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.

To run the tests, run the command `busted`

from within the exercise directory.

For more detailed information about the Lua track, including how to get help if you're having trouble, please visit the exercism.io Lua language 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.

```
local sieve = require('sieve')
describe('sieve', function()
local select = select or function(n, ...)
return table.pack(...)[n]
end
local function primes_from(co)
local primes = {}
while true do
local _, prime = coroutine.resume(co)
if prime == nil then return primes end
table.insert(primes, prime)
end
end
it('should return a coroutine that generates primes', function()
local co = sieve(9)
assert.equal(2, select(2, coroutine.resume(co)))
assert.equal(3, select(2, coroutine.resume(co)))
assert.equal(5, select(2, coroutine.resume(co)))
assert.equal(7, select(2, coroutine.resume(co)))
assert.is_nil(select(2, coroutine.resume(co)))
end)
it('should find primes up to 10', function()
assert.same({ 2, 3, 5, 7 }, primes_from(sieve(10)))
end)
it('should find primes up to 1000', function()
assert.same(
{ 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 },
primes_from(sieve(1000))
)
end)
it('should include the limit if it is prime', function()
assert.same({ 2, 3, 5, 7, 11, 13 }, primes_from(sieve(13)))
end)
it('should find the first prime', function()
assert.same({ 2 }, primes_from(sieve(2)))
end)
it('should not find any primes under 2', function()
assert.same({}, primes_from(sieve(1)))
end)
end)
```

```
local function find_smallest_nil(t, s, e)
for i=s,e do
if t[i] == nil then return i end
end
return nil
end
return function (n)
local prime_numbers = {}
local i = 2
return coroutine.create(
function()
repeat
local new_prime = find_smallest_nil(prime_numbers, i, n)
if new_prime == nil then break end
local j = 2
repeat
prime_numbers[new_prime * j] = true
j=j+1
until new_prime*j > n
i = (new_prime or n) + 1
coroutine.yield(new_prime)
until i > n
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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## Community comments

For really large n the number of prime numbers will be significantly smaller than the number of composite numbers. Right now you are storing all non-primes in your prime_numbers table, but if you could store only primes then you'd use less memory for really large sieves.

Looks good otherwise. Coroutines are really cool :)

yeah you are right.. but I am not sure how to less the memory usage. like your solution if we set 1~n to true and set it as false to non-prime numbers.. does that reduce the memory usage? having 'false' as a value of the key would take the memory space, wouldn't it?

Yeah I think you're right. I was thinking that you could mark them as true only once you knew for sure that they were prime and then leave the others nil, but you need to mark them true up front. My bad.