From 5d889313d96683e6a88ad3e3a9cd51c5bc88d2a5 Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Thu, 22 May 2025 20:11:38 -0400
Subject: [PATCH] add problem 69

---
 notebooks/problem0069.ipynb | 80 +++++++++++++++++++++++++++++++++++++
 1 file changed, 80 insertions(+)
 create mode 100644 notebooks/problem0069.ipynb

diff --git a/notebooks/problem0069.ipynb b/notebooks/problem0069.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d4c97c0c",
+   "metadata": {},
+   "source": [
+    "# [Totient Maximum](https://projecteuler.net/problem=69)\n",
+    "\n",
+    "This problem can be solved by hand.\n",
+    "\n",
+    "The [totient function](https://en.wikipedia.org/wiki/Euler%27s_totient_function) can be computed as\n",
+    "$$\\phi(n) = n\\prod_{p|n} \\left(1 - \\frac{1}{p}\\right)$$\n",
+    "Therefore\n",
+    "$$\\frac{n}{\\phi(n)} = \\prod_{p|n}\\left(\\frac{p}{p-1}\\right)$$\n",
+    "This implies that $n/\\phi(n)$ increases as the number of distinct prime factors of $n$ increases, so to maximize this quotient, we want $n$ to have as many distinct prime factors as possible.\n",
+    "\n",
+    "Furthermore, if $p < q$ for primes $p$ and $q$, then\n",
+    "$$\\frac{p}{p-1} > \\frac{q}{q-1}$$\n",
+    "Put simply, this means that smaller prime factors will lead to a larger $n/\\phi(n)$. These two facts suggest we should try numbers that are the product of the first several primes, which are called [primorials](https://en.wikipedia.org/wiki/Primorial). Our answer then is the largest primorial less than 1000000:\n",
+    "$$2 \\times 3 \\times 5 \\times 7 \\times 11 \\times 13 \\times 17 = 510510$$\n",
+    "\n",
+    "Alternatively, you can just ask SageMath - its implementation of $\\phi(n)$ is fast enough for this problem. However, you might run into difficulties writing your own implementation that's this fast."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "da203e05",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "510510"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "qs = {n: n/euler_phi(n) for n in range(1, 1000001)}\n",
+    "max(qs, key=qs.get)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5d99957e",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Totient: [A000010](https://oeis.org/A000010)\n",
+    "* Primorial numbers: [A002110](https://oeis.org/A002110)"
+   ]
+  }
+ ],
+ "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
+}