Figured out problem with Fermat test in problem 58.
parent
fe73c53c46
commit
45c7e862d3
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@ -33,7 +33,9 @@
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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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"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 get_last_corner_value(side_length):\n",
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@ -81,7 +83,7 @@
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" return (base * expmod(base, exp - 1, m) % m)\n",
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" \n",
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"\n",
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"def fermat_test(n):\n",
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"def fermat_test_sicp(n):\n",
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" if n < 40:\n",
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" return n in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]\n",
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" a = n - 3\n",
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@ -145,10 +147,10 @@
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" return False\n",
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" return trial_division(n)[0] == n\n",
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"\n",
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"assert(fermat_test(3))\n",
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"assert(fermat_test(107))\n",
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"assert(fermat_test(108) == False)\n",
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"assert(fermat_test(109))"
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"assert(fermat_test_sicp(3))\n",
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"assert(fermat_test_sicp(107))\n",
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"assert(fermat_test_sicp(108) == False)\n",
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"assert(fermat_test_sicp(109))"
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]
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},
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{
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@ -210,44 +212,33 @@
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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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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"import random\n",
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"\n",
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"def is_prime(n):\n",
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" if n == 1:\n",
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" return False\n",
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" for _ in range(100):\n",
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" a = n - random.randint(1, n - 1)\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" return True\n",
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"\n",
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"assert(fermat_test(3))\n",
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"assert(fermat_test(107))\n",
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"assert(fermat_test(108) == False)\n",
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"assert(fermat_test(109))"
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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": 6,
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"execution_count": 12,
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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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"n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913\n"
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"n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913\n",
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"26641\n",
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"Expected n is 26241.\n"
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]
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}
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],
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"source": [
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"s = get_solution()"
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"import random\n",
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"\n",
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"def is_prime(n):\n",
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" if n == 1:\n",
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" return False\n",
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" for _ in range(10):\n",
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" a = n - random.randint(1, n - 1)\n",
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" if not expmod(a, n, n) == a:\n",
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" return False\n",
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" return True\n",
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"\n",
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"print(get_solution())\n",
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"print(\"Expected n is 26241.\")"
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]
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},
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{
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@ -261,6 +252,65 @@
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"Try this algorithm instead [https://en.wikipedia.org/wiki/Miller–Rabin_primality_test](https://en.wikipedia.org/wiki/Miller–Rabin_primality_test)."
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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": 10,
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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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"n: 26241 count_total: 52481 count_primes: 5248 ratio: 0.09999809454850327\n",
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"26241\n",
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"For 561 sicp says True and real test says False.\n",
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"For 1105 sicp says True and real test says False.\n",
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"For 1729 sicp says True and real test says False.\n",
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"For 2465 sicp says True and real test says False.\n",
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"For 2821 sicp says True and real test says False.\n",
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"For 6601 sicp says True and real test says False.\n"
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]
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}
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],
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"source": [
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"carmichael_number = [561, 1105, 1729, 2465, 2821, 6601]\n",
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"\n",
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"def little_fermat(a, p): # SICP uses this\n",
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" return (a ** p) % p == a % p\n",
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"\n",
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"def little_fermat_simple(a, p): # Wikipedia uses this\n",
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" assert(a < p)\n",
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" return (a ** (p - 1)) % p == 1\n",
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"\n",
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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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" ps = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]\n",
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" if n <= ps[-1]:\n",
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" return n in ps\n",
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" for a in ps:\n",
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" #if not little_fermat_simple(a, n):\n",
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" if expmod(a, n - 1, n) != 1:\n",
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" #print(\"Fermat witness for {} is {}.\".format(n + 1, a))\n",
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" return False\n",
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" return True\n",
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"\n",
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"def is_prime(n):\n",
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" return fermat_test(n)\n",
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"\n",
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"print(get_solution())\n",
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"\n",
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"for n in carmichael_number:\n",
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" r_sicp = fermat_test_sicp(n)\n",
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" r = fermat_test(n)\n",
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" print(\"For {} sicp says {} and real test says {}.\".format(n, r_sicp, r))\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": null,
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@ -11846,7 +11846,7 @@ div#notebook {
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<span class="k">return</span> <span class="p">(</span><span class="n">base</span> <span class="o">*</span> <span class="n">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="k">def</span> <span class="nf">fermat_test_sicp</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="k">if</span> <span class="n">n</span> <span class="o"><</span> <span class="mi">40</span><span class="p">:</span>
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<span class="k">return</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">39</span><span class="p">]</span>
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<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span>
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@ -11910,10 +11910,10 @@ div#notebook {
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<span class="k">return</span> <span class="kc">False</span>
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<span class="k">return</span> <span class="n">trial_division</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">n</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">107</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">108</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">109</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test_sicp</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test_sicp</span><span class="p">(</span><span class="mi">107</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test_sicp</span><span class="p">(</span><span class="mi">108</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test_sicp</span><span class="p">(</span><span class="mi">109</span><span class="p">))</span>
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</pre></div>
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</div>
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</div>
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@ -11989,7 +11989,7 @@ div#notebook {
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [5]:</div>
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<div class="prompt input_prompt">In [12]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">random</span>
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@ -11997,27 +11997,14 @@ div#notebook {
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<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
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<span class="k">return</span> <span class="kc">False</span>
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<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
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<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
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<span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
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<span class="k">if</span> <span class="ow">not</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
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<span class="k">return</span> <span class="kc">False</span>
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<span class="k">return</span> <span class="kc">True</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">107</span><span class="p">))</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">108</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">109</span><span class="p">))</span>
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</pre></div>
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</div>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [6]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="n">get_solution</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">get_solution</span><span class="p">())</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Expected n is 26241."</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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@ -12028,6 +12015,8 @@ div#notebook {
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<div class="prompt"></div>
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<div class="output_subarea output_stream output_stdout output_text">
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<pre>n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913
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26641
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Expected n is 26241.
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</pre>
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</div>
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</div>
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@ -12046,6 +12035,70 @@ div#notebook {
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [10]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">carmichael_number</span> <span class="o">=</span> <span class="p">[</span><span class="mi">561</span><span class="p">,</span> <span class="mi">1105</span><span class="p">,</span> <span class="mi">1729</span><span class="p">,</span> <span class="mi">2465</span><span class="p">,</span> <span class="mi">2821</span><span class="p">,</span> <span class="mi">6601</span><span class="p">]</span>
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<span class="k">def</span> <span class="nf">little_fermat</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span> <span class="c1"># SICP uses this</span>
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<span class="k">return</span> <span class="p">(</span><span class="n">a</span> <span class="o">**</span> <span class="n">p</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="o">==</span> <span class="n">a</span> <span class="o">%</span> <span class="n">p</span>
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<span class="k">def</span> <span class="nf">little_fermat_simple</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span> <span class="c1"># Wikipedia uses this</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">a</span> <span class="o"><</span> <span class="n">p</span><span class="p">)</span>
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<span class="k">return</span> <span class="p">(</span><span class="n">a</span> <span class="o">**</span> <span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">%</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">1</span>
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<span class="k">def</span> <span class="nf">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
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<span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
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<span class="k">return</span> <span class="mi">1</span>
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<span class="k">if</span> <span class="p">(</span><span class="n">exp</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
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<span class="k">return</span> <span class="p">(</span><span class="n">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
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<span class="k">return</span> <span class="p">(</span><span class="n">base</span> <span class="o">*</span> <span class="n">expmod</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="n">ps</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">39</span><span class="p">]</span>
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<span class="k">if</span> <span class="n">n</span> <span class="o"><=</span> <span class="n">ps</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
|
||||
<span class="k">return</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">ps</span>
|
||||
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ps</span><span class="p">:</span>
|
||||
<span class="c1">#if not little_fermat_simple(a, n):</span>
|
||||
<span class="k">if</span> <span class="n">expmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="c1">#print("Fermat witness for {} is {}.".format(n + 1, a))</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="n">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_solution</span><span class="p">())</span>
|
||||
|
||||
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">carmichael_number</span><span class="p">:</span>
|
||||
<span class="n">r_sicp</span> <span class="o">=</span> <span class="n">fermat_test_sicp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="n">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"For </span><span class="si">{}</span><span class="s2"> sicp says </span><span class="si">{}</span><span class="s2"> and real test says </span><span class="si">{}</span><span class="s2">."</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r_sicp</span><span class="p">,</span> <span class="n">r</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
<div class="output_area">
|
||||
<div class="prompt"></div>
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>n: 26241 count_total: 52481 count_primes: 5248 ratio: 0.09999809454850327
|
||||
26241
|
||||
For 561 sicp says True and real test says False.
|
||||
For 1105 sicp says True and real test says False.
|
||||
For 1729 sicp says True and real test says False.
|
||||
For 2465 sicp says True and real test says False.
|
||||
For 2821 sicp says True and real test says False.
|
||||
For 6601 sicp says True and real test says False.
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [ ]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
|
|
|
|||
|
|
@ -11770,7 +11770,7 @@ div#notebook {
|
|||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Problem-Name-(Euler-Problem-xxx)">Problem Name (Euler Problem xxx)<a class="anchor-link" href="#Problem-Name-(Euler-Problem-xxx)">¶</a></h1><p><a href="/euler">Back to overview.</a></p>
|
||||
<h1 id="Magic-5-gon-ring-(Euler-Problem-68)">Magic 5-gon ring (Euler Problem 68)<a class="anchor-link" href="#Magic-5-gon-ring-(Euler-Problem-68)">¶</a></h1><p><a href="/euler">Back to overview.</a></p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
|
|||
Loading…
Reference in New Issue