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Finished 27 in ipython. Very nice one to be honest. Fuck yeah.

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Felix Martin
2018-02-08 23:10:09 +01:00
parent 03dda72069
commit 3f6995eabe
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ipython/EulerProblem027.html
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@@ -192,21 +192,91 @@ div#notebook {
<p>where |n| is the modulus/absolute value of n e.g. |11|=11 and |−4|=4.</p>
<p>Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n=0.</p>
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<p>Okay, bruteforcing this complete thing is definitely hard. The interesting thing is that euler provided two examples. If we calculate the primes for both terms we see that there is a certain overlap. This indicates that there is a relation between the two. Sure enough if we put $n-40$ into the first term we get the following.</p>
<p>$(n - 40)^2 + (n - 40) + 41 = n^2 - 80n + 1600 + n - 40 + 41 = n^2 - 79n +1601$</p>
<p>Let's assume that all incredible formulas can be derived by inserting $(n - p)$ into the formula. Of course, what ever value we choose for p the resulting terms must not exceed the boundaries for a or b. We calculate the boundaries.</p>
<p>$(n-p)^2 + (n-p) + 41 = n^2 + n(-2p + 1) + p^2 - p + 41$</p>
<p>Where $(-2p + 1) = a$ and $p^2 -p + 41 = b$. We can now calulate the bounds for a:</p>
<p>$-2p + 1 > -1000 \rightarrow p < 500.5$</p>
<p>$-2 p + 1 < 1000 \rightarrow p > -499.5$</p>
<p>And b:</p>
<p>$p^2 - p + 41 <= 1000$</p>
<p>$-30.472 < p < 31.472$</p>
<p>$p^2 - p + 41 >= 1000$ True for $\forall p \in \mathbb{N}$</p>
<p>So now we only have to check for the values p in range(-30, 32). Alternatively, for the example $p = 40$ was used, maybe the next smaller value $p = 31$ yields the correct solution:</p>
<p>$s = a\times b = (-2 * 31 + 1) * (31^2 - 31 + 41) = -61 \times 971 = -59231$</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">lru_cache</span>
<span class="nd">@lru_cache</span><span class="p">(</span><span class="n">maxsize</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</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">&lt;</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">sqrt</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">sqrt</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="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">if</span> <span class="n">s</span> <span class="o">*</span> <span class="n">s</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">assert</span><span class="p">(</span><span class="n">is_prime</span><span class="p">(</span><span class="mi">41</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">get_quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</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="n">p</span> <span class="o">-</span> <span class="n">p</span> <span class="o">+</span> <span class="mi">41</span>
<span class="n">n_max</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">p_max</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">30</span><span class="p">,</span> <span class="mi">32</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</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">10000</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">get_quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">n_max</span><span class="p">:</span>
<span class="n">n_max</span> <span class="o">=</span> <span class="n">n</span>
<span class="n">p_max</span> <span class="o">=</span> <span class="n">p</span>
<span class="k">break</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">p_max</span>
<span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">2</span> <span class="o">*</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="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">p</span> <span class="o">-</span> <span class="n">p</span> <span class="o">+</span> <span class="mi">41</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="o">-</span><span class="mi">59231</span><span class="p">)</span>
<span class="n">s</span>
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<pre>-59231</pre>
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