186 lines
4.7 KiB
Plaintext
186 lines
4.7 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Prime pair sets (Euler Problem 60)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"[https://projecteuler.net/problem=60](https://projecteuler.net/problem=60)\n",
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"\n",
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"The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.\n",
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"\n",
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"Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime.\n"
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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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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[3, 7], [7, 3], [3, 109], [109, 3], [3, 673], [673, 3], [7, 109], [109, 7], [7, 673], [673, 7], [109, 673], [673, 109]]\n"
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]
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}
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],
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"source": [
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"from itertools import combinations\n",
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"\n",
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"def concatenate_all(numbers):\n",
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" result = []\n",
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" for a, b in combinations(numbers, 2):\n",
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" result.append([a, b])\n",
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" result.append([b, a])\n",
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" return result\n",
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"\n",
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"print(concatenate_all([3, 7, 109, 673]))"
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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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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"def sieve_of_eratosthenes(limit):\n",
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" primes = []\n",
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" prospects = [n for n in range(2, limit)]\n",
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" while prospects:\n",
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" p = prospects[0]\n",
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" prospects = [x for x in prospects if x % p != 0]\n",
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" primes.append(p)\n",
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" if p * p > limit:\n",
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" break\n",
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" primes += prospects\n",
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" return primes\n",
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"\n",
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"primes = sieve_of_eratosthenes(100000)\n",
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"primes_set = set(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": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"def is_prime(n):\n",
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" if n % 2 == 0:\n",
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" return False\n",
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" while n % 2 == 0:\n",
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" n //= 2\n",
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" f = 3\n",
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" while f * f <= n:\n",
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" if n % f == 0:\n",
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" return False\n",
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" else:\n",
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" f += 2 \n",
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" return True\n",
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"\n",
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"def concatentation_is_prime(a, b):\n",
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" ab = int(str(a) + str(b))\n",
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" ba = int(str(b) + str(a))\n",
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" if is_prime(ab) and is_prime(ba):\n",
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" return True\n",
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" else:\n",
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" return False\n",
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" \n",
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"assert(concatentation_is_prime(673, 3))\n",
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"assert(concatentation_is_prime(673, 1) == False)"
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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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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[13, 5197, 5701, 6733, 8389]\n",
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"26033\n"
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]
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}
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],
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"source": [
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"potentials = []\n",
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"new_potentials = []\n",
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"done = False\n",
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"for p in primes:\n",
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" if done:\n",
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" break\n",
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" potentials += new_potentials\n",
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" new_potentials = []\n",
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" for potential in potentials:\n",
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" all_concatenations_prime = True\n",
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" for prime in potential:\n",
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" if not concatentation_is_prime(p, prime):\n",
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" all_concatenations_prime = False\n",
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" break\n",
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" if all_concatenations_prime:\n",
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" new_potential = list(potential)\n",
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" new_potential.append(p)\n",
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" if len(new_potential) > 4:\n",
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" print(new_potential)\n",
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" s = sum(new_potential)\n",
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" done = True\n",
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" break\n",
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" new_potentials.append(new_potential)\n",
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" if p < 15:\n",
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" potentials.append([p])\n",
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"\n",
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"print(s)\n",
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"assert(s == 26033)"
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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": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "Sun, 6 Jan 2019, 05:00",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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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": "3.6.5"
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},
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"tags": [
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"prime",
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"pairs"
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]
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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