Homework 4, Finding Primes.

Short Description:

Write a program that will find all the prime numbers between 2 and some maximum value.

Goals

When you finish this homework, you should have:

Written a program that employs arrays.

Formal Description

For this program we will implement a version of the Sieve f Eratosthenes. This is an algorithm that finds prime numbers by eliminating the numbers that are divisible by know primes.

The basic algorithm is


 Write down all of the numbers between 2 and n, the largest possible prime you are interested in
 Set the current prime to be 2.
 While the current prime is less than the square root of the largest possible prime
       Cross off all of the multiples of the current prime.
       Find the next number that has not been crossed off, this is the new current prime.
 List all of the numbers, 2 and higher, that have not been crossed off.  These are the prime numbers.

We will write this program in a series of guided steps. Please follow these steps. Your code is expected to be the same structure as mine.

Download the starting files
- Makefile
- sieve.cpp A shell for this program.
- list (a list of prime numbers less than 1000)

Declare the main array and initialize it.

Declare an array of booleans, size MAX_PRIME called sieve
Build a function void InitSieve(bool sieve[]);
- This function should fill the array with the value true.
- You may use MAX_PRIME as the array size.

Build a function void PrintSieve(const bool sieve[]);

Print the values of the array
- A T for true, a F for false
- Starting at 2
- Ending at MAX_PRIME -1,
- No spaces between characters
- A blank for the first value (sieve[1])
- 70 per line
- No space at the end of the lines.

Example:

 TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTT

In you main function put the following code:

InitSieve(sieve);
if (DEBUG) {
    PrintSieve(sieve);
    cout << endl;
}

Compile and run this.

Change DEBUG to be false
- Compile and run again.
- We will wrap a number of output statements inside a if (DEBUG) statement in this project.
- by changing the value of DEBUG you can switch the program between development mode and production mode.
- Some programmers make extensive use of this debugging technique.
- Add a comment above the declaration of the DEBUG constant to state what is is used for.

Add a single step sieve function
- Add a variable, current to main. This is an integer and should be initialized to 2, as the first prime number.
- Add a function void DoSieve(bool sieve[], int prime);
  - This takes the array and a prime number.
  - And crosses off (or sets to false) all multiples of that prime.
  - So sieve[4], sieve[6], ... will all be set to false when this is called with the prime set to 2.
  - sieve[prime*2], sieve[prime*3], ... in general.
- Add the following code to your main
- ```
        DoSieve(sieve,current);
        if (DEBUG) {
            PrintSieve(sieve);
            cout << endl;
        } 
```
- Compile and run the code with debugging turned on.
  - Make sure that the DoSieve function crosses off all multiples of two.
  - ```
   TTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF
  TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF
  TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF
          
```

Find the next available prime

Implement the function int FindNextPrime(const bool sieve[], int current);
- This function starts at the current prime value (2) and searches for the next true in the sieve array.
- This will be 3 after the first loop, 5, 7, ....
Change your main loop to be

    while (current < sqrt(MAX_PRIME)) {
        if (DEBUG) {
            cout << "The next prime is " << current << endl;
        }
        DoSieve(sieve,current);
        if (DEBUG) {
            PrintSieve(sieve);
            cout << endl;
        }
        current = FindNextPrime(sieve, current);
    }

With DEBUG set to true, compile and run this again.

Your output should be something like

[bennett@mirkwood me]$ sieve
 TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
                            ...

The next prime is 2
 TTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF
TFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTFTF
                            ...

The next prime is 3
 TTFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFF
TFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTFFFTFTF
                            ...

The next prime is 5
 TTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFF
TFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFTF
                            ...

The next prime is 7
 TTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFFFFFTFFFFFTFTFFFFFTFFF
TFTFFFFFTFFFTFFFFFTFFFFFFFTFFFTFTFFFTFTFFFTFFFFFFFTFFFFFTFFFTFFFFFTFTF
                            ...

The next prime is 11
 TTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFFFFFTFFFFFTFTFFFFFTFFF
TFTFFFFFTFFFTFFFFFTFFFFFFFTFFFTFTFFFTFTFFFTFFFFFFFFFFFFFTFFFTFFFFFTFTF
                            ...
.
.
.

The next prime is 31
 TTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFFFFFTFFFFFTFTFFFFFTFFF
TFTFFFFFTFFFTFFFFFTFFFFFFFTFFFTFTFFFTFTFFFTFFFFFFFFFFFFFTFFFTFFFFFTFTF
                            ...

Build a list of primes.
- In main, add two more variables
  - primes, an array of PRIMES_SIZE integers.
  - primeCount, an integer initialized to 0.
- Implement the function void FindPrimes(const bool sieve[], int primes[], int & primeCount);
  - This function will search the sieve array for the value of true greater than 1.
  - Every add the index where the value is greater than 2, add this to the primes array and increment primeCount
    - If there are more primes identified in sieve than PRIMES_SIZE, print the warning "Warning: there may be more primes, increase PRIMES_SIZE!".
- Implement the function void PrintPrimes(const int primes[], int primeCount);
  - This should print all of the primes in the array, one one line , separated by commas, with a newline at the end.
- Add the following to your main
```
    FindPrimes(sieve, primes, primeCount);
    PrintPrimes(primes, primeCount); 
```
- Compile and run your program.
- Adjust PRIMES_SIZE as necessary
- Continue to compile and run until you get the full list of primes.
Turn off debugging and test your program against the list of primes provided
- The following command should produce no output
- ```
 ./sieve | diff - list
    
```
Make sure you add proper documentation and submit your program.

Required Files

A single tar file containing the source code and makefile for this program.

Submission

Submit the assignment to the D2L folder homework 4 by the due date.