euler-048-self-power-20170113.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[Project Euler](https://projecteuler.net/)\n",
    "\n",
    "#### Distinct primes factors\n",
    "\n",
    "\n",
    "[Problem 48](https://projecteuler.net/problem=48)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The series, 1<sup>1</sup> + 2<sup>2</sup> + 3<sup>3</sup> + ... + 10<sup>10</sup> = 10405071317.  \n",
    "\n",
    "Find the last ten digits of the series, 1<sup>1</sup> + 2<sup>2</sup> + 3<sup>3</sup> + ... + 1000<sup>1000</sup>."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def foo(n, m):\n",
    "    return str(sum(i ** i for i in range(1, n + 1)))[-m:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0405071317'"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foo(10, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 14.5 ms per loop\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "'9110846700'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 1000\n",
    "m = 10\n",
    "%timeit foo(n, m)\n",
    "foo(n, m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I wonder if the third argument to pow() could help make the calculation faster by avoiding unnecessary calculations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 37.4 µs per loop\n",
      "1000000 loops, best of 3: 1.46 µs per loop\n"
     ]
    }
   ],
   "source": [
    "n = 1000\n",
    "m = 10\n",
    "%timeit n ** n\n",
    "%timeit pow(n, n, 10**m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def foo(n, m):\n",
    "    m10 = 10 ** m\n",
    "    return str(sum(pow(i, i, m10) for i in range(1, n + 1)))[-m:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'405071317'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foo(10, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 2.53 ms per loop\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "'9110846700'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 1000\n",
    "m = 10\n",
    "%timeit foo(n, m)\n",
    "foo(n, m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Try using modulo instead of str()[-m:] to get last m digits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def foo(n, m):\n",
    "    m10 = 10 ** m\n",
    "    return sum(pow(i, i, m10) for i in range(1, n + 1)) % m10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "405071317"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "foo(10, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 2.53 ms per loop\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9110846700"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 1000\n",
    "m = 10\n",
    "%timeit foo(n, m)\n",
    "foo(n, m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cell #1 has the straightforward brute force approach. It is correct.\n",
    "\n",
    "Cell #8 does not show leading zeroes, so it is not a good solution.\n",
    "\n",
    "Cell #5 does show leading zeroes, has reasonable readability, and is fast."
   ]
  }
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