add problem 37
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filifa
parent
4466680550
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53d201e591
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@ -0,0 +1,250 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "6fcde6bd",
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"metadata": {},
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"source": [
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"# [Truncatable Primes](https://projecteuler.net/problem=37)\n",
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"\n",
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"Apparently another term for these sorts of numbers is a [two-sided prime](https://en.wikipedia.org/wiki/Truncatable_prime).\n",
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"\n",
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"In case you're curious how we know there's only eleven of these primes (not including 2, 3, 5, or 7), this solution will outline it.\n",
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"\n",
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"We start by generating all the right-truncatable primes - the two-sided primes are necessarily a subset of these. If we know a number $n$ is *not* a right-truncatable prime, we know any number with $n$ as a prefix is *also* not a right-truncatable prime. This means we can start by looking at one-digit primes (2, 3, 5, and 7) and for each, try appending each of 1, 3, 7, and 9 and see which, if any, are also prime (appending any other digit would clearly result in a number divisible by 2 or 5). If we find another prime, we'll want to repeat the process of appending 1, 3, 7, or 9 to see if we can find more right-truncatable primes. We use a [depth-first search](https://en.wikipedia.org/wiki/Depth-first_search) for this."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "fecd8e1a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{23,\n",
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" 29,\n",
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" 31,\n",
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" 37,\n",
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" 53,\n",
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" 59,\n",
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" 71,\n",
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" 73,\n",
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" 79,\n",
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" 233,\n",
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" 239,\n",
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" 293,\n",
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" 311,\n",
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" 313,\n",
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" 317,\n",
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" 373,\n",
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" 379,\n",
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" 593,\n",
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" 599,\n",
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" 719,\n",
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" 733,\n",
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" 739,\n",
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" 797,\n",
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" 2333,\n",
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" 2339,\n",
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" 2393,\n",
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" 2399,\n",
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" 2939,\n",
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" 3119,\n",
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" 3137,\n",
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" 3733,\n",
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" 3739,\n",
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" 3793,\n",
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" 3797,\n",
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" 5939,\n",
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" 7193,\n",
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" 7331,\n",
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" 7333,\n",
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" 7393,\n",
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" 23333,\n",
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" 23339,\n",
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" 23399,\n",
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" 23993,\n",
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" 29399,\n",
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" 31193,\n",
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" 31379,\n",
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" 37337,\n",
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" 37339,\n",
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" 37397,\n",
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" 59393,\n",
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" 59399,\n",
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" 71933,\n",
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" 73331,\n",
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" 73939,\n",
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" 233993,\n",
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" 239933,\n",
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" 293999,\n",
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" 373379,\n",
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" 373393,\n",
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" 593933,\n",
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" 593993,\n",
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" 719333,\n",
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" 739391,\n",
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" 739393,\n",
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" 739397,\n",
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" 739399,\n",
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" 2339933,\n",
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" 2399333,\n",
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" 2939999,\n",
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" 3733799,\n",
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" 5939333,\n",
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" 7393913,\n",
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" 7393931,\n",
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" 7393933,\n",
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" 23399339,\n",
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" 29399999,\n",
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" 37337999,\n",
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" 59393339,\n",
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" 73939133}"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"right_truncatable_primes = set()\n",
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"\n",
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"visited = set()\n",
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"stack = [2, 3, 5, 7]\n",
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"while stack != []:\n",
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" n = stack.pop()\n",
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" if n in visited:\n",
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" continue\n",
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" visited.add(n)\n",
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" \n",
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" for d in (1,3,7,9):\n",
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" m = 10*n + d\n",
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" if is_prime(m):\n",
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" right_truncatable_primes.add(m)\n",
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" stack.append(m)\n",
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"\n",
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"right_truncatable_primes"
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]
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},
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{
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"cell_type": "markdown",
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"id": "95481066",
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"metadata": {},
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"source": [
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"Next, we write a function for checking if a prime is left-truncatable."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "6922aa89",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import count\n",
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"\n",
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"def is_left_truncatable_prime(p):\n",
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" for k in count(1):\n",
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" t = p % 10^k\n",
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" if not is_prime(t):\n",
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" return False\n",
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" if t == p:\n",
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" break\n",
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" \n",
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" return True"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f20fe991",
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"metadata": {},
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"source": [
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"Now just check all our right-truncatable primes to see if they are also left-truncatable."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "995c5372",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"{23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397}"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"two_sided_primes = {p for p in right_truncatable_primes if is_left_truncatable_prime(p)}\n",
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"two_sided_primes"
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]
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},
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{
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"cell_type": "markdown",
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"id": "97f60cbd",
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"metadata": {},
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"source": [
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"Sure enough, only eleven two-sided primes."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "3851ffee",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"748317"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"sum(two_sided_primes)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8f55a757",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Two-sided primes: [A020994](https://oeis.org/A020994)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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