{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Euler Problem 10\n",
    "\n",
    "The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\n",
    "\n",
    "Find the sum of all the primes below two million."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, reuse prime generator from 7 and go."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "61.8695481220002\n",
      "142913828922\n"
     ]
    }
   ],
   "source": [
    "import timeit\n",
    "\n",
    "def is_prime(n, smaller_primes):\n",
    "    for s in smaller_primes:\n",
    "        if n % s == 0:\n",
    "            return False\n",
    "        if s * s > n:\n",
    "            return True\n",
    "    return True\n",
    "\n",
    "def prime_generator_function():\n",
    "    primes = [2, 3, 5, 7]\n",
    "    for p in primes:\n",
    "        yield p\n",
    "    while True:\n",
    "        p += 2\n",
    "        if is_prime(p, primes):\n",
    "            primes.append(p)\n",
    "            yield p\n",
    "    \n",
    "def brute_force(): \n",
    "    ps = prime_generator_function()\n",
    "    s = 0\n",
    "    p = next(ps)\n",
    "    while p < 2000000:\n",
    "        s += p\n",
    "        p = next(ps)\n",
    "    return s\n",
    "    \n",
    "print(timeit.timeit(brute_force, number=10))\n",
    "assert(brute_force() == 142913828922)\n",
    "print(brute_force())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, here it may actually be way smarter to use a sieve. We implent it because the old one was shitty. Okay, I am actually interested in the time difference. So we use the old one first and then implement the optimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def get_primes_smaller(number):\n",
    "    primes = []\n",
    "    prospects = [n for n in range(2, number)]\n",
    "    while prospects:\n",
    "        p = prospects[0]\n",
    "        prospects = [x for x in prospects if x % p != 0]\n",
    "        primes.append(p)\n",
    "    return primes\n",
    "\n",
    "def brute_force():\n",
    "    return sum(get_primes_smaller(2000000))\n",
    "\n",
    "#print(timeit.timeit(brute_force, number=1))\n",
    "#print(brute_force())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This did not even terminate. We optimize the sieve by stopping when $p^2 > number$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "74.5315607270004\n",
      "142913828922\n"
     ]
    }
   ],
   "source": [
    "def sieve_of_eratosthenes(number):\n",
    "    primes = []\n",
    "    prospects = [n for n in range(2, number)]\n",
    "    while prospects:\n",
    "        p = prospects[0]\n",
    "        prospects = [x for x in prospects if x % p != 0]\n",
    "        primes.append(p)\n",
    "        if p * p > number:\n",
    "            break\n",
    "    return primes + prospects\n",
    "\n",
    "def brute_force():\n",
    "    return sum(sieve_of_eratosthenes(2000000))\n",
    "\n",
    "assert(sieve_of_eratosthenes(10) == [2, 3, 5, 7,])\n",
    "print(timeit.timeit(brute_force, number=10))\n",
    "assert(brute_force() == 142913828922)\n",
    "print(brute_force())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, I honestly did not expect this to be slower than our generator. Gotta keep that in mind for future prime related problems."
   ]
  }
 ],
 "metadata": {
  "completion_date": "Wed, 20 Aug 2014, 20:28",
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