From f69c2d4c001df1fedfd26e703b0ab6f391b3e481 Mon Sep 17 00:00:00 2001
From: filifa <filifa.breath652@8alias.com>
Date: Thu, 22 May 2025 00:19:28 -0400
Subject: [PATCH] add problem 63

---
 notebooks/problem0063.ipynb | 49 +++++++++++++++++++++++++++++++++++++
 1 file changed, 49 insertions(+)
 create mode 100644 notebooks/problem0063.ipynb

diff --git a/notebooks/problem0063.ipynb b/notebooks/problem0063.ipynb
new file mode 100644
index 0000000..0fa20b7
--- /dev/null
+++ b/notebooks/problem0063.ipynb
@@ -0,0 +1,49 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "829a3994",
+   "metadata": {},
+   "source": [
+    "# [Powerful Digit Counts](https://projecteuler.net/problem=63)\n",
+    "\n",
+    "This problem can be solved with a calculator.\n",
+    "\n",
+    "For $x^n$ to have $n$ digits, the following inequality must hold:\n",
+    "$$10^{n-1} \\leq x^n < 10^n$$\n",
+    "If we solve for $x$, we get\n",
+    "$$10^\\frac{n-1}{n} \\leq x < 10$$\n",
+    "In other words, if $10^\\frac{n-1}{n} \\leq x < 10$ for some $x$, then $x^n$ will have $n$ digits. Therefore, the number of $n$ digit numbers that are also $n$th powers is simply the number of integers between $10^\\frac{n-1}{n}$ and 10. This can be computed as\n",
+    "$$\\lfloor 10 - 10^\\frac{n-1}{n} \\rfloor$$\n",
+    "At $n=22$, the lower bound is greater than 9, so we only need to consider values between $n=1$ and $n=21$.\n",
+    "\n",
+    "Evaluating this expression for $n=1,2,3,\\ldots,21$ and summing, we get\n",
+    "$$9 + 6 + 5 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 1 \\times 11 = 49$$\n",
+    "\n",
+    "## Relevant sequences\n",
+    "* Numbers with $k$ digits that are also $k$th powers: [A132722](https://oeis.org/A132722)"
+   ]
+  }
+ ],
+ "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
+}