add problem 70

notebooks/problem0070.ipynb

@@ -0,0 +1,141 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "ec776294",
+   "metadata": {},
+   "source": [
+    "# [Totient Permutation](https://projecteuler.net/problem=70)\n",
+    "\n",
+    "SageMath's implementation of $\\phi(n)$ is fast enough that you could brute force this if you wanted, but if we're clever, we can solve more quickly.\n",
+    "\n",
+    "We'll write a simple function for determining if two numbers are digit permutations of each other."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "e6757165",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def is_permutation_pair(a, b):\n",
+    "    s, t = str(a), str(b)\n",
+    "    return sorted(s) == sorted(t)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bdeb8c77",
+   "metadata": {},
+   "source": [
+    "As in [problem 69](https://projecteuler.net/problem=69),\n",
+    "$$\\phi(n) = n\\prod_{p | n} \\left(1 - \\frac{1}{p}\\right)$$\n",
+    "\n",
+    "Rather than calculate the totients of every single number up to $10^7$, we'll start with just the primes - for a prime $p$, $\\phi(p) = p-1$. We'll store these primes in a [min-heap](https://en.wikipedia.org/wiki/Heap_(data_structure)) ordered by $\\frac{p}{\\phi(p)}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "244e05bd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import heapq\n",
+    "\n",
+    "limit = 10^7\n",
+    "\n",
+    "primes = prime_range(limit)\n",
+    "queue = [(p / (p - 1), (p, p - 1)) for p in primes]\n",
+    "heapq.heapify(queue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9a057da0",
+   "metadata": {},
+   "source": [
+    "Then we'll search to find the value $n$ with the smallest ratio that is also a digit permutation of its totient. We'll check composite values by pushing $np$ to the queue for each prime $p$.\n",
+    "\n",
+    "When we find a value that is a digit permutation of its totient, we'll know that it also has the smallest ratio and can stop."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "99c4267a",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "8319823"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "answer = None\n",
+    "visited = set()\n",
+    "while queue != []:\n",
+    "    _, (n, totient) = heapq.heappop(queue)\n",
+    "        \n",
+    "    if n in visited:\n",
+    "        continue\n",
+    "    visited.add(n)\n",
+    "        \n",
+    "    if is_permutation_pair(n, totient):\n",
+    "        answer = n\n",
+    "        break\n",
+    "        \n",
+    "    for p in primes:\n",
+    "        q = n * p\n",
+    "        if q >= limit:\n",
+    "            break\n",
+    "            \n",
+    "        if n % p != 0:\n",
+    "            new_totient = totient * (p - 1)\n",
+    "        else:\n",
+    "            new_totient = totient * p\n",
+    "        \n",
+    "        ratio = q / new_totient\n",
+    "        heapq.heappush(queue, (ratio, (q, new_totient)))\n",
+    "        \n",
+    "answer"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40cf1e01",
+   "metadata": {},
+   "source": [
+    "Note: lots of people in the problem thread make the assumption that the answer must be a [semiprime](https://en.wikipedia.org/wiki/Semiprime). However, Steendor points out that for certain upper bounds, this assumption does not hold. This solution avoids making that assumption."
+   ]
+  }
+ ],
+ "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
+}