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   "source": [
    "# [Prime Permutations](https://projecteuler.net/problem=49)\n",
    "\n",
    "An easy way to approach this problem is to first find [arithmetic progressions of primes](https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression), then check if any of those progressions have numbers that are permutations of each other.\n",
    "\n",
    "To start, we'll generate all the four-digit primes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d278b9e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "primes = prime_range(1000, 10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97dab665",
   "metadata": {},
   "source": [
    "Here's where we find arithmetic progressions. Given two primes $p < q$, we can calculate their difference $d = q - p$ and check if $q + d$ is prime - if it is, we have a three-number progression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0eaf16ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "from itertools import combinations\n",
    "\n",
    "progressions = set()\n",
    "for (p, q) in combinations(primes, 2):\n",
    "    d = q - p\n",
    "    r = q + d\n",
    "    if r < 10000 and is_prime(r):\n",
    "        progressions.add((p, q, r))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbbcab4f",
   "metadata": {},
   "source": [
    "Here's how many of these progressions there are."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "8c87fea2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "42994"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(progressions)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da3a48f3",
   "metadata": {},
   "source": [
    "Now we'll check each progression to find one where the digits of the numbers are permutations of each other."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "89cb3dd0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2969, 6299, 9629)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for (p, q, r) in progressions:\n",
    "    if p == 1487 and q == 4817 and r == 8147:\n",
    "        continue\n",
    "        \n",
    "    if sorted(p.digits()) == sorted(q.digits()) == sorted(r.digits()):\n",
    "        break\n",
    "        \n",
    "p, q, r"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "929e777d",
   "metadata": {},
   "source": [
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
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