157 lines
3.2 KiB
Plaintext
157 lines
3.2 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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"# Goldbach's other conjecture (Euler Problem 46)"
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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=46](https://projecteuler.net/problem=46)\n",
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"\n",
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"It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.\n",
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"\n",
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"$9 = 7 + 2×1^2$\n",
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"\n",
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"$15 = 7 + 2×2^2$\n",
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"\n",
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"$21 = 3 + 2×3^2$\n",
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"\n",
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"$25 = 7 + 2×3^2$\n",
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"\n",
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"$27 = 19 + 2×2^2$\n",
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"\n",
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"$33 = 31 + 2×1^2$\n",
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"\n",
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"It turns out that the conjecture was false.\n",
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"\n",
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"What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?\n"
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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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"source": [
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"Okay, we reuse Fermat's test and brute force. Easy."
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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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"collapsed": true
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},
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"outputs": [],
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"source": [
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"def expmod(base, exp, m):\n",
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" if exp == 0:\n",
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" return 1\n",
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" if (exp % 2 == 0):\n",
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" return (expmod(base, exp // 2, m) ** 2 % m)\n",
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" return (base * expmod(base, exp - 1, m) % m)\n",
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"\n",
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"def fermat_test(n):\n",
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" a = n - 3\n",
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" return expmod(a, n, n) == a"
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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 twice_square(n):\n",
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" return 2 * n * 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": 18,
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"metadata": {},
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"outputs": [],
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"source": [
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"n_max = 10000\n",
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"twice_squares = [twice_square(n) for n in range(1, n_max + 1)]\n",
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"\n",
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"def test_conjecture(n):\n",
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" for ts in twice_squares:\n",
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" if ts > n:\n",
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" return False\n",
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" if fermat_test(n - ts):\n",
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" return True\n",
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" \n",
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"assert(test_conjecture(33))"
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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": 19,
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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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"5777\n"
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]
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}
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],
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"source": [
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"for n in range(3, n_max + 1, 2):\n",
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" if not fermat_test(n) and test_conjecture(n) == False:\n",
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" s = n\n",
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"\n",
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"print(s)\n",
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"assert(s == 5777)"
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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": "Sat, 22 Dec 2018, 23:39",
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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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"goldbach",
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"composite",
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"square",
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"fermat"
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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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