An easy way to find prime numbers

idea

The definition of a prime number, I still remember, was taught in elementary school mathematics, "An integer greater than 1, except 1 and itself, cannot be divisible by any other number, it is called a prime number".

To find all prime numbers less than or equal to x, you can use recursive method, let f(x) be all prime numbers less than or equal to x. Then f(2) = [2] , when x > 2, f(x-1) is all prime numbers less than x. According to the definition of prime numbers, if all the prime numbers in the result of f(x-1) cannot divide x, then x is a prime number, otherwise it is not.

code

Due to the recursive algorithm, the direct implementation may generate a large number of repeated calculations, so we slightly modified it, using a map as a cache, and calculating from small to large, because large numbers will use the results of small numbers.

 defmodule Prime do
  
  def primes(x) when x >= 2 do
    primes(x, %{2 => [2]}, 3)
  end

  defp primes(x, m, n) when n > x, do: m[x] 
  defp primes(x, m, n) do
    m = case m[n] do
      nil -> Map.put(m ,n, do_primes(m, n))
      _ -> m
    end
    primes(x, m, n+1)
  end

  defp do_primes(m, n) do
    if Enum.all?(m[n-1], fn x -> rem(n, x) != 0 end) do
      [n | m[n-1]]
    else
      m[n-1]
    end
  end
end

have a test