Published at Jul 13 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.

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.

Execute the tests with:

```
$ elixir sieve_test.exs
```

In the test suites, all but the first test have been skipped.

Once you get a test passing, you can unskip the next one by
commenting out the relevant `@tag :pending`

with a `#`

symbol.

For example:

```
# @tag :pending
test "shouting" do
assert Bob.hey("WATCH OUT!") == "Whoa, chill out!"
end
```

Or, you can enable all the tests by commenting out the
`ExUnit.configure`

line in the test suite.

```
# ExUnit.configure exclude: :pending, trace: true
```

For more detailed information about the Elixir track, please see the help page.

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.

```
if !System.get_env("EXERCISM_TEST_EXAMPLES") do
Code.load_file("sieve.exs", __DIR__)
end
ExUnit.start()
ExUnit.configure(exclude: :pending, trace: true)
defmodule SieveTest do
use ExUnit.Case
# @tag :pending
test "a few primes" do
assert Sieve.primes_to(10) == [2, 3, 5, 7]
end
@tag :pending
test "primes to 1000" do
result = [
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
]
assert Sieve.primes_to(1000) == result
end
end
```

```
defmodule Sieve do
@doc """
Generates a list of primes up to a given limit.
"""
@spec primes_to(non_neg_integer) :: [non_neg_integer]
def primes_to(limit) do
do_primes_to(limit, 2, %{})
|> Enum.filter(&(elem(&1, 1)))
|> Enum.map(&(elem(&1, 0)))
|> Enum.sort
end
defp do_primes_to(limit, candidate, numbers) when candidate > limit do
numbers
end
defp do_primes_to(limit, candidate, numbers) do
next_numbers = if numbers[candidate] == false do # known composite
numbers
else
# kinda cheating here -- instead of leaving the primes
# unmarked, I'm marking them in a different way. could
# instead have put the marked ones in a set and subtracted
# them from a map of all or some such; this way just uses
# the same map to keep track of both.
mark_multiples(Map.put(numbers, candidate, true),
candidate * 2,
candidate,
limit)
end
do_primes_to(limit, candidate + 1, next_numbers)
end
defp mark_multiples(numbers, this_one, _, limit) when this_one > limit do
numbers
end
defp mark_multiples(numbers, this_one, candidate, limit) do
mark_multiples(Map.put(numbers, this_one, false),
this_one + candidate,
candidate,
limit)
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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