Solved problem 41.
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@ -13,6 +13,8 @@
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"collapsed": true
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"source": [
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"[https://projecteuler.net/problem=41]\n",
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"\n",
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"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",
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"\n",
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"What is the largest n-digit pandigital prime that exists?"
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@ -20,16 +22,78 @@
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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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"execution_count": 49,
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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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"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 expmod_naiv(base, exp, m):\n",
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" return base ** exp % 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": 50,
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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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"assert(expmod(2, 100000, 2143) == expmod_naiv(2, 100000, 2143))"
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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": 51,
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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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"assert(fermat_test(2143))"
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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": 54,
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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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"7652413\n"
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]
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}
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],
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"source": [
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"import itertools\n",
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"\n",
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"for pandigital in itertools.permutations(\"7654321\"):\n",
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" p = int(\"\".join(pandigital))\n",
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" if fermat_test(p):\n",
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" s = p\n",
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" break\n",
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" \n",
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"print(s)\n",
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"assert(s == 7652413)"
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]
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}
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],
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"metadata": {
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"completion_date": "",
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"completion_date": "Wed, 19 Dec 2018, 20:14",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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@ -49,7 +113,9 @@
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},
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"tags": [
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"pandigital",
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"prime"
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"prime",
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"fermat",
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"test"
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]
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},
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"nbformat": 4,
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@ -11778,6 +11778,7 @@ div#notebook {
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</div>
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<div class="inner_cell">
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<div class="text_cell_render border-box-sizing rendered_html">
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<p>[<a href="https://projecteuler.net/problem=41">https://projecteuler.net/problem=41</a>]</p>
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<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>
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<p>What is the largest n-digit pandigital prime that exists?</p>
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</div>
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@ -11785,15 +11786,80 @@ 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 [ ]:</div>
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<div class="prompt input_prompt">In [49]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span>
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<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>
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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">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>
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<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>
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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">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span>
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<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>
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</pre></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="prompt input_prompt">In [50]:</div>
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<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>
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</pre></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="prompt input_prompt">In [51]:</div>
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<div class="input_area">
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<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>
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</pre></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 [54]:</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">itertools</span>
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<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>
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<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>
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<span class="k">if</span> <span class="n">fermat_test</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
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<span class="n">s</span> <span class="o">=</span> <span class="n">p</span>
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<span class="k">break</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
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<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>
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</pre></div>
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<pre>7652413
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@ -645,16 +645,20 @@
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<tr class="table-warning">
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<tr>
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<td><a href="EulerProblem041.html">Problem 041</a></td>
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<td></td>
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<td>Wed, 19 Dec 2018, 20:14</td>
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<td>
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<kbd>pandigital</kbd>
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<kbd>prime</kbd>
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<kbd>fermat</kbd>
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<kbd>test</kbd>
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</td>
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</tr>
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