No29_Maths_3_42.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# First 2 functions are modified code snippets from https://code.sololearn.com/ckLyQdziTJHc/#py\n",
    "from numpy import log as ln\n",
    "from mpmath import li\n",
    "\n",
    "def f(x):\n",
    "    if x == 0:\n",
    "        return 1 \n",
    "    else:\n",
    "        total = 1\n",
    "        while(x>0):\n",
    "            total *= x\n",
    "            x -= 1\n",
    "        return total \n",
    "\n",
    "# Here we will use the wilsons theorem for prime numbers, stating that a natural number n > 1 is a prime number \n",
    "# if and only if the product of all the positive integers less than n is one less than a multiple of n.\n",
    "\n",
    "def wilson(x):\n",
    "    if(f(x-1)%x==(x-1) and x!=1):\n",
    "        return 1\n",
    "    else:\n",
    "        return 0 \n",
    "\n",
    "# Here i compute π(x)\n",
    "    \n",
    "def π(x):\n",
    "    summ=0\n",
    "    for i in range(1,x+1):\n",
    "        if(wilson(i)==1):\n",
    "            summ=summ+1\n",
    "    return summ\n",
    "\n",
    "# I use the li() function from mpmath to compute the offset logarithmic integral\n",
    "\n",
    "def lim(x):\n",
    "    return li(x, offset=True)\n",
    "\n",
    "xlnxArray=[]\n",
    "pxArray=[]\n",
    "liArray=[]\n",
    "iArray=[]\n",
    "\n",
    "for i in range (3,1000):\n",
    "    iArray.append(i)\n",
    "    xlnxArray.append(i/ln(i))\n",
    "    pxArray.append(π(i))\n",
    "    liArray.append(lim(i))                   \n",
    "    \n",
    "plt.plot(iArray, xlnxArray, 'r--', iArray, pxArray, 'b--', iArray, liArray, 'g--')\n",
    "plt.legend([\"x/ln(x)\",\"π(x)\",\"Li(x)\"])\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"f(x)\")\n",
    "plt.show()\n",
    "    \n",
    "# For values x>10, the formula X/ln(X) < π(X) < Li(x) is valid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No | x/lnx | π(x) | Li(x)\n",
      "-------------------------\n",
      "3  | 1.118 |  2   | 1.118\n",
      "4  | 1.922 |  2   | 1.922\n",
      "5  | 2.589 |  3   | 2.589\n",
      "6  | 3.177 |  3   | 3.177\n",
      "7  | 3.712 |  4   | 3.712\n",
      "8  | 4.209 |  4   | 4.209\n",
      "9  | 4.676 |  4   | 4.676\n",
      "10  | 5.120 |  4   | 5.120\n",
      "11  | 5.546 |  5   | 5.546\n",
      "12  | 5.955 |  5   | 5.955\n",
      "13  | 6.351 |  6   | 6.351\n",
      "14  | 6.736 |  6   | 6.736\n",
      "15  | 7.110 |  6   | 7.110\n",
      "16  | 7.475 |  6   | 7.475\n",
      "17  | 7.831 |  7   | 7.831\n",
      "18  | 8.181 |  7   | 8.181\n",
      "19  | 8.523 |  8   | 8.523\n",
      "20  | 8.860 |  8   | 8.860\n",
      "21  | 9.191 |  8   | 9.191\n",
      "22  | 9.517 |  8   | 9.517\n",
      "23  | 9.838 |  9   | 9.838\n",
      "24  | 10.155 |  9   | 10.155\n",
      "25  | 10.468 |  9   | 10.468\n",
      "26  | 10.777 |  9   | 10.777\n",
      "27  | 11.082 |  9   | 11.082\n",
      "28  | 11.383 |  9   | 11.383\n",
      "29  | 11.682 |  10   | 11.682\n",
      "30  | 11.977 |  10   | 11.977\n",
      "31  | 12.270 |  11   | 12.270\n",
      "32  | 12.560 |  11   | 12.560\n",
      "33  | 12.847 |  11   | 12.847\n",
      "34  | 13.132 |  11   | 13.132\n",
      "35  | 13.414 |  11   | 13.414\n",
      "36  | 13.695 |  11   | 13.695\n",
      "37  | 13.973 |  12   | 13.973\n",
      "38  | 14.248 |  12   | 14.248\n",
      "39  | 14.522 |  12   | 14.522\n",
      "40  | 14.794 |  12   | 14.794\n",
      "41  | 15.065 |  13   | 15.065\n",
      "42  | 15.333 |  13   | 15.333\n",
      "43  | 15.600 |  14   | 15.600\n",
      "44  | 15.865 |  14   | 15.865\n",
      "45  | 16.128 |  14   | 16.128\n",
      "46  | 16.390 |  14   | 16.390\n",
      "47  | 16.651 |  15   | 16.651\n",
      "48  | 16.910 |  15   | 16.910\n",
      "49  | 17.167 |  15   | 17.167\n",
      "50  | 17.424 |  15   | 17.424\n",
      "51  | 17.679 |  15   | 17.679\n",
      "52  | 17.932 |  15   | 17.932\n",
      "53  | 18.185 |  16   | 18.185\n",
      "54  | 18.436 |  16   | 18.436\n",
      "55  | 18.686 |  16   | 18.686\n",
      "56  | 18.935 |  16   | 18.935\n",
      "57  | 19.183 |  16   | 19.183\n",
      "58  | 19.430 |  16   | 19.430\n",
      "59  | 19.676 |  17   | 19.676\n",
      "60  | 19.920 |  17   | 19.920\n",
      "61  | 20.164 |  18   | 20.164\n",
      "62  | 20.407 |  18   | 20.407\n",
      "63  | 20.649 |  18   | 20.649\n",
      "64  | 20.890 |  18   | 20.890\n",
      "65  | 21.130 |  18   | 21.130\n",
      "66  | 21.369 |  18   | 21.369\n",
      "67  | 21.607 |  19   | 21.607\n",
      "68  | 21.844 |  19   | 21.844\n",
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     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "274  | 62.694 |  58   | 62.694\n",
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      "824  | 150.734 |  143   | 150.734\n",
      "825  | 150.883 |  143   | 150.883\n",
      "826  | 151.032 |  143   | 151.032\n",
      "827  | 151.181 |  144   | 151.181\n",
      "828  | 151.329 |  144   | 151.329\n",
      "829  | 151.478 |  145   | 151.478\n",
      "830  | 151.627 |  145   | 151.627\n",
      "831  | 151.776 |  145   | 151.776\n",
      "832  | 151.925 |  145   | 151.925\n",
      "833  | 152.073 |  145   | 152.073\n",
      "834  | 152.222 |  145   | 152.222\n",
      "835  | 152.371 |  145   | 152.371\n",
      "836  | 152.519 |  145   | 152.519\n",
      "837  | 152.668 |  145   | 152.668\n",
      "838  | 152.816 |  145   | 152.816\n",
      "839  | 152.965 |  146   | 152.965\n",
      "840  | 153.114 |  146   | 153.114\n",
      "841  | 153.262 |  146   | 153.262\n",
      "842  | 153.410 |  146   | 153.410\n",
      "843  | 153.559 |  146   | 153.559\n",
      "844  | 153.707 |  146   | 153.707\n",
      "845  | 153.856 |  146   | 153.856\n",
      "846  | 154.004 |  146   | 154.004\n",
      "847  | 154.152 |  146   | 154.152\n",
      "848  | 154.301 |  146   | 154.301\n",
      "849  | 154.449 |  146   | 154.449\n",
      "850  | 154.597 |  146   | 154.597\n",
      "851  | 154.746 |  146   | 154.746\n",
      "852  | 154.894 |  146   | 154.894\n",
      "853  | 155.042 |  147   | 155.042\n",
      "854  | 155.190 |  147   | 155.190\n",
      "855  | 155.338 |  147   | 155.338\n",
      "856  | 155.486 |  147   | 155.486\n",
      "857  | 155.634 |  148   | 155.634\n",
      "858  | 155.783 |  148   | 155.783\n",
      "859  | 155.931 |  149   | 155.931\n",
      "860  | 156.079 |  149   | 156.079\n",
      "861  | 156.227 |  149   | 156.227\n",
      "862  | 156.375 |  149   | 156.375\n",
      "863  | 156.522 |  150   | 156.522\n",
      "864  | 156.670 |  150   | 156.670\n",
      "865  | 156.818 |  150   | 156.818\n",
      "866  | 156.966 |  150   | 156.966\n",
      "867  | 157.114 |  150   | 157.114\n",
      "868  | 157.262 |  150   | 157.262\n",
      "869  | 157.410 |  150   | 157.410\n",
      "870  | 157.557 |  150   | 157.557\n",
      "871  | 157.705 |  150   | 157.705\n",
      "872  | 157.853 |  150   | 157.853\n",
      "873  | 158.000 |  150   | 158.000\n",
      "874  | 158.148 |  150   | 158.148\n",
      "875  | 158.296 |  150   | 158.296\n",
      "876  | 158.443 |  150   | 158.443\n",
      "877  | 158.591 |  151   | 158.591\n",
      "878  | 158.738 |  151   | 158.738\n",
      "879  | 158.886 |  151   | 158.886\n",
      "880  | 159.033 |  151   | 159.033\n",
      "881  | 159.181 |  152   | 159.181\n",
      "882  | 159.328 |  152   | 159.328\n",
      "883  | 159.476 |  153   | 159.476\n",
      "884  | 159.623 |  153   | 159.623\n",
      "885  | 159.771 |  153   | 159.771\n",
      "886  | 159.918 |  153   | 159.918\n",
      "887  | 160.065 |  154   | 160.065\n",
      "888  | 160.213 |  154   | 160.213\n",
      "889  | 160.360 |  154   | 160.360\n",
      "890  | 160.507 |  154   | 160.507\n",
      "891  | 160.654 |  154   | 160.654\n",
      "892  | 160.802 |  154   | 160.802\n",
      "893  | 160.949 |  154   | 160.949\n",
      "894  | 161.096 |  154   | 161.096\n",
      "895  | 161.243 |  154   | 161.243\n",
      "896  | 161.390 |  154   | 161.390\n",
      "897  | 161.537 |  154   | 161.537\n",
      "898  | 161.684 |  154   | 161.684\n",
      "899  | 161.831 |  154   | 161.831\n",
      "900  | 161.978 |  154   | 161.978\n",
      "901  | 162.125 |  154   | 162.125\n",
      "902  | 162.272 |  154   | 162.272\n",
      "903  | 162.419 |  154   | 162.419\n",
      "904  | 162.566 |  154   | 162.566\n",
      "905  | 162.713 |  154   | 162.713\n",
      "906  | 162.860 |  154   | 162.860\n",
      "907  | 163.007 |  155   | 163.007\n",
      "908  | 163.154 |  155   | 163.154\n",
      "909  | 163.301 |  155   | 163.301\n",
      "910  | 163.447 |  155   | 163.447\n",
      "911  | 163.594 |  156   | 163.594\n",
      "912  | 163.741 |  156   | 163.741\n",
      "913  | 163.888 |  156   | 163.888\n",
      "914  | 164.034 |  156   | 164.034\n",
      "915  | 164.181 |  156   | 164.181\n",
      "916  | 164.328 |  156   | 164.328\n",
      "917  | 164.474 |  156   | 164.474\n",
      "918  | 164.621 |  156   | 164.621\n",
      "919  | 164.767 |  157   | 164.767\n",
      "920  | 164.914 |  157   | 164.914\n",
      "921  | 165.060 |  157   | 165.060\n",
      "922  | 165.207 |  157   | 165.207\n",
      "923  | 165.353 |  157   | 165.353\n",
      "924  | 165.500 |  157   | 165.500\n",
      "925  | 165.646 |  157   | 165.646\n",
      "926  | 165.793 |  157   | 165.793\n",
      "927  | 165.939 |  157   | 165.939\n",
      "928  | 166.085 |  157   | 166.085\n",
      "929  | 166.232 |  158   | 166.232\n",
      "930  | 166.378 |  158   | 166.378\n",
      "931  | 166.524 |  158   | 166.524\n",
      "932  | 166.671 |  158   | 166.671\n",
      "933  | 166.817 |  158   | 166.817\n",
      "934  | 166.963 |  158   | 166.963\n",
      "935  | 167.109 |  158   | 167.109\n",
      "936  | 167.255 |  158   | 167.255\n",
      "937  | 167.402 |  159   | 167.402\n",
      "938  | 167.548 |  159   | 167.548\n",
      "939  | 167.694 |  159   | 167.694\n",
      "940  | 167.840 |  159   | 167.840\n",
      "941  | 167.986 |  160   | 167.986\n",
      "942  | 168.132 |  160   | 168.132\n",
      "943  | 168.278 |  160   | 168.278\n",
      "944  | 168.424 |  160   | 168.424\n",
      "945  | 168.570 |  160   | 168.570\n",
      "946  | 168.716 |  160   | 168.716\n",
      "947  | 168.862 |  161   | 168.862\n",
      "948  | 169.008 |  161   | 169.008\n",
      "949  | 169.154 |  161   | 169.154\n",
      "950  | 169.299 |  161   | 169.299\n",
      "951  | 169.445 |  161   | 169.445\n",
      "952  | 169.591 |  161   | 169.591\n",
      "953  | 169.737 |  162   | 169.737\n",
      "954  | 169.883 |  162   | 169.883\n",
      "955  | 170.028 |  162   | 170.028\n",
      "956  | 170.174 |  162   | 170.174\n",
      "957  | 170.320 |  162   | 170.320\n",
      "958  | 170.466 |  162   | 170.466\n",
      "959  | 170.611 |  162   | 170.611\n",
      "960  | 170.757 |  162   | 170.757\n",
      "961  | 170.902 |  162   | 170.902\n",
      "962  | 171.048 |  162   | 171.048\n",
      "963  | 171.194 |  162   | 171.194\n",
      "964  | 171.339 |  162   | 171.339\n",
      "965  | 171.485 |  162   | 171.485\n",
      "966  | 171.630 |  162   | 171.630\n",
      "967  | 171.776 |  163   | 171.776\n",
      "968  | 171.921 |  163   | 171.921\n",
      "969  | 172.067 |  163   | 172.067\n",
      "970  | 172.212 |  163   | 172.212\n",
      "971  | 172.357 |  164   | 172.357\n",
      "972  | 172.503 |  164   | 172.503\n",
      "973  | 172.648 |  164   | 172.648\n",
      "974  | 172.793 |  164   | 172.793\n",
      "975  | 172.939 |  164   | 172.939\n",
      "976  | 173.084 |  164   | 173.084\n",
      "977  | 173.229 |  165   | 173.229\n",
      "978  | 173.375 |  165   | 173.375\n",
      "979  | 173.520 |  165   | 173.520\n",
      "980  | 173.665 |  165   | 173.665\n",
      "981  | 173.810 |  165   | 173.810\n",
      "982  | 173.955 |  165   | 173.955\n",
      "983  | 174.100 |  166   | 174.100\n",
      "984  | 174.246 |  166   | 174.246\n",
      "985  | 174.391 |  166   | 174.391\n",
      "986  | 174.536 |  166   | 174.536\n",
      "987  | 174.681 |  166   | 174.681\n",
      "988  | 174.826 |  166   | 174.826\n",
      "989  | 174.971 |  166   | 174.971\n",
      "990  | 175.116 |  166   | 175.116\n",
      "991  | 175.261 |  167   | 175.261\n",
      "992  | 175.406 |  167   | 175.406\n",
      "993  | 175.551 |  167   | 175.551\n",
      "994  | 175.696 |  167   | 175.696\n",
      "995  | 175.840 |  167   | 175.840\n",
      "996  | 175.985 |  167   | 175.985\n",
      "997  | 176.130 |  168   | 176.130\n",
      "998  | 176.275 |  168   | 176.275\n",
      "999  | 176.420 |  168   | 176.420\n"
     ]
    }
   ],
   "source": [
    "print(\"No | x/lnx | π(x) | Li(x)\")            \n",
    "print(\"-------------------------\")\n",
    "\n",
    "# I re-append all the values so that the cell can be run independendly\n",
    "\n",
    "xlnxArray=[]\n",
    "pxArray=[]\n",
    "liArray=[]\n",
    "iArray=[]\n",
    "\n",
    "for i in range (3,1000):\n",
    "    xlnxArray.append(i/ln(i))\n",
    "    pxArray.append(π(i))\n",
    "    liArray.append(lim(i))                    # For testing purposes i print the function values below\n",
    "    \n",
    "    print(str(i) + \"  | \" + str(\"%.3f\" %(liArray[i-3])) + \" |  \" + str(pxArray[i-3]) + \"   | \" + str(\"%.3f\" %liArray[i-3])) "
   ]
  }
 ],
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