Figured out problem with Fermat test in problem 58.

ipython/EulerProblem058.ipynb

@@ -33,7 +33,9 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "metadata": {},
+   "metadata": {
+    "collapsed": true
+   },
    "outputs": [],
    "source": [
     "def get_last_corner_value(side_length):\n",
@@ -81,7 +83,7 @@
     "    return (base * expmod(base, exp - 1, m) % m)\n",
     "    \n",
     "\n",
-    "def fermat_test(n):\n",
+    "def fermat_test_sicp(n):\n",
     "    if n < 40:\n",
     "        return n in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]\n",
     "    a = n - 3\n",
@@ -145,10 +147,10 @@
     "        return False\n",
     "    return trial_division(n)[0] == n\n",
     "\n",
-    "assert(fermat_test(3))\n",
+    "assert(fermat_test_sicp(3))\n",
-    "assert(fermat_test(107))\n",
+    "assert(fermat_test_sicp(107))\n",
-    "assert(fermat_test(108) == False)\n",
+    "assert(fermat_test_sicp(108) == False)\n",
-    "assert(fermat_test(109))"
+    "assert(fermat_test_sicp(109))"
    ]
   },
   {
@@ -210,44 +212,33 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 12,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": [
-    "import random\n",
-    "\n",
-    "def is_prime(n):\n",
-    "    if n == 1:\n",
-    "        return False\n",
-    "    for _ in range(100):\n",
-    "        a = n - random.randint(1, n - 1)\n",
-    "        if not expmod(a, n, n) == a:\n",
-    "            return False\n",
-    "    return True\n",
-    "\n",
-    "assert(fermat_test(3))\n",
-    "assert(fermat_test(107))\n",
-    "assert(fermat_test(108) == False)\n",
-    "assert(fermat_test(109))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913\n"
+      "n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913\n",
+      "26641\n",
+      "Expected n is 26241.\n"
      ]
     }
    ],
    "source": [
-    "s = get_solution()"
+    "import random\n",
+    "\n",
+    "def is_prime(n):\n",
+    "    if n == 1:\n",
+    "        return False\n",
+    "    for _ in range(10):\n",
+    "        a = n - random.randint(1, n - 1)\n",
+    "        if not expmod(a, n, n) == a:\n",
+    "            return False\n",
+    "    return True\n",
+    "\n",
+    "print(get_solution())\n",
+    "print(\"Expected n is 26241.\")"
    ]
   },
   {
@@ -261,6 +252,65 @@
     "Try this algorithm instead [https://en.wikipedia.org/wiki/Miller–Rabin_primality_test](https://en.wikipedia.org/wiki/Miller–Rabin_primality_test)."
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "n: 26241 count_total: 52481 count_primes: 5248 ratio: 0.09999809454850327\n",
+      "26241\n",
+      "For 561 sicp says True and real test says False.\n",
+      "For 1105 sicp says True and real test says False.\n",
+      "For 1729 sicp says True and real test says False.\n",
+      "For 2465 sicp says True and real test says False.\n",
+      "For 2821 sicp says True and real test says False.\n",
+      "For 6601 sicp says True and real test says False.\n"
+     ]
+    }
+   ],
+   "source": [
+    "carmichael_number = [561, 1105, 1729, 2465, 2821, 6601]\n",
+    "\n",
+    "def little_fermat(a, p): # SICP uses this\n",
+    "    return (a ** p) % p == a % p\n",
+    "\n",
+    "def little_fermat_simple(a, p): # Wikipedia uses this\n",
+    "    assert(a < p)\n",
+    "    return (a ** (p - 1)) % p == 1\n",
+    "\n",
+    "def expmod(base, exp, m):\n",
+    "    if exp == 0:\n",
+    "        return 1\n",
+    "    if (exp % 2 == 0):\n",
+    "        return (expmod(base, exp // 2, m) ** 2 % m)\n",
+    "    return (base * expmod(base, exp - 1, m) % m)\n",
+    "\n",
+    "def fermat_test(n):\n",
+    "    ps = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39]\n",
+    "    if n <= ps[-1]:\n",
+    "        return n in ps\n",
+    "    for a in ps:\n",
+    "        #if not little_fermat_simple(a, n):\n",
+    "        if expmod(a, n - 1, n) != 1:\n",
+    "            #print(\"Fermat witness for {} is {}.\".format(n + 1, a))\n",
+    "            return False\n",
+    "    return True\n",
+    "\n",
+    "def is_prime(n):\n",
+    "    return fermat_test(n)\n",
+    "\n",
+    "print(get_solution())\n",
+    "\n",
+    "for n in carmichael_number:\n",
+    "    r_sicp = fermat_test_sicp(n)\n",
+    "    r = fermat_test(n)\n",
+    "    print(\"For {} sicp says {} and real test says {}.\".format(n, r_sicp, r))\n"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,

ipython/html/EulerProblem058.html

@@ -11846,7 +11846,7 @@ div#notebook {
     <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>
     
 
-<span class="k">def</span> <span class="nf">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
+<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>
     <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">40</span><span class="p">:</span>
         <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>
     <span class="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span>
@@ -11910,10 +11910,10 @@ div#notebook {
         <span class="k">return</span> <span class="kc">False</span>
     <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>
 
-<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>
+<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>
-<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>
+<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>
-<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>
+<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>
-<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>
+<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>
 </pre></div>
 </div>
 </div>
@@ -11989,7 +11989,7 @@ div#notebook {
 </div>
 <div class="cell border-box-sizing code_cell rendered">
 <div class="input">
-<div class="prompt input_prompt">In [5]:</div>
+<div class="prompt input_prompt">In [12]:</div>
 <div class="inner_cell">
 <div class="input_area">
 <div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">random</span>
@@ -11997,27 +11997,14 @@ div#notebook {
 <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">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
         <span class="k">return</span> <span class="kc">False</span>
-    <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>
+    <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>
         <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>
         <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>
             <span class="k">return</span> <span class="kc">False</span>
     <span class="k">return</span> <span class="kc">True</span>
 
-<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>
+<span class="nb">print</span><span class="p">(</span><span class="n">get_solution</span><span class="p">())</span>
-<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>
+<span class="nb">print</span><span class="p">(</span><span class="s2">"Expected n is 26241."</span><span class="p">)</span>
-<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>
-<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>
-</pre></div>
-</div>
-</div>
-</div>
-</div>
-<div class="cell border-box-sizing code_cell rendered">
-<div class="input">
-<div class="prompt input_prompt">In [6]:</div>
-<div class="inner_cell">
-<div class="input_area">
-<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>
 </pre></div>
 </div>
 </div>
@@ -12028,6 +12015,8 @@ div#notebook {
 <div class="prompt"></div>
 <div class="output_subarea output_stream output_stdout output_text">
 <pre>n: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913
+26641
+Expected n is 26241.
 </pre>
 </div>
 </div>
@@ -12046,6 +12035,70 @@ div#notebook {
 </div>
 <div class="cell border-box-sizing code_cell rendered">
 <div class="input">
+<div class="prompt input_prompt">In [10]:</div>
+<div class="inner_cell">
+<div class="input_area">
+<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>
+
+<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>
+    <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>
+
+<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>
+    <span class="k">assert</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="n">p</span><span class="p">)</span>
+    <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>
+
+<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>
+    <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="k">return</span> <span class="mi">1</span>
+    <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>
+        <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>
+    <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>
+
+<span class="k">def</span> <span class="nf">fermat_test</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
+    <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>
+    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</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">

ipython/html/EulerProblem068.html

@@ -11770,7 +11770,7 @@ div#notebook {
 </div>
 <div class="inner_cell">
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-<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>
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