{
 "cells": [
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   "cell_type": "markdown",
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   "source": [
    "# Prime pair sets (Euler Problem 60)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "[https://projecteuler.net/problem=60](https://projecteuler.net/problem=60)\n",
    "\n",
    "The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.\n",
    "\n",
    "Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[3, 7], [7, 3], [3, 109], [109, 3], [3, 673], [673, 3], [7, 109], [109, 7], [7, 673], [673, 7], [109, 673], [673, 109]]\n"
     ]
    }
   ],
   "source": [
    "from itertools import combinations\n",
    "\n",
    "def concatenate_all(numbers):\n",
    "    result = []\n",
    "    for a, b in combinations(numbers, 2):\n",
    "        result.append([a, b])\n",
    "        result.append([b, a])\n",
    "    return result\n",
    "\n",
    "print(concatenate_all([3, 7, 109, 673]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def sieve_of_eratosthenes(limit):\n",
    "    primes = []\n",
    "    prospects = [n for n in range(2, limit)]\n",
    "    while prospects:\n",
    "        p = prospects[0]\n",
    "        prospects = [x for x in prospects if x % p != 0]\n",
    "        primes.append(p)\n",
    "        if p * p > limit:\n",
    "            break\n",
    "    primes += prospects\n",
    "    return primes\n",
    "\n",
    "primes = sieve_of_eratosthenes(100000)\n",
    "primes_set = set(primes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_prime(n):\n",
    "    if n % 2 == 0:\n",
    "        return False\n",
    "    while n % 2 == 0:\n",
    "        n //= 2\n",
    "    f = 3\n",
    "    while f * f <= n:\n",
    "        if n % f == 0:\n",
    "            return False\n",
    "        else:\n",
    "            f += 2   \n",
    "    return True\n",
    "\n",
    "def concatentation_is_prime(a, b):\n",
    "    ab = int(str(a) + str(b))\n",
    "    ba = int(str(b) + str(a))\n",
    "    if is_prime(ab) and is_prime(ba):\n",
    "        return True\n",
    "    else:\n",
    "        return False\n",
    "    \n",
    "assert(concatentation_is_prime(673, 3))\n",
    "assert(concatentation_is_prime(673, 1) == False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[13, 5197, 5701, 6733, 8389]\n",
      "26033\n"
     ]
    }
   ],
   "source": [
    "potentials = []\n",
    "new_potentials = []\n",
    "done = False\n",
    "for p in primes:\n",
    "    if done:\n",
    "        break\n",
    "    potentials += new_potentials\n",
    "    new_potentials = []\n",
    "    for potential in potentials:\n",
    "        all_concatenations_prime = True\n",
    "        for prime in potential:\n",
    "            if not concatentation_is_prime(p, prime):\n",
    "                all_concatenations_prime = False\n",
    "                break\n",
    "        if all_concatenations_prime:\n",
    "            new_potential = list(potential)\n",
    "            new_potential.append(p)\n",
    "            if len(new_potential) > 4:\n",
    "                print(new_potential)\n",
    "                s = sum(new_potential)\n",
    "                done = True\n",
    "                break\n",
    "            new_potentials.append(new_potential)\n",
    "    if p < 15:\n",
    "        potentials.append([p])\n",
    "\n",
    "print(s)\n",
    "assert(s == 26033)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "completion_date": "Sun, 6 Jan 2019, 05:00",
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  "tags": [
   "prime",
   "pairs"
  ]
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