142 lines
3.8 KiB
Plaintext
142 lines
3.8 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "ec776294",
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"metadata": {},
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"source": [
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"# [Totient Permutation](https://projecteuler.net/problem=70)\n",
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"\n",
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"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",
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"\n",
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"We'll write a simple function for determining if two numbers are digit permutations of each other."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "e6757165",
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"metadata": {},
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"outputs": [],
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"source": [
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"def is_permutation_pair(a, b):\n",
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" s, t = str(a), str(b)\n",
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" return sorted(s) == sorted(t)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "bdeb8c77",
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"metadata": {},
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"source": [
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"As in [problem 69](https://projecteuler.net/problem=69),\n",
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"$$\\phi(n) = n\\prod_{p | n} \\left(1 - \\frac{1}{p}\\right)$$\n",
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"\n",
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"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 $\\frac{p}{\\phi(p)}$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "244e05bd",
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"metadata": {},
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"outputs": [],
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"source": [
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"import heapq\n",
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"\n",
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"limit = 10^7\n",
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"\n",
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"primes = prime_range(limit)\n",
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"queue = [(p / (p - 1), (p, p - 1)) for p in primes]\n",
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"heapq.heapify(queue)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9a057da0",
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"metadata": {},
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"source": [
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"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",
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"\n",
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"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."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "99c4267a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"8319823"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"answer = None\n",
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"visited = set()\n",
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"while queue != []:\n",
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" _, (n, totient) = heapq.heappop(queue)\n",
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" \n",
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" if n in visited:\n",
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" continue\n",
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" visited.add(n)\n",
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" \n",
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" if is_permutation_pair(n, totient):\n",
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" answer = n\n",
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" break\n",
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" \n",
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" for p in primes:\n",
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" q = n * p\n",
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" if q >= limit:\n",
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" break\n",
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" \n",
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" if n % p != 0:\n",
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" new_totient = totient * (p - 1)\n",
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" else:\n",
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" new_totient = totient * p\n",
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" \n",
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" ratio = q / new_totient\n",
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" heapq.heappush(queue, (ratio, (q, new_totient)))\n",
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" \n",
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"answer"
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]
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},
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{
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"cell_type": "markdown",
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"id": "40cf1e01",
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"metadata": {},
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"source": [
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"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."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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