Programs related to number theory written in C++ when I was a high school student. I don't guarantee that they're error-free.

EulersCriterion.cpp

#include <iostream>
#include <fstream>
#include <sstream>
#include <string>
#include <cstring>
#include <string.h>
#include <vector>
#include <stdio.h>
#include <stdlib.h>
using namespace std;

int main() {
    //Explain the statement of Euler's Criterion
    string statement("This program helps user to understand Euler's Criterion:\n");
    statement = statement + "For any odd prime 𝑝 and natural number 𝑎 such that (𝑎,𝑝) = 1,\n";
    statement = statement + "Let (𝑎|𝑝) ≡ 𝑎^[(𝑝 − 1)/2] (mod 𝑝) be the Legendre symbol, then\n\n";
    statement = statement + "⎧ (𝑎|𝑝) =  1 if 𝑎 is a quadratic residue modulo 𝑝\n⎨\n";
    statement = statement + "⎩ (𝑎|𝑝) = −1 if 𝑎 is a quadratic non-residue modulo 𝑝\n\n";

    //Obtain input
    int a = int();
    int p = int();
    int root = 0;
    cout << statement << "Enter the value of odd prime 𝑝: ";
    cin >> p;
    cout << "Enter the value of 𝑎 which is BETWEEN 1 AND 𝑝 − 1: ";
    cin >> a;
    std::stringstream ss;

    //Check whether a suits the format
    if (a % p == 0 || p < 0 || a < 0) {
    cerr << "Enter two coprime positive integers for the values of 𝑎 and 𝑝." << endl;
    return 1;
    }

    //Create a list of numbers {1,2,...,p - 1}
    vector<int> list;
    for (int i = 1; i < p; i++) {
    list.push_back(i);
    }

    //Sort the numbers into 𝑏 𝑏′≡ 𝑎 (mod 𝑝)
    string str = "";//Contains the thing to be print
    string listbody = "\nResults: \n";
    cout << listbody;
    while (!list.empty()) {
    int b = list.at(0);
    int originalSize = list.size();//Helps check if 𝑝 is prime
    for (int i = 0; i < list.size(); i++) {
        int instantHalf = list.at(i);
        int instantProduct = b * instantHalf;
        int remainder = instantProduct % p;
        if (remainder == a) {
        if (instantHalf == b) {
            root = b;
            ss << b << "² ≡ " << instantProduct << " ≡ " << remainder << " (mod " << p <<")\n";
            list.erase(list.begin());
        } else {
            ss << b << " × " << instantHalf << "≡ " << instantProduct << " ≡ " << remainder << " (mod " << p << ")\n";
            list.erase(list.begin()+i);
            list.erase(list.begin());
        }
        str = ss.str();
        cout << str;
        listbody += str;
        ss.str("");
        str.clear();
        break;
        }
    }

    //Terminate the program if 𝑝 is not prime to avoid infinite loop
    if (originalSize == list.size()) {
        cerr << "Enter an odd prime for the value of 𝑝." << endl;
        return 1;
    }
    }

    //Print out the finding
    if (root > 0) {
    root = p - root;//so that the smaller number is shown
    ss << "\n∵ " << root << "² ≡ " << root*root << " ≡ " << a << " (mod " << p << ")\n∴ " << a << " is a quadratic residue modulo " << p << ".\n";
    
    } else {
    ss << "\n∵ From the above list, there does NOT exist an integer 𝑥 such that 𝑥² ≡ " << a << " (mod " << p << ")\n∴ " << a << " is a quadratic non-residue modulo " << p << ".\n";
    }
    string finding = ss.str();
    ss.str("");
    cout << finding;

    //Print out the explanation
    ss << "\nBy the L.H.S. of the above list,\n  product of {1,2,...,𝑝 − 1}\n≡ (" << p << " − " << 1 << ")! (mod " << p << ")\n≡ −1 (mod " << p << ") (Wilson's Thm.)\n";
    string explanationL = ss.str();
    ss.str("");
    cout << explanationL;
    string explanationR = "\nBy the R.H.S. of the above list,\n";
    if (root == 0) {
    explanationR += "  product of {1,2,...,𝑝 − 1}\n≡ 𝑎^[(𝑝 − 1)/2]\n";
    } else {
    ss << explanationR << "∵ " << root << " × " << p - root << " ≡ " << root << " × −" << root << " ≡ −𝑎 (mod 𝑝)\n∴ product of {1,2,...,𝑝 − 1}\n≡ −𝑎^[(𝑝 − 1)/2] (mod 𝑝)\n";
    explanationR = ss.str();
    ss.str("");
    }
    cout << explanationR;
    string explanation = "\nAfter equating both sides,\n";
    /**Temporary value of the Legendre Symbol (If 𝑎 is QNR mod 𝑝, it'll be multiplied by -1)*/
    int legendre = 1;
    if (root == 0) {
    explanation += "𝑎^[(𝑝 − 1)/2]≡ −1(mod 𝑝)";
    legendre = -1;
    } else {
    explanation += "∵ −𝑎^[(𝑝 − 1)/2]≡ −1(mod 𝑝)";
    explanation += "∴ 𝑎^[(𝑝 − 1)/2]≡ 1(mod 𝑝)";
    }
    explanation += "\nBy the definition of Legendre symbol,";
    explanation += "(𝑎|𝑝) ≡ 𝑎^[(𝑝 − 1)/2] (mod 𝑝) = ";
    ss << legendre;
    explanation += ss.str();
    ss.str("");
    cout << explanation << endl;

    //Check if the user needs to save the list
    cout << "Would you like to save the above list? (Enter 'y' if yes. Otherwise, enter nothing.)";
    string input;
    cin >> input;
    if (input != "y") {
    return 0;
    }

    //Check if the file exists
    char fileName [80];
    int n;
    n = sprintf(fileName,"QR%dmod%d.txt",a,p);
    ifstream fin(fileName);
    if (fin) {
    // file exists
    cout << fileName << " already exists. Please refer to that." << endl;
    fin.close();
    return 0;
    }
    else {
    // file does not exist
    fin.close();
    }
    ofstream fout(fileName);// creates file for output and destroys file contents if the file already exists.
    fout << statement << listbody << finding << explanationL << explanationR << explanation;//Write contents to the file.
    fout.close();//Close the file

    return 0;
}

EulersCriterion_Win.cpp

#include <iostream>
#include <fstream>
#include <sstream>
#include <string>
#include <cstring>
#include <string.h>
#include <vector>
#include <stdio.h>
#include <stdlib.h>
using namespace std;

int main() {
    //Explain the statement of Euler's Criterion
    string statement("This program helps user to understand Euler's Criterion:\n");
    statement = statement + "For any odd prime p and natural number a such that (a,p) = 1,\n";
    statement = statement + "Let (a|p) = a^[(p - 1)/2] (mod p) be the Legendre symbol, then\n\n";
    statement = statement + "(a|p) =  1 if a is a quadratic residue modulo p\n";
    statement = statement + "(a|p) = -1 if a is a quadratic non-residue modulo p\n\n";

    //Obtain input
    int a = int();
    int p = int();
    int root = 0;
    cout << statement << "Enter the value of odd prime p: ";
    cin >> p;
    cout << "Enter the value of a which is BETWEEN 1 AND p - 1: ";
    cin >> a;
    std::stringstream ss;

    //Check whether a suits the format
    if (a % p == 0 || p < 0 || a < 0) {
        cerr << "Enter two coprime positive integers for the values of a and p." << endl;
        return 1;
    }

    //Create a list of numbers {1,2,...,p - 1}
    vector<int> list;
    for (int i = 1; i < p; i++) {
        list.push_back(i);
    }

    //Sort the numbers into b b′= a (mod p)
    string str = "";//Contains the thing to be print
    string listbody = "\nResults: \n";
    cout << listbody;
    while (!list.empty()) {
        int b = list.at(0);
        int originalSize = list.size();//Helps check if p is prime
        for (int i = 0; i < list.size(); i++) {
            int instantHalf = list.at(i);
            int instantProduct = b * instantHalf;
            int remainder = instantProduct % p;
            if (remainder == a) {
                if (instantHalf == b) {
                    root = b;
                    ss << b << "^2 = " << instantProduct << " = " << remainder << " (mod " << p <<")\n";
                    list.erase(list.begin());
                } else {
                    ss << b << " * " << instantHalf << "= " << instantProduct << " = " << remainder << " (mod " << p << ")\n";
                    list.erase(list.begin()+i);
                    list.erase(list.begin());
                }
                str = ss.str();
                cout << str;
                listbody += str;
                ss.str("");
                str.clear();
                break;
            }
        }

        //Terminate the program if p is not prime to avoid infinite loop
        if (originalSize == list.size()) {
            cerr << "Enter an odd prime for the value of p." << endl;
            return 1;
        }
    }

    //Print out the finding
    if (root > 0) {
        root = p - root;//so that the smaller number is shown
        ss << "\nSince " << root << "^2 = " << root*root << " = " << a << " (mod " << p << ")\ni.e. " << a << " is a quadratic residue modulo " << p << ".\n";

    } else {
        ss << "\nSince from the above list, there does NOT exist an integer x such that x^2 = " << a << " (mod " << p << ")\ni.e. " << a << " is a quadratic non-residue modulo " << p << ".\n";
    }
    string finding = ss.str();
    ss.str("");
    cout << finding;

    //Print out the explanation
    ss << "\nBy the L.H.S. of the above list,\n  product of {1,2,...,p - 1}\n= (" << p << " - " << 1 << ")! (mod " << p << ")\n= -1 (mod " << p << ") (Wilson's Thm.)\n";
    string explanationL = ss.str();
    ss.str("");
    cout << explanationL;
    string explanationR = "\nBy the R.H.S. of the above list,\n";
    if (root == 0) {
        explanationR += "  product of {1,2,...,p - 1}\n= a^[(p - 1)/2]\n";
    } else {
        ss << explanationR << "Since " << root << " * " << p - root << " = " << root << " * -" << root << " = -a (mod p)\ni.e. product of {1,2,...,p - 1}\n= -a^[(p - 1)/2] (mod p)\n";
        explanationR = ss.str();
        ss.str("");
    }
    cout << explanationR;
    string explanation = "\nAfter equating both sides,\n";
    /**Temporary value of the Legendre Symbol (If a is QNR mod p, it'll be multiplied by -1)*/
    int legendre = 1;
    if (root == 0) {
        explanation += "a^[(p - 1)/2]= -1(mod p)";
        legendre = -1;
    } else {
        explanation += "Since -a^[(p - 1)/2]= -1(mod p)";
        explanation += "i.e. a^[(p - 1)/2]= 1(mod p)";
    }
    explanation += "\nBy the definition of Legendre symbol,";
    explanation += "(a|p) = a^[(p - 1)/2] (mod p) = ";
    ss << legendre;
    explanation += ss.str();
    ss.str("");
    cout << explanation << endl;

    //Check if the user needs to save the list
    cout << "Would you like to save the above list? (Enter 'y' if yes. Otherwise, enter nothing.)";
    string input;
    cin >> input;
    if (input != "y") {
        return 0;
    }

    //Check if the file exists
    char fileName [80];
    int n;
    n = sprintf(fileName,"QR%dmod%d.txt",a,p);
    ifstream fin(fileName);
    if (fin) {
        // file exists
        cout << fileName << " already exists. Please refer to that." << endl;
        fin.close();
        return 0;
    }
    else {
        // file does not exist
        fin.close();
    }
    ofstream fout(fileName);// creates file for output and destroys file contents if the file already exists.
    fout << statement << listbody << finding << explanationL << explanationR << explanation;//Write contents to the file.
    fout.close();//Close the file

    return 0;
}

GausssLemma.cpp

#include <iostream>
#include <stdio.h>
#include <stdlib.h>
using namespace std;

//Return the least +ve residue
int congruence(int number, int N);
int gcd(int a, int b);
void printS(int number, int prime);
int Partition(int low,int high,int arr[]);
void Quick_sort(int low,int high,int arr[]);

int main(void) {
    int a,p;
    cout << "This program helps user to understand Gauss's Lemma:" << endl;
    cout << "Let p be an ODD prime and a be any integer." << endl;
    cout << "Supose S = {a,2a,3a,...,[(p-1)/2]*a}" << endl;
    cout << "and n be the number of elements in S that is greater than p/2" << endl;
    cout << "Then (a|p) = (-1)^n, where (a|p) is the Legendre symbol." << endl;
    cout << "Enter an ODD prime p: ";
    cin >> p;
    cout << "Enter an integer a: ";
    cin >> a;
    if (congruence(a,p) != a) {
        cout << a << " = " << congruence(a,p) << " (mod " << p << ")" << endl;
        a = congruence(a,p);
    }
    if (gcd(a,p) != 1) {
        char quit;
        cerr << "Enter an ODD prime for the value of p!" << endl;
        cout << "Press any key to continue.";
        cin >> quit;
        return 1;
    }
    printS(a,p);

    return 0;
}

int congruence(int number, int N) {
    if (N >= 0) {
        return number % N;
    } else {
        return congruence(-(number/N-1)*N+number,N);
    }
}

int gcd(int a, int b) {
    if (a % b == 0) {
        return b;
    } else {
        return gcd(b,a % b);
    }
}

void printS(int number, int prime) {
    cout << "S = {" << number;
    int *list = new int[(prime - 1) / 2];
    list[0] = number;
    for (int i = 2; i <= (prime - 1) / 2; i++) {
        list[i - 1] = number * i;
        cout << "," << list[i - 1];
    }
    cout << "}" << endl;

    cout << "Product of all elements in S" << endl;
    cout << "= " << list[0];
    for (int i = 1; i < (prime - 1) / 2; i++) {
        cout << " * " << list[i];
    }
    cout << "\n= (1 * " << list[0] << ")";
    for (int i = 1; i < (prime - 1) / 2; i++) {
        cout << " * (" << i + 1 << " * " << number << ")";
    }
    cout << "\n= " << (prime - 1) / 2 << "! * " << number << "^" << (prime - 1) / 2 << endl;
    cout << "= [(" << prime << " - 1) / 2]! * " << number << "^[(" << prime << " - 1) / 2]" << endl;
    cout << "= [(p - 1) / 2]! * a^[(p - 1) / 2]" << endl;
    cout << "= [(p - 1) / 2]! * (a|p) (mod p) (1)" << endl;
    cout << "Figure out [(p - 1) / 2]! (mod p):" << endl;

    cout << "Replace each element in S by its least +ve residue." << endl;
    cout << "S' = {" << number;
    for (int i = 2; i <= (prime - 1) / 2; i++) {
        list[i - 1] = congruence(list[i - 1],prime);
        cout << "," << list[i - 1];
    }
    cout << "}" << endl;

    cout << "It's trivial that product of all elements in S = that in S' (mod p)" << endl;
    cout << "i.e. Product of all elements in S' = [(p - 1) / 2]! * (a|p) (mod p) (2)" << endl;

    cout << "Rearrange them in acsending order: " << endl;
    Quick_sort(0,(prime - 1) / 2 - 1,list);
    cout << "S' = {" << list[0];
    for (int i = 1; i < (prime - 1) / 2; i++) {
        cout << "," << list[i];
    }
    cout << "}" << endl;

    int n = 0;
    cout << "Replace every element b in S' that is greater than p/2 by p - b." << endl;
    cout << "S\" = {" << list[0];
    for (int i = 2; i <= (prime - 1) / 2; i++) {
        if (list[i - 1] > prime / 2) {
            list[i - 1] = prime - list[i - 1];
            n++;
        }
        cout << "," << list[i - 1];
    }
    cout << "}" << endl;

    cout << "The above list is actually a list of all natural numbers within " << "(p - 1) / 2 = (" << prime << " - 1) / 2 = " << (prime - 1)/2 << "." << endl;
    cout << "n = " << n << endl;

    cout << "  Product of all elements in S\"" << endl;
    cout << "= [(p - 1) / 2]! (3a)" << endl;
    cout << "= " << (prime - 1)/2 << "!" << endl;
    cout << "= " << list[0];
    for (int i = 1; i < (prime - 1) / 2; i++) {
        cout << " * " << list [i];
    }
    cout << "\n= " << list[0];
    for (int i = 1; i < (prime - 1) / 2 - n; i++) {
        cout << " * " << list [i];
    }
    for (int i = (prime - 1) / 2 - n; i < (prime - 1) / 2; i++) {
        list[i] = prime - list [i];
        cout << " * (" << prime << " - " << list [i] << ")";
    }
    cout << " (Write the product in terms of elements in S')" << endl;
    cout << "= " << list[0];
    for (int i = 1; i < (prime - 1) / 2 - n; i++) {
        cout << " * " << list [i];
    }
    for (int i = (prime - 1) / 2 - n; i < (prime - 1) / 2; i++) {
        cout << " * (- " << list [i] << ")";
    }
    cout << " (mod " << prime << ")\n= (-1)^" << n;
    for (int i = 0; i < (prime - 1) / 2; i++) {
        cout << " * " << list [i];
    }
    cout << " (mod " << prime << ")\n= (-1)^" << n << " * product of all elements in S' (mod " << prime << ")" << endl;
    cout << "= (-1)^n * [(p - 1) / 2]! * (a|p) (mod p) (3b) (from (2))" << endl;
    cout << "Since (3a) and (3b) represent the same thing mod p," << endl;
    cout << "[(p - 1) / 2]! = (-1)^n * [(p - 1) / 2]! * (a|p) (mod p)" << endl;
    cout << "(a|p) = (-1)^n" << endl;
}

int Partition(int low,int high,int arr[])
{ int i,high_vac,low_vac,pivot/*,itr*/;
    pivot=arr[low];
    while(high>low)
    { high_vac=arr[high];

        while(pivot<high_vac)
        {
            if(high<=low) break;
            high--;
            high_vac=arr[high];
        }

        arr[low]=high_vac;
        low_vac=arr[low];
        while(pivot>low_vac)
        {
            if(high<=low) break;
            low++;
            low_vac=arr[low];
        }
        arr[high]=low_vac;
    }
    arr[low]=pivot;
    return low;
}

void Quick_sort(int low,int high,int arr[])
{
    int Piv_index,i;
    if(low<high)
    {
        Piv_index=Partition(low,high,arr);
        Quick_sort(low,Piv_index-1,arr);
        Quick_sort(Piv_index+1,high,arr);
    }
}

GPrimeList.cpp

#include <iostream>
#include <fstream>
#include <math.h>
#include <time.h>
#include <assert.h>
#include <gmpxx.h>
#include <stdio.h>
#include <stdlib.h>
using namespace std;

int main() {
    time_t start;//Record the time at the beginning
    int numPrime;//Number of primes wanted
    int primePerRow = 10;//Number of primes printed in each line
    cout << "Enter the number of primes: ";
    cin >> numPrime;
    cout << "Enter the number of primes in each line: ";
    cin >> primePerRow;
    time (&start);//Record the time at the beginning
    assert (numPrime > 0 & primePerRow > 1);//Abort the program if input isn't +ve
    time_t end;//Record the time at the end
    double timeElasped;//Total time elapsed
    mpz_t* primeList = new mpz_t[numPrime];//Array of primes
    //Initialize the first 2 elements with the first 2 primes
    mpz_init_set_str(primeList[0],"2",10);
    mpz_init_set_str(primeList[1],"3",10);
    //Initialize the other elements of primeList to avoid segmentation fault
    for (int i = 2; i < numPrime; i++) {
        mpz_init(primeList[i]);
    }
    int sizePrimeList = 2;//Number of non-zero elements of primeList
    //Print out what we have at first
    cout << "The list of primes:" << endl;
    cout << primeList[0] << "," << primeList[1];
    mpz_t zero,two;
    mpz_init_set_str(zero,"0",10);
    mpz_init_set_str(two,"2",10);
    mpz_t number;
    mpz_t sqrtNumber;
    mpz_init_set_str(number,"5",10);//Test primality of numbers starting from 5
    mpz_init(sqrtNumber);
    while (sizePrimeList < numPrime) {
        mpz_sqrt(sqrtNumber,number);
        bool isPrime = true;//Decide primality of number
        for (int i = 0; mpz_cmp(primeList[i],sqrtNumber) < 1; i++) {
            if (mpz_congruent_p(number,zero,primeList[i]) != 0) {
                //number has a prime divisor
                isPrime = false;
                break;
            }
        }
        if (isPrime) {
            mpz_set(primeList[sizePrimeList],number);//Index in array is zero-based
            sizePrimeList++;//One more element is added to primeList
            if (sizePrimeList % primePerRow == 1) {
                cout << endl;
            } else {
                cout << ",";
            }
            cout << number;
        }
        mpz_add(number,number,two);//Don't need to test even numbers
    }
    mpz_sub(number,number,two);//2 is added to number when it exits the while loop
    char count [80];
    char duration [80];
    int s;
    s = gmp_sprintf(count,"\n\nThere are %d primes between 1 and %Zd.",sizePrimeList,number);
    cout << count << endl;
    time (&end);//Record the time at the end
    timeElasped = difftime (end,start);//Calculate the total time elapsed
    s = gmp_sprintf(duration,"It took %f seconds to generate the list of primes.\n", timeElasped);
    cout << duration << endl;

    //Check if the user needs to save the list
    cout << "Would you like to save the above list? (Enter 'y' if yes. Otherwise, enter nothing.)";
    char input;
    cin >> input;
    if (input != 'y') {
        return 0;
        delete [] primeList;
        primeList = NULL;
    }

    //Check if the file exists
    char fileName [80];
    int n;
    n = gmp_sprintf(fileName,"PrimeList%Zd.txt",number);
    ifstream fin(fileName);
    if (fin) {
        // file exists
        cout << fileName << " already exists. Please refer to that." << endl;
        fin.close();
        return 0;
        delete [] primeList;
        primeList = NULL;
    }
    else {
        // file does not exist
        fin.close();
    }
    ofstream fout(fileName);// creates file for output and destroys file
    //Print out the content
    fout << "The list of primes:";
    for (int i = 0; i < numPrime; i++) {
        if (i % primePerRow == 0) {
            fout << endl;
        } else {
            fout << ",";
        }
        fout << primeList[i];
    }
    fout << count << endl;
    fout << duration;

    //Clean up work
    delete [] primeList;//When done, free memory pointed to by primeList
    primeList = NULL;//Clear primeList to prevent using invalid memory reference

    return 0;
}

PrimeList.cpp

#include <iostream>
#include <fstream>
#include <math.h>
#include <time.h>
#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
using namespace std;

int main() {
    time_t start,end;//Record the time at the beginning and the end
    double timeElasped;//Total time elapsed
    int* primeList = NULL;   //Pointer to int, initialize to nothing
    int numPrime;//Number of primes wanted
    int primePerRow = 10;//Number of primes printed in each line
    cout << "Enter the number of primes wanted: ";
    cin >> numPrime;
    cout << "Enter the number of primes in each line: ";
    cin >> primePerRow;
    time (&start);//Record the time at the beginning
    assert (numPrime > 0 & primePerRow > 1);//Abort the program if input isn't +ve
    primeList = new int[numPrime];//Array of primes
    //Initialize the first 2 elements with the first 2 primes
    primeList[0] = 2;
    primeList[1] = 3;
    int sizePrimeList = 2;//Number of non-zero elements of primeList

    //Print out what we have at first
    cout << "The list of primes:" << endl;
    cout << primeList[0] << "," << primeList[1];
    int number = 5;//Test primality of numbers starting from 5

    while (sizePrimeList < numPrime) {
        bool isPrime = true;//Decide primality of number
        for (int i = 0; primeList[i] <= sqrt(number); i++) {
            if (number % primeList[i] == 0) {
                //number has a prime divisor
                isPrime = false;
                break;
            }
        }
        if (isPrime) {
            primeList[sizePrimeList] = number;//Index in array is zero-based
            sizePrimeList++;//One more element is added to primeList
            if (sizePrimeList % primePerRow == 1) {
                cout << endl;
            } else {
                cout << ",";
            }
            cout << number;
        }
        number += 2;//Don't need to test even numbers
    }
    char count [80];
    char duration [80];
    int s;
    s = sprintf(count,"\n\nThere are %d primes between 1 and %d.",sizePrimeList,(number-2));
    cout << count << endl;
    time (&end);//Record the time at the end
    timeElasped = difftime (end,start);//Calculate the total time elapsed
    s = sprintf(duration,"It took %f seconds to generate the list of primes.\n", timeElasped);
    cout << duration << endl;

    //Check if the user needs to save the list
    cout << "Would you like to save the above list? (Enter 'y' if yes. Otherwise, enter nothing.)";
    char input;
    cin >> input;
    if (input != 'y') {
        return 0;
        delete [] primeList;
        primeList = NULL;
    }

    //Check if the file exists
    char fileName [80];
    int n;
    n = sprintf(fileName,"PrimeList%d.txt",(number-2));
    ifstream fin(fileName);
    if (fin) {
        // file exists
        cout << fileName << " already exists. Please refer to that." << endl;
        fin.close();
        return 0;
        delete [] primeList;
        primeList = NULL;
    }
    else {
        // file does not exist
        fin.close();
    }
    ofstream fout(fileName);// creates file for output and destroys file
    //Print out the content
    fout << "The list of primes:";
    for (int i = 0; i < numPrime; i++) {
        if (i % primePerRow == 0) {
            fout << endl;
        } else {
            fout << ",";
        }
        fout << primeList[i];
    }
    fout << count << endl;
    fout << duration;

    //Clean up work
    delete [] primeList;//When done, free memory pointed to by primeList
    primeList = NULL;//Clear primeList to prevent using invalid memory reference

    return 0;
}