Solved problem 58.
This commit is contained in:
@@ -11780,13 +11780,15 @@ div#notebook {
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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=58">https://projecteuler.net/problem=58</a></p>
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<p>Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.</p>
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<p>37 36 35 34 33 32 31
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<pre><code>37 36 35 34 33 32 31
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38 17 16 15 14 13 30
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39 18 5 4 3 12 29
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40 19 6 1 2 11 28
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41 20 7 8 9 10 27
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42 21 22 23 24 25 26
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43 44 45 46 47 48 49</p>
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43 44 45 46 47 48 49
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</code></pre>
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<p>It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.</p>
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<p>If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?</p>
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</div>
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@@ -11794,6 +11796,256 @@ 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 [1]:</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="k">def</span> <span class="nf">get_last_corner_value</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
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<span class="k">return</span> <span class="n">side_length</span> <span class="o">*</span> <span class="n">side_length</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="mi">9</span><span class="p">)</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="mi">25</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">get_corner_values</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
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<span class="k">if</span> <span class="n">side_length</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="p">[</span><span class="mi">1</span><span class="p">]</span>
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<span class="k">return</span> <span class="p">[</span><span class="n">get_last_corner_value</span><span class="p">(</span><span class="n">side_length</span><span class="p">)</span> <span class="o">-</span> <span class="n">i</span> <span class="o">*</span> <span class="p">(</span><span class="n">side_length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
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<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)][::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</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">9</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</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">21</span><span class="p">,</span> <span class="mi">25</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_corner_values</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span> <span class="o">==</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">43</span><span class="p">,</span> <span class="mi">49</span><span class="p">])</span>
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<span class="k">def</span> <span class="nf">get_diagonal_values</span><span class="p">(</span><span class="n">side_length</span><span class="p">):</span>
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<span class="k">return</span> <span class="p">[</span><span class="n">corner_value</span>
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<span class="k">for</span> <span class="n">length</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">side_length</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
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<span class="k">for</span> <span class="n">corner_value</span> <span class="ow">in</span> <span class="n">get_corner_values</span><span class="p">(</span><span class="n">length</span><span class="p">)</span>
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<span class="p">]</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</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">9</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_diagonal_values</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</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">9</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">21</span><span class="p">,</span> <span class="mi">25</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 [2]:</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="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="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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<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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">5</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">7</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">11</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">13</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">17</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">19</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">23</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">29</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">31</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">37</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="n">a</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">39</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">def</span> <span class="nf">trial_division</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="p">[]</span>
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<span class="k">if</span> <span class="n">n</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="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
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<span class="k">while</span> <span class="n">n</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="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
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<span class="n">f</span> <span class="o">=</span> <span class="mi">3</span>
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<span class="k">while</span> <span class="n">f</span> <span class="o">*</span> <span class="n">f</span> <span class="o"><=</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="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
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<span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
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<span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
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<span class="n">n</span> <span class="o">//=</span> <span class="n">f</span>
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<span class="k">else</span><span class="p">:</span>
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<span class="n">f</span> <span class="o">+=</span> <span class="mi">2</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="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
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<span class="k">return</span> <span class="n">a</span>
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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">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>
|
||||
<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">
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||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_solution</span><span class="p">():</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">count_primes</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="n">count_total</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
|
||||
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">get_corner_values</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">count_total</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
<span class="k">if</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
|
||||
<span class="n">count_primes</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
<span class="n">ratio</span> <span class="o">=</span> <span class="n">count_primes</span> <span class="o">/</span> <span class="n">count_total</span>
|
||||
<span class="k">if</span> <span class="n">ratio</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">ratio</span> <span class="o"><</span> <span class="mf">0.10</span><span class="p">:</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"n: </span><span class="si">{}</span><span class="s2"> count_total: </span><span class="si">{}</span><span class="s2"> count_primes: </span><span class="si">{}</span><span class="s2"> ratio: </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">count_total</span><span class="p">,</span> <span class="n">count_primes</span><span class="p">,</span> <span class="n">ratio</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="n">n</span>
|
||||
<span class="n">n</span> <span class="o">+=</span> <span class="mi">2</span>
|
||||
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">get_solution</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
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></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">26241</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>26241
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I actually got different results for the Fermat test and for the prime test which relies on reliable computation. I will actually try to solve the problem with ramdonized Fermat test now.</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</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>
|
||||
|
||||
<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="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="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</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>
|
||||
</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: 26641 count_total: 53281 count_primes: 5328 ratio: 0.09999812315834913
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Something seems to be off with my Fermat test...</p>
|
||||
<p>Seems like there are systematic errors with the Fermat tests. Certain primes cannot be deteced.</p>
|
||||
<p>Try this algorithm instead <a href="https://en.wikipedia.org/wiki/Miller–Rabin_primality_test">https://en.wikipedia.org/wiki/Miller–Rabin_primality_test</a>.</p>
|
||||
</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">
|
||||
|
||||
Reference in New Issue
Block a user