nuber theory

Author: Nadim-Mahmud

Source/Number Theory/eulers totient sieve.cpp

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+/// ***Eulers totients Sieve [n*n]
+/// Number of Co-prime upto a number
+
+bool mr[MX];
+int pi[MX];
+
+void phi(){
+    int i,j;
+    pi[1] = 1;
+    for(i=2;i<MX;i++){
+        if(!mr[i]){
+            for(j=i;j<MX;j+=i){
+                if(pi[j]==0) pi[j] = j;
+                mr[j] = 1;
+                pi[j] = pi[j]/i*(i-1);
+            }
+        }
+    }
+}
+

Source/Number Theory/miller rabin preimality test.cpp

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+
+/// ***Miller-Rabin Primality test [120]
+
+/** usigned long long is not working sometimes
+*   left shift == mult by 2 right shift == divide bye 2
+*   multiplying two numbers (a*b)%c avoiding overflow
+*/
+ll mulmod(ll a, ll b, ll mod){
+    ll x = 0,y = a % mod;
+    while (b > 0){
+        if (b&1) x = (x + y) % mod;
+        y = (y<<1) % mod;
+        b >>= 1;
+    }
+    return x % mod;
+}
+
+//Bigmod
+ll modulo(ll n, ll r, ll mod){
+    ll x = 1;
+    while (r > 0){
+        if (r&1) x = mulmod(x,n,mod);
+        n = mulmod(n,n,mod);
+        r >>= 1;
+    }
+    return x % mod;
+}
+/// higher value of "it" ensure higher percision [recomendation 7]
+
+bool MillerRabin(ll p,int it)
+{
+    if (p < 2) return 0;
+    if (p != 2 && p % 2==0) return 0;
+
+    ll i,a,tmp,mod,s=p-1;
+
+    while(s%2==0){
+        s>>=1;
+    }
+    for(i=0;i<it;i++){
+        a = rand()%(p-1)+1;
+        tmp = s;
+        mod = modulo(a, tmp, p);
+        if(mod==1 || mod == p-1)continue;
+        while (tmp!=p-1&&mod!=1&&mod!=p-1){
+            mod = mulmod(mod, mod, p);
+            tmp <<= 1;
+        }
+        if(mod!=p-1) return 0;
+    }
+    return 1;
+}
+
+/** this body of miller rabin giving correct ans upto 10^9
+*   with lower time complexity
+*/
+
+for(i=1;i<=it;i++){
+    a = rand() % (p - 1) + 1, tmp = s;
+    mod = mod_pow(a, tmp, p);
+    while(tmp != p - 1 && mod != 1 && mod != p - 1) {
+        mod = mod_mul(mod, mod, p);
+        tmp *= 2;
+    }
+    if (mod != p - 1 && tmp % 2 == 0) return false;
+}
+

Source/Number Theory/moduler inverse.cpp

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+
+/// *** Modular Inverse [ log(n) ]
+
+/**  [ large division = (upper%M)*modi(low)) ]
+*    [ after modular division value should be moded ]
+*    [ mainly it returns an iverse and multiplicable value ]
+*    [ for loop == first loop throw all number(multiply) with simple mod -]
+*    [- then modular inverse ]
+*/
+
+#define M 1000000007
+#define pii pair<ll,ll>
+pii extnuc(ll a,ll b)
+{
+    if(b==0)return pii(1,0);
+    pii d=extnuc(b,a%b);
+    return pii(d.second,d.first-d.second*(a/b));
+}
+
+ll modi(ll n) /// n must be moded value
+{
+    pii d=extnuc(n,M);
+    return ((d.first%M)+M)%M;
+}
+
+///Now factorial & inverse factorial with MOD
+void fact()
+{
+    ll i;
+    fr[0]=fi[0]=1;
+    for(i=1;i<2000007;i++){
+        fr[i]=(fr[i-1]*i)%M;
+        fi[i]=(fi[i-1]*modi(i))%M;
+        //P(fr[i])
+    }
+}
+

Source/Number Theory/number of divisor in qube root.cpp

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+
+/// ***Number of divisors [Quberoot(n)]
+
+/** There can be only two prime after qube root
+*   so we factorize upto quberoot by the trial divison then
+*   handle ramainig <=2 prime
+*   Miller Rabin needed
+*/
+
+ll NOD(ll n){
+    ll i,j,r,nod=1,cn;
+    for(i=1;i<=in&&(pr[i]*pr[i]*pr[i])<=n;i++){
+        cn = 1;
+        while(n%pr[i]==0){
+            n /= pr[i];
+            cn++;
+        }
+        nod *= cn;
+    }
+
+    r = sqrtl(n);
+    while((r+1)*(r+1)<=n) r++;
+
+    if(MillerRabin(n,8)) nod *= 2;
+    else if(r*r==n&&MillerRabin(r,8)) nod *= 3;
+    else if(n!=1) nod *= 4;
+
+    return nod;
+}
+