eulerbooks/notebooks/problem0039.ipynb

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    "# [Integer Right Triangles](https://projecteuler.net/problem=39)\n",
    "\n",
    "If a right triangle has integer side lengths, the side lengths are a [Pythagorean triple](https://en.wikipedia.org/wiki/Pythagorean_triple). In [problem 9](https://projecteuler.net/problem=9), we wrote a generator for primitive Pythagorean triples based off of Euclid's formula. We can modify that generator to cut off after the triplets have passed a maximum perimeter. Note that a triangle with side lengths generated by Euclid's formula will have perimeter $2m^2 + 2mn$."
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    "from itertools import count\n",
    "\n",
    "def primitive_pythagorean_triplets(max_perim):\n",
    "    for m in count(2):\n",
    "        if 2*m^2 + 2*m > max_perim:\n",
    "            break\n",
    "\n",
    "        for n in range(1, m):\n",
    "            if not ((m % 2) ^^ (n % 2)) or gcd(m, n) != 1:\n",
    "                continue\n",
    "            \n",
    "            a = m^2 - n^2\n",
    "            b = 2*m*n\n",
    "            c = m^2 + n^2\n",
    "            \n",
    "            if a + b + c > max_perim:\n",
    "                break\n",
    "            \n",
    "            yield (a, b, c)"
   ]
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   "source": [
    "Now we can just iterate through our new generator and count the perimeters. We also count multiples of the perimeters to include non-primitive triplets."
   ]
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     "data": {
      "text/plain": [
       "840"
      ]
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   "source": [
    "from collections import Counter\n",
    "\n",
    "max_perim = 1000\n",
    "perimeters = Counter()\n",
    "for (a, b, c) in primitive_pythagorean_triplets(max_perim):\n",
    "    for k in count(1):\n",
    "        perimeter = k * (a + b + c)\n",
    "        if perimeter > max_perim:\n",
    "            break\n",
    "        \n",
    "        perimeters[perimeter] += 1\n",
    "\n",
    "max(perimeters, key=perimeters.get)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cb1b692",
   "metadata": {},
   "source": [
    "## Related sequences\n",
    "* Number of integer right triangles with perimeter $n$: [A024155](https://oeis.org/A024155)"
   ]
  }
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