85 lines
3.0 KiB
Markdown
85 lines
3.0 KiB
Markdown
---
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id: 5900f4051000cf542c50ff18
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title: 'Problem 153: Investigating Gaussian Integers'
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challengeType: 5
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forumTopicId: 301784
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dashedName: problem-153-investigating-gaussian-integers
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---
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# --description--
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As we all know the equation $x^2 = -1$ has no solutions for real $x$.
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If we however introduce the imaginary number $i$ this equation has two solutions: $x = i$ and $x = -i$.
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If we go a step further the equation ${(x - 3)}^2 = -4$ has two complex solutions: $x = 3 + 2i$ and $x = 3 - 2i$, which are called each others' complex conjugate.
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Numbers of the form $a + bi$ are called complex numbers.
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In general $a + bi$ and $a − bi$ are each other's complex conjugate. A Gaussian Integer is a complex number $a + bi$ such that both $a$ and $b$ are integers.
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The regular integers are also Gaussian integers (with $b = 0$).
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To distinguish them from Gaussian integers with $b ≠ 0$ we call such integers "rational integers."
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A Gaussian integer is called a divisor of a rational integer $n$ if the result is also a Gaussian integer.
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If for example we divide 5 by $1 + 2i$ we can simplify in the following manner:
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Multiply numerator and denominator by the complex conjugate of $1 + 2i$: $1 − 2i$.
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The result is:
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$$\frac{5}{1 + 2i} = \frac{5}{1 + 2i} \frac{1 - 2i}{1 - 2i} = \frac{5(1 - 2i)}{1 - {(2i)}^2} = \frac{5(1 - 2i)}{1 - (-4)} = \frac{5(1 - 2i)}{5} = 1 - 2i$$
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So $1 + 2i$ is a divisor of 5.
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Note that $1 + i$ is not a divisor of 5 because:
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$$\frac{5}{1 + i} = \frac{5}{2} - \frac{5}{2}i$$
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Note also that if the Gaussian Integer ($a + bi$) is a divisor of a rational integer $n$, then its complex conjugate ($a − bi$) is also a divisor of $n$. In fact, 5 has six divisors such that the real part is positive: {1, 1 + 2i, 1 − 2i, 2 + i, 2 − i, 5}.
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The following is a table of all of the divisors for the first five positive rational integers:
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| n | Gaussian integer divisors with positive real part | Sum s(n) of these divisors |
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|---|---------------------------------------------------|----------------------------|
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| 1 | 1 | 1 |
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| 2 | 1, 1 + i, 1 - i, 2 | 5 |
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| 3 | 1, 3 | 4 |
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| 4 | 1, 1 + i, 1 - i, 2, 2 + 2i, 2 - 2i, 4 | 13 |
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| 5 | 1, 1 + 2i, 1 - 2i, 2 + i, 2 - i, 5 | 12 |
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For divisors with positive real parts, then, we have: $\displaystyle\sum_{n=1}^5 s(n) = 35$.
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For $1 ≤ n ≤ {10}^5$, $\displaystyle\sum_{n = 1}^{{10}^5} s(n) = 17924657155$.
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What is $\displaystyle\sum_{n=1}^{{10}^8} s(n)$?
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# --hints--
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`sumGaussianIntegers()` should return `17971254122360636`.
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```js
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assert.strictEqual(sumGaussianIntegers(), 17971254122360636);
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```
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# --seed--
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## --seed-contents--
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```js
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function sumGaussianIntegers() {
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return true;
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}
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sumGaussianIntegers();
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```
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# --solutions--
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```js
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// solution required
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```
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