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     "source": [
      "###The hyperexponentiation of a number\n",
      "####Problem 188\n",
      "\n",
      "The hyperexponentiation or tetration of a number $a$ by a positive integer $b$, denoted by $a\u2191\u2191b$ or $^ba$, is recursively defined by:\n",
      "\n",
      "$$a\u2191\u21911 = a,$$\n",
      "$$a\u2191\u2191(k+1) = a^{(a\u2191\u2191k)}.$$\n",
      "\n",
      "Thus we have e.g. $3\u2191\u21912 = 3^3 = 27$, hence $3\u2191\u21913 = 3^{27} = 7625597484987$ and $3\u2191\u21914$ is roughly $10^{3.6383346400240996*10^{12}}$.\n",
      "\n",
      "Find the last $8$ digits of $1777\u2191\u21911855$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "open Euler\n",
      "\n",
      "let p188() =\n",
      "    let rec loop (n:bigint) k = \n",
      "        match k with\n",
      "        |1 -> n\n",
      "        |_ -> bigint.ModPow(n,loop n (k-1),100000000I)\n",
      "    loop 1777I 1855\n",
      "\n",
      "Euler.Timer(p188)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 1,
       "text": [
        "val it : bigint * float = (95962097, 0.0109785)"
       ]
      }
     ],
     "prompt_number": 1
    },
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