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    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "4782\n"
     ],
     "output_type": "stream"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "φ = (1 + np.sqrt(5))/2\n",
    "\n",
    "def digits(k): # number of digits of the kth Fibonacci number\n",
    "    if k < 2:\n",
    "        return 1\n",
    "    return np.ceil(k* np.log10(φ) - np.log10(5)/2) + 1\n",
    "\n",
    "def smallestFib(n): # smallest Fibonacci number with n digits\n",
    "    if n < 2:\n",
    "        return 1\n",
    "    return np.ceil((n + np.log10(5) / 2 - 1) / np.log10(φ))\n",
    "\n",
    "print(int(smallestFib(1000)))"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n",
     "is_executing": false
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## b)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "443839\n"
     ],
     "output_type": "stream"
    }
   ],
   "source": [
    "sumNumbers = 0\n",
    "\n",
    "def powerSum(n): # sum of the fifths powers of the digits\n",
    "    pow = 0\n",
    "    while n > 0:\n",
    "        pow += (n%10)**5\n",
    "        n = n // 10\n",
    "    return pow\n",
    "\n",
    "for k in range(2,354294):\n",
    "    if powerSum(k) == k:\n",
    "        sumNumbers += k\n",
    "    \n",
    "print(sumNumbers)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n",
     "is_executing": false
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## c)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n",
     "is_executing": false
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "55\n"
     ],
     "output_type": "stream"
    }
   ],
   "source": [
    "N = 1000000\n",
    "sieve = np.array(range(N))\n",
    "for k in sieve[2:int(np.sqrt(N))+1]:\n",
    "    if k != 0: sieve[2*k::k] = 0\n",
    "\n",
    "primes = list(map(str,sieve[sieve != 0][1:]))\n",
    "\n",
    "def rotations(p):\n",
    "    return [p[k:] + p[:k] for k in range(len(p))]\n",
    "    \n",
    "circular = ['2']\n",
    "notcircular = []\n",
    "\n",
    "for p in primes:    \n",
    "    if p in circular or p in notcircular:\n",
    "        continue    \n",
    "    if any(a in p for a in ['0','2','4','5','6','8']):\n",
    "        continue    \n",
    "    rots = rotations(p)\n",
    "    if all(a in primes for a in rots):\n",
    "        circular += rots\n",
    "    else:\n",
    "        notcircular += rots\n",
    "print(len(circular))    \n"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n",
     "is_executing": false
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "outputs": [
    {
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mIndexError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-14-dd01c87031bf>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      3\u001b[0m         \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mz\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m         \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mintersect1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m         \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0ma\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mIndexError\u001b[0m: index 0 is out of bounds for axis 0 with size 0"
     ],
     "ename": "IndexError",
     "evalue": "index 0 is out of bounds for axis 0 with size 0",
     "output_type": "error"
    }
   ],
   "source": [
    "for n in range(12):\n",
    "    for z in range(n):\n",
    "        a,b = list(str(n)), list(str(z))\n",
    "        c = np.intersect1d(a,b)\n",
    "        print(a[a != c[0]])"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n",
     "is_executing": false
    }
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