20150201_Euler500.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Lien vers le probl\u00e8me : "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import IPython.display"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "IPython.display.IFrame(src='https://projecteuler.net/problem=500', width=800, height=300)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "        <iframe\n",
        "            width=\"800\"\n",
        "            height=300\"\n",
        "            src=\"https://projecteuler.net/problem=500\"\n",
        "            frameborder=\"0\"\n",
        "            allowfullscreen\n",
        "        ></iframe>\n",
        "        "
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "<IPython.lib.display.IFrame at 0x1050db470>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Explication pour 120 :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2**3 * 3 * 5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Le nombre de diviseurs diff\u00e9rents est de $4 * 2 * 2$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "4 * 2 * 2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "16"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si on avait une puissance de 2 avec autant de diviseurs ce serait :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 ** 15"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "32768"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Donc effectivement, c'est plus compact de rajouter des nombres premiers pour un total de diviseurs \u00e9gal \u00e0 16. En fait, on peut toujours r\u00e9duire le nombre de diviseurs en rajoutant une puissance d'un autre nombre premier plus grand."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 ** 14 * 3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "49152"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "En fait l\u00e0 on a maintenant 15 * 2 = 30 diviseurs. Du coup, si on en veut 16, il faut mettre 2 \u00e0 la puissance 7."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 ** 7 * 3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "384"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Effectivement, c'est d\u00e9j\u00e0 beaucoup plus petit. Et on vient d'apprendre la r\u00e8gle que quand on rajoute un nombre premier, on doit baisser le nombre de diviseurs de 2 d'un facteur 2 et rajouter un 1 sur le nombre suivant."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 ** 3 * 3 * 5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 ** 1 * 3 * 5 * 7"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "210"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "210 * 4 / 7"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "120.0"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 * 3 * 5 * 7 * 11"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "2310"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Autre propri\u00e9t\u00e9 : si les exposants sont tous 1 alors pour avoir 16 diviseurs, il y a maximum 4 nombres premiers dans l'\u00e9criture recherch\u00e9e car $2 * 2 * 2 * 2 = 16$. Dans le cas de l'expos\u00e9, il y a au maximum $2^n = 2^{500500}$ c'est \u00e0 dire $n = 500500$ nombres premiers dans l'\u00e9criture du nombre cherch\u00e9. On pourrait donc d\u00e9j\u00e0 calculer une table de nombres premiers de cette taille l\u00e0.\n",
      "\n",
      "La strat\u00e9gie pourrait donc \u00eatre : \n",
      "\n",
      "- on conna\u00eet le nombre maximums de nombres premiers\n",
      "- on cherche \u00e0 en \u00e9liminer, sachant qu'on a un tradeoff : \n",
      "    - si je diminue l'exposant, je dois en augmenter un autre, j'ai tout int\u00e9r\u00eat \u00e0 augmenter un petit exposant\n",
      "    - si on gagne dans cette op\u00e9ration, alors il faut la faire\n",
      "    - sinon, on a trouv\u00e9 le bon nombre"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Formellement : le nombre cherch\u00e9 s'\u00e9crit \u00e0 l'aide d'un certain nombre de nombres premiers $n$ (inconnu a priori)\n",
      "\n",
      "$$\n",
      "x = \\prod_{i=1}^{n} p_i^{w_i}\n",
      "$$\n",
      "\n",
      "avec \n",
      "\n",
      "$$\n",
      "\\prod_{i=1}^{n} (w_i + 1) = 2^{500500}\n",
      "$$\n",
      "\n",
      "On sait que $n$ vaut au maximum $500500$.\n",
      "\n",
      "Les r\u00e8gles d'optimisation sont que si on baisse l'exposant du nombre le plus grand $n$, et que l'on incr\u00e9mente l'exposant du nombre $k$ alors on obtient un nouveau nombre qui est multipli\u00e9 par \n",
      "\n",
      "$$\\alpha = \\frac{p_k^2}{p_n}$$\n",
      "\n",
      "Si $\\alpha < 1$ alors l'op\u00e9ration est b\u00e9n\u00e9fique. \n",
      "\n",
      "Par contre, on veut conserver le nombre de diviseurs total, qui passe de $\\prod_{i=1}^{n} (w_i + 1) = 2^{500500}$ \u00e0 $\\prod_{i=1 i \\neq k}^{n-1} (w_i + 1) * (w_k + 3)$. Il est donc multipli\u00e9 par \n",
      "\n",
      "$$\n",
      "\\frac{w_k + 3}{(w_n + 1)(w_k + 1)}\n",
      "$$\n",
      "\n",
      "Si on veut que le changement du nombre de diviseurs soit \u00e9gale \u00e0 1, c'est-\u00e0-dire qu'il y ait le m\u00eame nombre de diviseurs au d\u00e9but comme \u00e0 la fin, il faut que :\n",
      "\n",
      "$$\n",
      "w_n (w_k + 1) = 2\n",
      "$$\n",
      "\n",
      "\n",
      "Donc en fait ce qu'il faudrait conna\u00eetre c'est la d\u00e9composition en nombres premiers de 500500 pour savoir combien on peut incr\u00e9menter de nombre sur ce principe. Car on ne peut pas faire toutes les transformations que l'on veut !\n",
      "\n",
      "Bonne nouvelle, [Wolfram Alpha](http://www.wolframalpha.com/input/?i=prime+decomposition+of+500500) nous indique que : \n",
      "\n",
      "$$500500 = 2^2 * 5^3 * 7 * 11 * 13$$\n",
      "\n",
      "Donc \n",
      "\n",
      "$$2^{500500} = 2^{2^2 * 5^3 * 7 * 11 * 13}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si on red\u00e9marre dans le cas 120 avec "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2 * 3 * 5 * 7"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "210"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2**3 * 3**3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "216"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "2**7 * 3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "384"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Le nombre de diviseurs est le bon : $2^4$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Tentative d'\u00e9criture d'un algorithme"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "L'id\u00e9e est la suivante : on donne les poids et on essaye de modifier it\u00e9rativement les poids au fur et \u00e0 mesure.\n",
      "\n",
      "Exemple, on boucle sur k en prenant l'indice le plus bas. Si la condition d'incr\u00e9mentation est v\u00e9rifi\u00e9e, on calcule le facteur $\\alpha$ obtenu. Si on gagne effectivement quelque chose, alors on le fait, sinon on passe."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "L'algorithme sur les nombres premiers est tir\u00e9 de [http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n/3035188#3035188](http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n/3035188#3035188)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def rwh_primes1(n):\n",
      "    # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188\n",
      "    \"\"\" Returns  a list of primes < n \"\"\"\n",
      "    sieve = [True] * int(n/2)\n",
      "    for i in range(3,int(n**0.5)+1,2):\n",
      "        if sieve[int(i/2)]:\n",
      "            sieve[int(i*i/2)::i] = [False] * int((n-i*i-1)/(2*i)+1)\n",
      "    return [2] + [2*i+1 for i in range(1, int(n/2)) if sieve[i]]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[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]\n"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import log\n",
      "\n",
      "class euler500_solver():\n",
      "    def __init__(self, number_of_divisors, primes):\n",
      "        self.number_of_divisors = number_of_divisors\n",
      "        self.max_primes = int(log(number_of_divisors) / log(2))\n",
      "        self.primes = primes[:self.max_primes]\n",
      "        self.weights = [1] * self.max_primes\n",
      "    \n",
      "    def step(self):\n",
      "        max_index = [(ind, v) for (ind, v) in enumerate(self.weights) if v!= 0][-1][0]\n",
      "        for i in range(max_index + 1):\n",
      "            if self.weights[max_index] * (self.weights[i] + 1) == 2:\n",
      "                if self.primes[i]**2 / self.primes[max_index] < 1:\n",
      "                    self.weights[i] += 2\n",
      "                    self.weights[max_index] -= 1\n",
      "                    return True\n",
      "        return False\n",
      "        \n",
      "    def compute_current_number(self, modulo=None):\n",
      "        factors = ([p**w for (p, w) in zip(self.primes, self.weights)])\n",
      "\n",
      "        if modulo == None:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product *= f\n",
      "            return product\n",
      "        else:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product = (product * f) % modulo\n",
      "            return product"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 97
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(16, rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 98
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.primes"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 99,
       "text": [
        "[2, 3, 5, 7]"
       ]
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 100,
       "text": [
        "[1, 1, 1, 1]"
       ]
      }
     ],
     "prompt_number": 100
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 101,
       "text": [
        "210"
       ]
      }
     ],
     "prompt_number": 101
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 102,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 102
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 103,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 104,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 104
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 105,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 105
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On trouve la bonne solution. Comment \u00e9tendre \u00e7a au cas qui nous int\u00e9resse ? Il faudrait id\u00e9alement un cas interm\u00e9diaire qui ne fasse pas exploser l'algorithme."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "primes = rwh_primes1(10000000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "len(primes) / 500500"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 78,
       "text": [
        "1.3278301698301698"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Okay, on a assez de nombres premiers."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(2**500500, primes)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 106
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 107,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 108,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 108
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 109,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 109
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "last_step = e.step()\n",
      "while last_step:\n",
      "    last_step = e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 110
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number(modulo=500500507)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 111,
       "text": [
        "493807392"
       ]
      }
     ],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "from pylab import plot, array, xlim"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 115
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(e.weights)\n",
      "xlim(0, 10000)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 116,
       "text": [
        "(0, 10000)"
       ]
      },
      {
       "metadata": {
        "png": {
         "height": 255,
         "width": 382
        }
       },
       "output_type": "display_data",
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jbUX38YXDPfv7JOcneU2S/5Lk5qH8LyZc88tJvpvktiTvTPL64Z7vTfKBZb7O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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10f89dc18>"
       ]
      }
     ],
     "prompt_number": 116
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(array(e.weights).cumsum())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 96,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10c9a3470>]"
       ]
      },
      {
       "metadata": {
        "png": {
         "height": 255,
         "width": 408
        }
       },
       "output_type": "display_data",
       "png": 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Pe+2LJA2Trv2aj4i/iIiR8kL4m4CPAucCn21abXY5vatan5nPAOsobvu8Z7nN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\n6QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10c362978>"
       ]
      }
     ],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "max(e.weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 112,
       "text": [
        "3"
       ]
      }
     ],
     "prompt_number": 112
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Exemple avec 32 diviseurs"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def compute_number_of_divisors(weights):\n",
      "    prod = 1\n",
      "    for weight in weights:\n",
      "        if weight != 0:\n",
      "            prod *= (weight + 1)\n",
      "    return prod"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def compute_current_number(primes, weights):\n",
      "    product = 1\n",
      "    for p, w in zip(primes, weights):\n",
      "        if w != 0:\n",
      "            product *= p ** w\n",
      "    return product"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 125
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On initialise :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = primes[:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 118,
       "text": [
        "[2, 3, 5, 7, 11]"
       ]
      }
     ],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights = [1, 1, 1, 1, 1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 154
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notre position de d\u00e9part est :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors(weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 155,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 155
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 156,
       "text": [
        "2310"
       ]
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On veut am\u00e9liorer ce r\u00e9sultat. On essaye donc de se d\u00e9barasser du plus haut nombre premier :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights[4] = 0\n",
      "weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 157,
       "text": [
        "[1, 1, 1, 1, 0]"
       ]
      }
     ],
     "prompt_number": 157
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ceci r\u00e9duit le nombre de diviseurs :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors(weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 158,
       "text": [
        "16"
       ]
      }
     ],
     "prompt_number": 158
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      " compute_current_number(p, weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 159,
       "text": [
        "210"
       ]
      }
     ],
     "prompt_number": 159
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On peut augmenter n'importe lequel des facteurs premiers dans ce cas l\u00e0 :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights[0] += 2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 160
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 161,
       "text": [
        "[3, 1, 1, 1, 0]"
       ]
      }
     ],
     "prompt_number": 161
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors(weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 162,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 162
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 163,
       "text": [
        "840"
       ]
      }
     ],
     "prompt_number": 163
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On se retrouve au status quo. Et si on diminuait encore l'exposant le plus grand ?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights[3] = 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 164
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 165,
       "text": [
        "[3, 1, 1, 0, 0]"
       ]
      }
     ],
     "prompt_number": 165
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors(weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 167,
       "text": [
        "16"
       ]
      }
     ],
     "prompt_number": 167
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 166,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 166
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On peut envisager deux possibilit\u00e9s : \n",
      "\n",
      "- soit augmenter le poids de 2, qui est d\u00e9j\u00e0 3\n",
      "- soit augmenter le poids de 3, qui est 1\n",
      "\n",
      "Comparons les deux cas."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors([7, 1, 1, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 172,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 172
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, [7, 1, 1, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 173,
       "text": [
        "1920"
       ]
      }
     ],
     "prompt_number": 173
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors([3, 3, 1, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 174,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 174
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, [3, 3, 1, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 176,
       "text": [
        "1080"
       ]
      }
     ],
     "prompt_number": 176
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Donc l\u00e0, on a plut\u00f4t int\u00e9r\u00eat \u00e0 augmenter le facteur 3!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "weights = [3, 3, 1, 0, 0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 177
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On peut encore essayer d'aller plus loin."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors([7, 3, 0, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 179,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 179
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_current_number(p, [7, 3, 0, 0, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 178,
       "text": [
        "3456"
       ]
      }
     ],
     "prompt_number": 178
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Conclusions sur l'exemple avec 32 : j'ai appris que les poids ne pouvaient \u00eatre que des puissances de 2 moins 1. Et que du coup pour choisir correctement, il fallait toujours v\u00e9rifier le prochain voisin de poids diff\u00e9rents des poids initiaux."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Nouvelle tentative"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class euler500_solver():\n",
      "    def __init__(self, number_of_divisors, primes):\n",
      "        self.number_of_divisors = number_of_divisors\n",
      "        self.max_primes = int(log(number_of_divisors) / log(2))\n",
      "        self.primes = primes[:self.max_primes]\n",
      "        self.weights = [1] * self.max_primes\n",
      "    \n",
      "    def step(self):\n",
      "        max_index = [(ind, v) for (ind, v) in enumerate(self.weights) if v!= 0][-1][0]\n",
      "        max_prime = self.primes[max_index]\n",
      "        for current_index in range(max_index + 1):\n",
      "            current_weight = self.weights[current_index]\n",
      "            current_prime = self.primes[current_index]\n",
      "            next_weight = 2 * current_weight + 1\n",
      "            delta_weight = next_weight - current_weight\n",
      "            if current_prime ** delta_weight / max_prime < 1:\n",
      "                self.weights[max_index] -= 1\n",
      "                self.weights[current_index] = next_weight\n",
      "                return True\n",
      "                break\n",
      "        return False\n",
      "        \n",
      "    def compute_current_number(self, modulo=None):\n",
      "        factors = ([p**w for (p, w) in zip(self.primes, self.weights)])\n",
      "\n",
      "        if modulo == None:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product *= f\n",
      "            return product\n",
      "        else:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product = (product * f) % modulo\n",
      "            return product"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 235
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On teste le nouvel algorithme sur le cas 16 :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(16, rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 181
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 182,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 182
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 183,
       "text": [
        "120"
       ]
      }
     ],
     "prompt_number": 183
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 184,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 184
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Sur le cas 32 :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(32, rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 185
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 186,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 186
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 187,
       "text": [
        "840"
       ]
      }
     ],
     "prompt_number": 187
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 189,
       "text": [
        "[3, 1, 1, 1, 0]"
       ]
      }
     ],
     "prompt_number": 189
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "compute_number_of_divisors(e.weights)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 192,
       "text": [
        "32"
       ]
      }
     ],
     "prompt_number": 192
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Le r\u00e9sultat attendu semble marcher."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(128, rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 193
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 194,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 194
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 195,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 196,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.weights"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 197,
       "text": [
        "[3, 3, 1, 1, 1, 0, 0]"
       ]
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "On peut maintenant tester le vrai cas :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver(2**500500, primes)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 236
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%time\n",
      "last_step = e.step()\n",
      "while last_step:\n",
      "    last_step = e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "CPU times: user 1min 23s, sys: 18.5 s, total: 1min 42s\n",
        "Wall time: 1min 50s\n"
       ]
      }
     ],
     "prompt_number": 237
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.weights[:20]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 240,
       "text": [
        "[31, 15, 15, 15, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3, 3, 3, 3]"
       ]
      }
     ],
     "prompt_number": 240
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number(modulo=500500507)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 239,
       "text": [
        "35407281"
       ]
      }
     ],
     "prompt_number": 239
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Solution optimis\u00e9e Numpy"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = np.array([0, 1, 1, 0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 208
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a.nonzero()[0][-1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 212,
       "text": [
        "2"
       ]
      }
     ],
     "prompt_number": 212
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "from math import log\n",
      "\n",
      "def rwh_primes1(n):\n",
      "    # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188\n",
      "    \"\"\" Returns  a list of primes < n \"\"\"\n",
      "    sieve = [True] * int(n/2)\n",
      "    for i in range(3,int(n**0.5)+1,2):\n",
      "        if sieve[int(i/2)]:\n",
      "            sieve[int(i*i/2)::i] = [False] * int((n-i*i-1)/(2*i)+1)\n",
      "    return [2] + [2*i+1 for i in range(1, int(n/2)) if sieve[i]]\n",
      "\n",
      "class euler500_solver_numpy():\n",
      "    def __init__(self, number_of_divisors, primes):\n",
      "        self.number_of_divisors = number_of_divisors\n",
      "        self.max_primes = int(log(number_of_divisors) / log(2))\n",
      "        self.primes = np.array(primes[:self.max_primes])\n",
      "        self.weights = np.array([1] * self.max_primes, dtype=np.int16)\n",
      "    \n",
      "    def step(self):\n",
      "        max_index = self.weights.nonzero()[0][-1]\n",
      "        max_prime = self.primes[max_index]\n",
      "        \n",
      "        #current_weights = self.weights\n",
      "        #next_weights = 2 * self.weights + 1\n",
      "        #delta_weights = next_weights - current_weights\n",
      "        \n",
      "        factors = np.power(self.primes, self.weights + 1) / max_prime\n",
      "        candidates = factors < 1\n",
      "        if factors[candidates].size > 0:\n",
      "            new_index = candidates.argmax()\n",
      "            self.weights[max_index] = 0\n",
      "            self.weights[new_index] = 2 * self.weights[new_index] + 1\n",
      "            return True\n",
      "        else:\n",
      "            return False\n",
      "        \n",
      "    def iterate(self):\n",
      "        last_step = self.step()\n",
      "        while last_step:\n",
      "            last_step = self.step()\n",
      "        return last_step\n",
      "        \n",
      "    def compute_current_number(self, modulo=None):\n",
      "        factors = ([p**w for (p, w) in zip(self.primes, self.weights)])\n",
      "\n",
      "        if modulo == None:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product *= f\n",
      "            return product\n",
      "        else:\n",
      "            product = 1\n",
      "            for f in factors:\n",
      "                product = (product * f) % modulo\n",
      "            return product"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e = euler500_solver_numpy(32, rwh_primes1(100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.step()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e.compute_current_number()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "840"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "primes = rwh_primes1(10000000)\n",
      "e2 = euler500_solver_numpy(2**500500, primes)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%time\n",
      "e2.iterate()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "CPU times: user 12 s, sys: 643 ms, total: 12.7 s\n",
        "Wall time: 12.7 s\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "result = 1\n",
      "for p, w in zip(e2.primes, e2.weights): \n",
      "    result = (result * int(p) ** int(w)) % 500500507\n",
      "result"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "35407281"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e2.compute_current_number(modulo=500500507)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "-c:55: RuntimeWarning: overflow encountered in long_scalars\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "19538647"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e2.weights[:20]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 232,
       "text": [
        "array([31, 15, 15, 15,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  3,  3,\n",
        "        3,  3,  3], dtype=int16)"
       ]
      }
     ],
     "prompt_number": 232
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}