Published at Dec 02 2018
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Instructions

Test suite

Solution

Use the Sieve of Eratosthenes to find all the primes from 2 up to a given number.

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e. not prime) the multiples of each prime, starting with the multiples of 2. It does not use any division or remainder operation.

Create your range, starting at two and continuing up to and including the given limit. (i.e. [2, limit])

The algorithm consists of repeating the following over and over:

- take the next available unmarked number in your list (it is prime)
- mark all the multiples of that number (they are not prime)

Repeat until you have processed each number in your range.

When the algorithm terminates, all the numbers in the list that have not been marked are prime.

The wikipedia article has a useful graphic that explains the algorithm: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

Notice that this is a very specific algorithm, and the tests don't check that you've implemented the algorithm, only that you've come up with the correct list of primes. A good first test is to check that you do not use division or remainder operations (div, /, mod or % depending on the language).

Follow the setup instructions for Crystal here:

http://exercism.io/languages/crystal

More help installing can be found here:

http://crystal-lang.org/docs/installation/index.html

Execute the tests with:

```
$ crystal spec
```

In each test suite all but the first test have been skipped.

Once you get a test passing, you can unskip the next one by changing `pending`

to `it`

.

Sieve of Eratosthenes at Wikipedia http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

It's possible to submit an incomplete solution so you can see how others have completed the exercise.

```
require "spec"
require "../src/*"
describe "Primes" do
describe "#sieve" do
it "no primes under two" do
Primes.sieve(1).should eq [] of Int32
end
pending "find first prime" do
Primes.sieve(2).should eq [2]
end
pending "find primes up to 10" do
Primes.sieve(10).should eq [2, 3, 5, 7]
end
pending "limit is prime" do
Primes.sieve(13).should eq [2, 3, 5, 7, 11, 13]
end
pending "find primes up to 1000" do
Primes.sieve(1000).should eq [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]
end
end
end
```

```
require "bit_array"
module Primes
def self.sieve(max : Int) : Array(Int32)
b = BitArray.new(max, true)
# 1 s'not prime
b[0] = false
(2..b.size).each { |i|
case b[i-1]
when false
next
when true
j = i * 2
while j <= max
b[j-1] = false
j = j + i
end
end
}
a = [] of Int32
b.each.with_index.reject{|e| e[0] == false}.each{|e| a.push e[1]+1}
a
end
end
```

A huge amount can be learned from reading other people’s code. This is why we wanted to give exercism users the option of making their solutions public.

Here are some questions to help you reflect on this solution and learn the most from it.

- What compromises have been made?
- Are there new concepts here that you could read more about to improve your understanding?

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