From 877c4798318b3a5fdf15fefe021b99501b60205e Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Wed, 21 May 2025 00:49:13 -0400
Subject: [PATCH] add problem 64

---
 notebooks/problem0064.ipynb | 143 ++++++++++++++++++++++++++++++++++++
 1 file changed, 143 insertions(+)
 create mode 100644 notebooks/problem0064.ipynb

diff --git a/notebooks/problem0064.ipynb b/notebooks/problem0064.ipynb
new file mode 100644
index 0000000..5ab2650
--- /dev/null
+++ b/notebooks/problem0064.ipynb
@@ -0,0 +1,143 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "721eef2a",
+   "metadata": {},
+   "source": [
+    "# [Odd Period Square Roots](https://projecteuler.net/problem=64)\n",
+    "\n",
+    "SageMath can compute the periods of these continued fractions if you ask it nicely, but it's pretty slow since we have to construct a new quadratic field on every loop."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d9f0863f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1322"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "odd_period_sqrts = set()\n",
+    "for N in range(1, 10001):\n",
+    "    if is_square(N):\n",
+    "        continue\n",
+    "        \n",
+    "    K.<s> = QuadraticField(N)\n",
+    "    if continued_fraction(s).period_length() % 2 == 1:\n",
+    "        odd_period_sqrts.add(N)      \n",
+    "        \n",
+    "len(odd_period_sqrts)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a67c8448",
+   "metadata": {},
+   "source": [
+    "We can speed it up if we apply an algorithm for computing the [repetend of continued fractions of square roots](https://en.wikipedia.org/wiki/Periodic_continued_fraction)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "94fd3a59",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def repetend_sqrt(S):\n",
+    "    m, d, a0 = 0, 1, isqrt(S)\n",
+    "    \n",
+    "    repetend = []\n",
+    "    a = a0\n",
+    "    while a != 2 * a0:\n",
+    "        m = d * a - m\n",
+    "        d = (S - m^2) / d\n",
+    "        a = floor((a0 + m) / d)\n",
+    "        repetend.append(a)\n",
+    "        \n",
+    "    return repetend"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "67b9827d",
+   "metadata": {},
+   "source": [
+    "To gain an intuition for this algorithm (without getting too deep into the weeds), it's helpful to consider the more general problem of finding the continued fraction of a [quadratic irrational](https://en.wikipedia.org/wiki/Quadratic_irrational_number). To do this, consider the equation\n",
+    "$$\\frac{m+\\sqrt{S}}{d} = a_0 + \\frac{1}{r_1}$$\n",
+    "The first term of the continued fraction, $a_0$, is simply the floor of the left hand side, and the following terms are just the terms of the continued fraction of $r_1$. We can then use some algebra to solve for $r_1$.\n",
+    "\n",
+    "$$r_1 = \\frac{d\\left(da_0 - m + \\sqrt{S}\\right)}{S-(m-da_0)^2}$$\n",
+    "\n",
+    "This shows that $r_1$ is *also* a quadratic irrational, so we can apply this process repeatedly to determine successive terms of the continued fraction.\n",
+    "\n",
+    "It is also known that the last term in the period of the continued fraction of a square root is twice the initial term, which provides a stopping condition."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "b8490f58",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1322"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "odd_period_sqrts = {N for N in range(1, 10001) if not is_square(N) and len(repetend_sqrt(N)) % 2 == 1}\n",
+    "len(odd_period_sqrts)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f6ba21d9",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Periods of continued fractions of $\\sqrt{n}$: [A003285](https://oeis.org/A003285)"
+   ]
+  }
+ ],
+ "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
+}