Finding prime numbers

prime_sieve.ipynb

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  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "def indivisible(x, primes):\n",
    "  for prime in primes:\n",
    "    if x % prime == 0:\n",
    "      return False\n",
    "  return True\n",
    "\n",
    "def naive_sieve(n):\n",
    "  \"\"\"This is widely known as trial division.\"\"\"\n",
    "  primes = [2]\n",
    "  for number in range(2, n+1):\n",
    "    if indivisible(number, primes):\n",
    "      primes.append(number)\n",
    "  return primes\n",
    "\n",
    "def squared_sieve(n):\n",
    "  largest_square_ = int(np.ceil(np.sqrt(n)))\n",
    "  primes = naive_sieve(largest_square_)\n",
    "  candidate_primes = [i for i in range(1, n+1)]\n",
    "  for prime in primes:\n",
    "    m = prime-1\n",
    "    while m < n:\n",
    "      candidate_primes[m] = None\n",
    "      m += prime\n",
    "  \n",
    "  primes.extend([prime for prime in candidate_primes if prime is not None][1:])\n",
    "  return primes\n",
    "\n",
    "def doubly_squared_sieve(n):\n",
    "  largest_square_ = int(np.ceil(np.sqrt(n)))\n",
    "  primes = squared_sieve(largest_square_)\n",
    "  candidate_primes = [i for i in range(1, n+1)]\n",
    "  for prime in primes:\n",
    "    m = prime-1\n",
    "    while m < n:\n",
    "      candidate_primes[m] = None\n",
    "      m += prime\n",
    "  \n",
    "  primes.extend([prime for prime in candidate_primes if prime is not None][1:])\n",
    "  return primes"
   ]
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "16.3 s ± 3.39 s per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "naive_sieve(100_000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "138 ms ± 22 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "squared_sieve(100_000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "126 ms ± 4.52 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "doubly_squared_sieve(100_000)"
   ]
  }
 ]
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