# The Sieve of Eratosthenes

The goal of this assignment is to design a program which will produce lists of prime numbers. The method is based on the one discovered by Erastosthenes (276 - 196 BC). His method goes like this. First, write down a list of integers
```
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
```
Then mark all multiples of 2:
```
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
x   x   x    x     x     x     x     x     x
```
Move to the next unmarked number, which in this case is 3, then mark all its multiples:
```
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
x   x   x x  x     x     x  x  x     x     x
```
Continue in this fashion, marking all multiples of the next unmarked number until there are no new unmarked numbers. The numbers which survive this marking process (the Sieve of Eratosthenses) are primes. (You should complete this marking process yourself).

The new part of the C language that you will have to learn in order to do this program is the array.

## Problem 1

Design a program to make a list of primes less than N = 100 using Eratosthenes' method:
``` PSEUDOPROGRAM:

Set up an array of integers z[N]:
for i from 0 to N, set z[i] = i.

for i from 2 to N, mark all multiples
of i by setting them to zero

for i from 2 to N,
print all unmarked multiples.
```
Your program should be as short and elegant as possible, while still being clearly understandable. Below is an outline for the program
```
#define N 20 /* use small number for testing */

main(){

int z[N];  /* array of integers, z, ... z[N-1] */

for ( i = 0; i < N; i++ )
initialize z[i];

for ( i = 2; i < N; i++ )
mark the multiples of i;
/* this marking can be done with
one for and one if statement */

for ( i = 2; i < N; i++ )
print the unmarked entries of z;
}
```

## Problem 2

Modifiy your program so that the output goes to a file.

## Problem 3

Investigate the function
```   p(n) = Number of primes less than n
```

## Problem 4

Investigate the infinite series
```  1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + ...
```
constructed by adding the reciprocals of the primes.
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