{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pandigital prime (Euler Problem 41)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "[https://projecteuler.net/problem=41]\n",
    "\n",
    "We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.\n",
    "\n",
    "What is the largest n-digit pandigital prime that exists?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def expmod(base, exp, m):\n",
    "    if exp == 0:\n",
    "        return 1\n",
    "    if (exp % 2 == 0):\n",
    "        return (expmod(base, exp // 2, m) ** 2 % m)\n",
    "    return (base * expmod(base, exp - 1, m) % m)\n",
    "    \n",
    "def expmod_naiv(base, exp, m):\n",
    "    return base ** exp % m\n",
    "\n",
    "def fermat_test(n):\n",
    "    a = n - 3\n",
    "    return expmod(a, n, n) == a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert(expmod(2, 100000, 2143) == expmod_naiv(2, 100000, 2143))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert(fermat_test(2143))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7652413\n"
     ]
    }
   ],
   "source": [
    "import itertools\n",
    "\n",
    "for pandigital in itertools.permutations(\"7654321\"):\n",
    "    p = int(\"\".join(pandigital))\n",
    "    if fermat_test(p):\n",
    "        s = p\n",
    "        break\n",
    "        \n",
    "print(s)\n",
    "assert(s == 7652413)"
   ]
  }
 ],
 "metadata": {
  "completion_date": "Wed, 19 Dec 2018, 20:14",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python3.6",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  },
  "tags": [
   "pandigital",
   "prime",
   "fermat",
   "test"
  ]
 },
 "nbformat": 4,
 "nbformat_minor": 2
}