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Euler has design of a homepage and there is a link back to the overview for each solution.

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Felix Martin
2018-06-14 20:11:26 -04:00
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<title>EulerProblem027</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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<h1 id="Euler-Problem-27">Euler Problem 27<a class="anchor-link" href="#Euler-Problem-27">&#182;</a></h1><p>Euler discovered the remarkable quadratic formula:</p>
<h1 id="Euler-Problem-27">Euler Problem 27<a class="anchor-link" href="#Euler-Problem-27"></a></h1><p><a href="/euler">Back to overview.</a></p><p>Euler discovered the remarkable quadratic formula:</p>
<p>$n^2 + n + 41$</p>
<p>It turns out that the formula will produce 40 primes for the consecutive integer values 0≤n≤39. However, when $n=40$, $40^2 + 40 + 41 = 40(40+1)+41$ is divisible by $41$, and certainly when $n=41,41^2+41+41$ is clearly divisible by 41.</p>
<p>The incredible formula $n^2−79n+1601$ was discovered, which produces 80 primes for the consecutive values 0≤n≤79. The product of the coefficients, $−79$ and $1601$, is $−126479$.</p>
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<p>$n^2 + an +b$, where |a|&lt;1000 and |b|≤1000</p>
<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>$p^2 - p + 41 &gt;= 1000$ 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>
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<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>
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<pre>-59231</pre>
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