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  {
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   "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",
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   "id": "27ac9cdb",
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    {
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      "text/plain": [
       "153"
      ]
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   "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)"
   ]
  },
  {
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   "id": "8cc002c4",
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   "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": {},
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   "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)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
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