Solved problem 41.

ipython/EulerProblem041.ipynb

@@ -13,6 +13,8 @@
     "collapsed": true
    },
    "source": [
+    "[https://projecteuler.net/problem=41]\n",
+    "\n",
     "We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.\n",
     "\n",
     "What is the largest n-digit pandigital prime that exists?"
@@ -20,16 +22,78 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 49,
    "metadata": {
     "collapsed": true
    },
    "outputs": [],
-   "source": []
+   "source": [
+    "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 expmod_naiv(base, exp, m):\n",
+    "    return base ** exp % m\n",
+    "\n",
+    "def fermat_test(n):\n",
+    "    a = n - 3\n",
+    "    return expmod(a, n, n) == a"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 50,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "assert(expmod(2, 100000, 2143) == expmod_naiv(2, 100000, 2143))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "assert(fermat_test(2143))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 54,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "7652413\n"
+     ]
+    }
+   ],
+   "source": [
+    "import itertools\n",
+    "\n",
+    "for pandigital in itertools.permutations(\"7654321\"):\n",
+    "    p = int(\"\".join(pandigital))\n",
+    "    if fermat_test(p):\n",
+    "        s = p\n",
+    "        break\n",
+    "        \n",
+    "print(s)\n",
+    "assert(s == 7652413)"
+   ]
   }
  ],
  "metadata": {
-  "completion_date": "",
+  "completion_date": "Wed, 19 Dec 2018, 20:14",
   "kernelspec": {
    "display_name": "Python 3",
    "language": "python3.6",
@@ -49,7 +113,9 @@
   },
   "tags": [
    "pandigital",
-   "prime"
+   "prime",
+   "fermat",
+   "test"
   ]
  },
  "nbformat": 4,

ipython/html/EulerProblem041.html

@@ -11778,6 +11778,7 @@ div#notebook {
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
+<p>[<a href="https://projecteuler.net/problem=41">https://projecteuler.net/problem=41</a>]</p>
 <p>We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.</p>
 <p>What is the largest n-digit pandigital prime that exists?</p>
 </div>
@@ -11785,15 +11786,80 @@ div#notebook {
 </div>
 <div class="cell border-box-sizing code_cell rendered">
 <div class="input">
-<div class="prompt input_prompt">In [ ]:</div>
+<div class="prompt input_prompt">In [49]:</div>
 <div class="inner_cell">
 <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span> 
+<div class=" highlight hl-ipython3"><pre><span></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">expmod_naiv</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">return</span> <span class="n">base</span> <span class="o">**</span> <span class="n">exp</span> <span class="o">%</span> <span class="n">m</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">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span>
+    <span class="k">return</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>
 </pre></div>
 </div>
 </div>
 </div>
 </div>
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+<div class="prompt input_prompt">In [50]:</div>
+<div class="inner_cell">
+<div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span><span class="p">(</span><span class="n">expmod</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">100000</span><span class="p">,</span> <span class="mi">2143</span><span class="p">)</span> <span class="o">==</span> <span class="n">expmod_naiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">100000</span><span class="p">,</span> <span class="mi">2143</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 [51]:</div>
+<div class="inner_cell">
+<div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span><span class="p">(</span><span class="n">fermat_test</span><span class="p">(</span><span class="mi">2143</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 [54]:</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">itertools</span>
+
+<span class="k">for</span> <span class="n">pandigital</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span><span class="p">(</span><span class="s2">"7654321"</span><span class="p">):</span>
+    <span class="n">p</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">pandigital</span><span class="p">))</span>
+    <span class="k">if</span> <span class="n">fermat_test</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
+        <span class="n">s</span> <span class="o">=</span> <span class="n">p</span>
+        <span class="k">break</span>
+        
+<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
+<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">7652413</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>7652413
+</pre>
+</div>
+</div>
+</div>
+</div>
+</div>
 </div>
 </div>
 </body>

ipython/html/index.html

@@ -645,16 +645,20 @@
                 </tr>
                 
                 
-                <tr class="table-warning">
+                <tr>
                 
                   <td><a href="EulerProblem041.html">Problem 041</a></td>
-                  <td></td>
+                  <td>Wed, 19 Dec 2018, 20:14</td>
                   <td>
                   
                     <kbd>pandigital</kbd>
                   
                     <kbd>prime</kbd>
                   
+                    <kbd>fermat</kbd>
+                  
+                    <kbd>test</kbd>
+                  
                   </td>
                 </tr>