{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f118733f",
   "metadata": {},
   "source": [
    "# [Circular Primes](https://projecteuler.net/problem=35)\n",
    "\n",
    "We're only looking for [circular primes](https://en.wikipedia.org/wiki/Circular_prime) below one million, which makes this search not too strenuous. Obviously, being prime is a precondition for being a circular prime, so we can start our search by only checking primes below one million.\n",
    "\n",
    "We can then use Python's [`deque` class](https://docs.python.org/3/library/collections.html) to write a function that cycles through a prime number's digits to check if it is a circular prime."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9d4833c5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "55\n"
     ]
    }
   ],
   "source": [
    "from collections import deque\n",
    "\n",
    "def is_circular_prime(p):\n",
    "    d = deque(str(p))\n",
    "    while True:\n",
    "        d.rotate()\n",
    "        rp = int(\"\".join(d))\n",
    "        if rp == p:\n",
    "            break\n",
    "        elif not is_prime(rp):\n",
    "            return False\n",
    "    \n",
    "    return True\n",
    "\n",
    "\n",
    "circular_primes = set(p for p in prime_range(1000000) if is_circular_prime(p))\n",
    "print(len(circular_primes))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e21d050",
   "metadata": {},
   "source": [
    "There are other optimizations you can make - for instance, every multi-digit circular prime can only be composed of the digits 1, 3, 7, and 9 (since having another digit would eventually result in a rotation that ends in an even number or 5, which obviously isn't prime) - but why bother (for this problem) when it's already this fast?\n",
    "\n",
    "Another note - there aren't that many known circular primes, and all of the known ones greater than one million are [repunits](https://en.wikipedia.org/wiki/Repunit).\n",
    "\n",
    "## Relevant sequences\n",
    "* Circular primes: [A068652](https://oeis.org/A068652)\n",
    "\n",
    "#### 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)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.5",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}