Prime Numbers.cpp

#include<iostream>
#include<cmath>
#include<algorithm>
#include<vector>
#include<bitset>
using namespace std;

/* 

NOTE: The below Benchmark only show how efficient a method is for lots of prime number tests
      However, each method mentioned above has it own special usecase (example: some are fast
      for less number of prime tests, some are fast for lots of prime tests).
        • is_prime_simple(i)  - best for few prime tests
        • SieveOfEratosthenes - best for lots of prime tests

➜ Performance Benchmarks for 0 <= i <= 100'000'000 (i.e. 1e8) when
  compiled using `g++ -O2 "Prime Numbers.cpp"` on Intel i5-5200U
➜ Bash command: `command time ./a.out`

•    SieveOfEratosthenes   ->   0.817 s (and ~11.9 MB of RAM)
•  dp_prime_factors[i]==i  ->   1.737 s (and ~381.6 MB of RAM)
•         is_prime(i)      ->  12.077 s
•       is_prime_dp(i)     ->  23.710 s
•     is_prime_simple(i)   -> 150.718 s

*/

//--------------------------------------------------------------------------------------------------------------------

// NOTE: Tested and works perfectly
// Time Complexity = O(sqrt(n))
// Space Complexity = O(1)
template<typename T>
bool is_prime_simple(const T &n) {
    // This funtion will return FALSE for ALL NEGATIVE numbers
    // Corner cases
    if (n < 2) return false;
    if (n < 4) return true;

    // This is checked so that we can skip middle five numbers in below loop
    if ((n % 2) == 0 or (n % 3) == 0) return false;

    auto nSqrt = static_cast<uint64_t>(sqrt(abs(n)));
    // REFER: https://www.geeksforgeeks.org/sieve-of-eratosthenes/
    //          Factors of 2 and 3, https://media.geeksforgeeks.org/wp-content/cdn-uploads/SieveofEratosthenes3.jpg
    // NOTE : We use `i+=6` to skip all factors of 2 and 3
    for (uint64_t i = 5; i <= nSqrt; i += 6)
        if ((n % i) == 0 or (n % (i + 2)) == 0)
            return false;

    return true;
}

//--------------------------------------------------------------------------------------------------------------------

// NOTE: Tested and works perfectly
// The sieve of Eratosthenes is one of the most efficient ways to find
// all primes smaller than n when n is smaller than 10 million or so
// Can check all number from (-∞,N]
// Time Complexity = O(n*log(log(n)))
// Space Complexity = O(n)
template<uint64_t N>
struct SieveOfEratosthenes{
    // Constructs a bitset with all bits set to zero
    bitset<N+1> dp_primes;

    constexpr SieveOfEratosthenes() {
        dp_primes.set();  // set all bits to `true`
        for(uint64_t i = 2; i*i <= N; ++i){
            if(dp_primes.test(i)){
                for(uint64_t j = i*i; j <= N; j+=i)
                    dp_primes.reset(j);
            }
        }
    }

    template<typename T>
    inline bool isprime(const T &num) const {
        if(num <= 1 || N < num) return false;
        return this->dp_primes.test(num);
    }
};

// NOTE: `SieveOfEratosthenes` MUST be global, otherwise it gives segmentation fault
// NOTE: uses just ~11.9 MB of RAM
SieveOfEratosthenes<100'000'000> sieveOfEratosthenes;

//--------------------------------------------------------------------------------------------------------------------

// Smallest factor for numbers from 1 to "N" are stored in "dp_prime_factors"
// Store smallest prime factor for all numbers in the range [0, N]
struct SmallestPrimeFactor{
    std::vector<uint32_t> dp_prime_factors;
    SmallestPrimeFactor(const uint32_t &N){
        dp_prime_factors.resize(N+1, 0);
        dp_prime_factors[0] = dp_prime_factors[1] = 1;
        uint64_t i = 2;
        for(; i*i <= N; ++i){
            if(dp_prime_factors[i]==0){
                dp_prime_factors[i] = i;
                for(uint64_t j = i*i; j <= N; j+=i)
                    if(dp_prime_factors[j]==0)
                        dp_prime_factors[j] = i;
            }
        }
        for(; i <= N; ++i) 
            if(dp_prime_factors[i]==0) dp_prime_factors[i] = i;
    }

    template<typename T>
    inline bool isprime(const T &num){
        if(num <= 1 || dp_prime_factors.size() <= num) return false;
        return num == dp_prime_factors[num];
    }
};

// NOTE: `SmallestPrimeFactor` MUST be global, otherwise it gives segmentation fault
// NOTE: uses ~381.6 MB of RAM
// SmallestPrimeFactor smallestPrimeFactor(100'000'000);

//--------------------------------------------------------------------------------------------------------------------

// NOTE: Tested and works perfectly
// Correct answer for all n <= ( 1073741824 = power(2,30) )
// Time Complexity = O(log2(n))
// Space Complexity = O(1)

// sizeof(composite_numbers) / sizeof(composite_numbers[0]) = 189
const int32_t composite_numbers[] = {587861, 873181, 2035153, 2508013, 3828001, 4335241, 6189121, 6733693, 8725753, 8902741, 9439201, 10024561, 10267951, 10403641, 12032021, 13773061, 14469841, 14676481, 15247621, 17098369, 17236801, 17316001, 19328653, 19384289, 23382529, 24904153, 25276421, 26280073, 29111881, 30058381, 31405501, 34657141, 35703361, 37439201, 37964809, 50201089, 53711113, 56052361, 60957361, 62289541, 62756641, 64377991, 67194401, 68154001, 79411201, 79624621, 80918281, 82929001, 82995421, 84350561, 90014653, 90698401, 92625121, 96895441, 99036001, 99830641, 101649241, 106485121, 108596953, 110135821, 114910489, 115039081, 116151661, 116682721, 118901521, 119327041, 124630273, 127664461, 133800661, 134696801, 138736153, 139952671, 143106133, 145206361, 146843929, 148910653, 153589801, 155203361, 163954561, 171454321, 171679561, 172947529, 173401621, 174479729, 178482151, 178837201, 180115489, 186393481, 188516329, 189941761, 200753281, 207132481, 212027401, 214852609, 216821881, 217123069, 221884001, 228842209, 230357761, 230630401, 235476697, 258634741, 261703417, 263428181, 275283401, 279377281, 288120421, 299736181, 308448649, 313338061, 319053281, 320454751, 326695141, 328573477, 329153653, 329769721, 361307521, 362569201, 363245581, 366532321, 366652201, 382304161, 395044651, 405739681, 413138881, 414395701, 416964241, 418226581, 467491753, 478317601, 481239361, 483006889, 483029821, 492559141, 507960001, 524151253, 531681281, 532758241, 542497201, 551672221, 555465601, 558977761, 569332177, 579606301, 580565233, 595405201, 595590841, 597717121, 605221501, 612816751, 620169409, 625060801, 625482001, 630888481, 644004817, 652969351, 686059921, 702683101, 710382401, 739444021, 743404663, 771043201, 775368901, 784966297, 794937601, 804978721, 822531841, 824389441, 842960981, 851703301, 855734401, 863984881, 876850801, 891706861, 918661501, 922845241, 932148253, 935794081, 939947009, 940123801, 941056273, 944832533, 951941161, 954732853, 955134181, 958735681, 962442001, 985052881, 1027334881};

int64_t pow(int64_t, uint64_t, uint64_t);

bool is_prime(const int32_t &n){
    if(n==2 or n==3 or n==5) 
        return true;
    if(n < 2 or (n % 2) == 0 or (n % 3) == 0 or (n % 5) == 0)
        return false;
    if(n < 49)
        return true;
    if((n % 7) == 0 or (n % 11) == 0 or (n % 13) == 0 or (n % 17) == 0 or
        (n % 19) == 0 or (n % 23) == 0 or (n % 29) == 0 or (n % 31) == 0 or
        (n % 37) == 0 or (n % 41) == 0 or (n % 43) == 0 or (n % 47) == 0)
        return false;
    if(n < 2809)  // 53*53 == 2809
        return true;
    if(binary_search(begin(composite_numbers), end(composite_numbers), n))
        return false;
    if((pow(2, n, n) == 2) and (pow(74623, n, n) == (74623 % n)) and (pow(10659, n, n) == (10659 % n)))
        return true;
    return false;
}

// To compute base^exp under modulo "mod"
int64_t pow(int64_t base, uint64_t exp, uint64_t mod) {
    // Generally mod = 1000000007
    // valid ONLY for +ve exponent
    int64_t result = 1;
    base %= mod;
    while (exp > 0) {
        if ((exp & 1) == 1) {
            result = (result * base) % mod; // result = simple_product(result, base, mod);
        }
        base = (base * base) % mod; // base = simple_product(base, base, mod);
        exp >>= 1; //exp = exp / 2;
    }
    return result;
}

//--------------------------------------------------------------------------------------------------------------------

// NOTE: Tested and works perfectly
// largest prime tested in dp_primes = 10005
// largest number that can be tested with DP = 100100025 OR 1e8
// dp_primes.size() = 1229
std::vector<int32_t> dp_primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,9941,9949,9967,9973};

template<typename T>
bool is_prime_dp(T n) {
    // This funtion will return FALSE for ALL NEGATIVE numbers
    // Corner cases
    if (n < 2) return false;
    if (n < 4) return true;

    // This is checked so that we can skip middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0) return false;

    auto dp_primes_len = dp_primes.size();
    auto nSqrt = static_cast<uint64_t>(sqrt(abs(n)));
    uint64_t i = 0;

    for (; i < dp_primes_len && dp_primes[i] <= nSqrt; ++i)
        if(n % dp_primes[i] == 0)
            return false;

    // this is done to fit i in the series: 5,11,17,23,...
    auto &last_prime = dp_primes[dp_primes_len - 1];
    i = last_prime - ((last_prime - 5) % 6);

    for (; i <= nSqrt; i += 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// WARNING: NOT Tested
void append_till_n_dp_primes(int n){
    if(dp_primes.size() >= n) return;

    auto &last_prime = dp_primes[dp_primes.size() - 1];
    auto new_primes = last_prime - ((last_prime - 5) % 6) + 6; // this is done to fit i in the series: 5,11,17,23,...
    for(int count = dp_primes.size(); count < n; new_primes += 6){
        if(is_prime_dp(new_primes)){
            dp_primes.push_back(new_primes);
            count++;
        }
    }
}

//--------------------------------------------------------------------------------------------------------------------

int main(){
    for(int i = 0; i < 100'000'000; ++i){
        if(
            sieveOfEratosthenes.isprime(i)
            // smallestPrimeFactor.isprime(i)
            // is_prime(i)
            // is_prime_dp(i)
            // is_prime_simple(i)
        )
        {
            // cout << i << ", ";
        }
    }

    cout << "\n\n";
    return 0;
}