From ce226c4d5b1099029523937f320b78c12ce4415c Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Sat, 24 May 2025 12:20:33 -0400
Subject: [PATCH] add problem 71

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "e7323b89",
+   "metadata": {},
+   "source": [
+    "# [Ordered Fractions](https://projecteuler.net/problem=71)\n",
+    "\n",
+    "This problem can be solved by hand.\n",
+    "\n",
+    "The concept the problem is describing is called a [Farey sequence](https://en.wikipedia.org/wiki/Farey_sequence). The example given in the problem is $F_8$, and we are tasked with finding the numerator of the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
+    "\n",
+    "It turns out there is a very simple method for determining this. Whenever you have two neighbors $\\frac{a}{b}$ and $\\frac{c}{d}$ in a Farey sequence, the next term that will appear between them in a subsequent Farey sequence is simply $\\frac{a+c}{b+d}$, called the [mediant](https://en.wikipedia.org/wiki/Mediant_(mathematics)) of the two neighbors. For example, since we're given that the left neighbor of $\\frac{3}{7}$ in $F_8$ is $\\frac{2}{5}$, the next fraction to appear between the two will be\n",
+    "$$\\frac{2+3}{5+7} = \\frac{5}{12}$$\n",
+    "Naturally, this fraction will first appear in $F_{12}$, meaning $\\frac{5}{12}$ is the left neighbor of $\\frac{3}{7}$ in that Farey sequence. We could then, in turn, find the mediant of $\\frac{5}{12}$ and $\\frac{3}{7}$ to find the next left neighbor of $\\frac{3}{7}$ ($\\frac{8}{19}$, appearing in $F_{19}$).\n",
+    "\n",
+    "We can repeat this process until we find a mediant with a denominator greater than 1000000. At that point, we know that mediant will *not* be in $F_{1000000}$, so whatever left neighbor we just used to calculate it must be the left neighbor of $\\frac{3}{7}$ in $F_{1000000}$.\n",
+    "\n",
+    "To compute this by hand, observe that the $n$th mediant computed in this manner is simply\n",
+    "$$\\frac{2 + 3n}{5 + 7n}$$\n",
+    "Therefore, we just need to find the largest $n$ such that $5 + 7n \\leq 1000000$, which is $n=142856$. This gives us a numerator of 428570.\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* Numerators of Farey sequences: [A007305](https://oeis.org/A007305)"
+   ]
+  }
+ ],
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+   "language": "sage",
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