add problem 57

notebooks/problem0057.ipynb

@@ -0,0 +1,175 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a22e7878",
+   "metadata": {},
+   "source": [
+    "# [Square Root Convergents](https://projecteuler.net/problem=57)\n",
+    "\n",
+    "Stop me if you've heard this one before: easy with SageMath."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "27ac9cdb",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "153"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "convergents = continued_fraction(sqrt(2)).convergents()\n",
+    "\n",
+    "cs = []\n",
+    "for c in convergents[1:1001]:\n",
+    "    n, d = c.as_integer_ratio()\n",
+    "    if len(n.digits()) > len(d.digits()):\n",
+    "        cs.append(c)\n",
+    "    \n",
+    "len(cs)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8cc002c4",
+   "metadata": {},
+   "source": [
+    "Here's how to work this yourself.\n",
+    "\n",
+    "If you were to look up the [square root of 2](https://en.wikipedia.org/wiki/Square_root_of_2), you would discover that the denominators of successive convergents of $\\sqrt{2}$ form a sequence called the [Pell numbers](https://en.wikipedia.org/wiki/Pell_number). The numerators are half of a related sequence called the Pell-Lucas numbers. We can easily make generators for these sequences from their definitions."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "58c359a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def pell_numbers():\n",
+    "    a, b = 0, 1\n",
+    "    while True:\n",
+    "        yield a\n",
+    "        a, b = 2*a + b, a\n",
+    "\n",
+    "\n",
+    "def pell_lucas_numbers():\n",
+    "    a, b = 2, 2\n",
+    "    yield a\n",
+    "    while True:\n",
+    "        yield a\n",
+    "        a, b = 2*a + b, a"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c8588bfa",
+   "metadata": {},
+   "source": [
+    "With these generators, we can make a generator of the convergents of $\\sqrt{2}$. We'll skip the first generated value since the first Pell number is 0."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "308eaa91",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(1, 0)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "convergents = ((p//2, q) for (p, q) in zip(pell_lucas_numbers(), pell_numbers()))\n",
+    "next(convergents)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9f7817c1",
+   "metadata": {},
+   "source": [
+    "Now we just iterate over the convergents and check the digits."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "ee53511d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "153"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "digits = lambda n: floor(1 + log(n, 10))\n",
+    "\n",
+    "cs = []\n",
+    "for (i, (p, q)) in enumerate(convergents):\n",
+    "    if i >= 1000:\n",
+    "        break\n",
+    "        \n",
+    "    if digits(p) > digits(q):\n",
+    "        cs.append((p, q))\n",
+    "        \n",
+    "len(cs)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "88596f4c",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Numerators of convergents of $\\sqrt{2}$: [A001333](https://oeis.org/A001333)\n",
+    "* Pell numbers (denominators of convergents of $\\sqrt{2}$): [A000129](https://oeis.org/A000129)"
+   ]
+  }
+ ],
+ "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
+}