2018-02-08 18:09:51 +01:00
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< h1 id = "Euler-Problem-27" > Euler Problem 27< a class = "anchor-link" href = "#Euler-Problem-27" > ¶ < / a > < / h1 > < 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− − − < / p >
< p > Considering quadratics of the form:< / p >
< p > $n^2 + an +b$, where |a|< 1000 and |b|≤1000< / p >
< p > where |n| is the modulus/absolute value of n e.g. |11|=11 and |− < / 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 >
2018-02-08 23:10:09 +01:00
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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 >
2018-02-08 18:09:51 +01:00
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2018-02-08 23:10:09 +01:00
< div class = "prompt input_prompt" > In [1]:< / div >
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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 >
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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 = "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 > < / div >
< / div >
< / div >
< / div >
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< div class = "output_wrapper" >
< div class = "output" >
< div class = "output_area" > < div class = "prompt output_prompt" > Out[1]:< / div >
< div class = "output_text output_subarea output_execute_result" >
< pre > -59231< / pre >
< / div >
< / div >
< / div >
< / div >
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< / div >
< / div >
< / div >
< / body >
< / html >
