, B)

void E_GCD (ll A, ll B, LL &d, LL &x, LL &y) {    if(b==0) {         1 0; D = A;    }      Else {        E_GCD (b, a%B, D, y, x);        Y-= x* (A/b);    }}

3. Chinese remainder theorem

Congruence equations

x ≡a1 (mod m1)

x ≡a2 (mod m2)

... ...

x ≡ak (mod mk)

The set of all solutions to a set of equations is:

x1 = n1*a1 + n2*a2 + ... + nk*akwhere Ni mod mi = 1,ni = ai * ti, can be used Euclidean expansion theorem to seek ti. where m = m1*m2*m3 *MN;

#include <iostream>using namespacestd; //extended Euclidean theorem with parameters that can be negative    voidEXOJLD (intAintBint&x,int&y) {          //According to Euclid's theorem.        if(b = =0){//the greatest common divisor of any number with 0 is its own. x =1; Y=0; }Else{              intx1, y1; EXOJLD (b, a%B, x1, y1); if(A*b <0){//The opposite of the different numberx =-Y1; Y= A/b*y1-X1; }Else{//Same numberx =Y1; Y= x1-a/b*Y1; }          }      }      //residual theorem    intCALSYDL (intA[],intM[],intk) {          intN[K];//This can be deleted        intMM =1;//least common multiple        intresult =0;  for(inti =0; I < K; i++) {mm*=M[i]; }           for(intj =0; J < K; J + +){              intL, J; EXOJLD (mm/M[J],-M[j], L, J); N[J]= M[j] * j +1;//1N[J] = mm/m[j] * L;//2 "Note" 1 and 2 values should be equal. Result + = n[j]*A[j]; }          return(Result% mm + mm)% mm;//fall between (0, MM), this is written to prevent result initial negative, in this case can not be negative may be directly written: return result%mm;     }                  intMain () {inta[3] = {2,3,2}; intm[3] = {3,5,7}; cout<<"Results:"<<calsydl (A, M,3) <<Endl; }  

4. Euler functions(The number of all the numbers in front of a number with this number coprime)  Euler function expression formula: Euler (x) =x (1-1/P1) (1-1/P2) (1-1/P3) (1-1/P4) ... (1-1/PN), where P1,P2......PN is the all-factor of x, and X is an integer that is not 0. Euler (1) =1 (the number of unique and 1 coprime is 1 itself).

    //solving Euler functions directly    intEulerintN) {//return Euler (n)         intRes=n,a=O;  for(intI=2; i*i<=a;i++){               if(a%i==0) {res=res/i* (I-1);//Division first to prevent overflow of intermediate data                  while(a%i==0) a/=i; }           }           if(a>1) res=res/a* (A-1); returnRes; }            //selection method to play the European pull function table    #defineMax 1000001intEuler[max]; voidInit () {euler[1]=1;  for(intI=2; i<max;i++) Euler[i]=i;  for(intI=2; i<max;i++)              if(euler[i]==i) for(intj=i;j<max;j+=i) euler[j]=euler[j]/i* (I-1);//Division first to prevent overflow of intermediate data}

Number theory (Primer)