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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "4c2c6cf3-da5e-4fef-b37b-cbed145f4eea",
+   "metadata": {},
+   "source": [
+    "[COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-72/issue-6/Counterexample-to-Eulers-conjecture-on-sums-of-like-powers/bams/1183528522.full)\n",
+    "\n",
+    "![](https://i.sstatic.net/VQL6D.png)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "id": "7de18ad5-b328-4618-911d-32c61ddab13d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def sum_of_powers(M: int, n=5) -> list[tuple[int]]:\n",
+    "    \"\"\"Return all (a, b, c, d, e), all < M, such that a^n + b^n +  c^n + d^n = e^n.\n",
+    "    This is O(M^3) in time and O(M^2) in space.\"\"\"\n",
+    "    table = {a ** n + b ** n: (a, b)\n",
+    "             for (a, b) in combinations(range(M), 2)}\n",
+    "    results = []\n",
+    "    for (c, d, e) in combinations(range(M), 3):\n",
+    "        an_plus_bn = e ** n - (c ** n + d ** n)\n",
+    "        if an_plus_bn in table:\n",
+    "            a, b = table[an_plus_bn]\n",
+    "            if a <= b <= c <= d: # Don't want duplicates\n",
+    "                results.append((a, b, c, d, e))\n",
+    "    return results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "id": "00dc3732-ed29-4137-a3f8-fc7921e28d08",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[(27, 84, 110, 133, 144)]"
+      ]
+     },
+     "execution_count": 59,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sum_of_powers(150)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 56,
+   "id": "8c603613-b561-4a97-9779-c4d94269331a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 3.39 s, sys: 5.23 ms, total: 3.4 s\n",
+      "Wall time: 3.4 s\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[[27, 84, 110, 133, 144], [54, 168, 220, 266, 288]]"
+      ]
+     },
+     "execution_count": 56,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "%time sum_of_powers(300)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "id": "6e9de2d8-abf2-482d-adeb-c2ac01d10b4e",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 1min 9s, sys: 227 ms, total: 1min 9s\n",
+      "Wall time: 1min 11s\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[[27, 84, 110, 133, 144],\n",
+       " [54, 168, 220, 266, 288],\n",
+       " [81, 252, 330, 399, 432],\n",
+       " [108, 336, 440, 532, 576],\n",
+       " [135, 420, 550, 665, 720],\n",
+       " [162, 504, 660, 798, 864]]"
+      ]
+     },
+     "execution_count": 57,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "%time sum_of_powers(1000)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "4b0ee9d9-d38d-4a63-8ccf-7baed27967f1",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.13.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}