102 lines
2.3 KiB
Markdown
102 lines
2.3 KiB
Markdown
---
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id: 5900f4051000cf542c50ff18
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challengeType: 5
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title: 'Problem 153: Investigating Gaussian Integers'
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forumTopicId: 301784
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---
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## Description
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<section id='description'>
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As we all know the equation x2=-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.
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x=3+2i and x=3-2i 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.
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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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.
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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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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.
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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
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with positive real partSum s(n) of these
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divisors111
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21, 1+i, 1-i, 25
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31, 34
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41, 1+i, 1-i, 2, 2+2i, 2-2i,413
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51, 1+2i, 1-2i, 2+i, 2-i, 512
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For divisors with positive real parts, then, we have: .
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For 1 ≤ n ≤ 105, ∑ s(n)=17924657155.
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What is ∑ s(n) for 1 ≤ n ≤ 108?
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>euler153()</code> should return 17971254122360636.
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testString: assert.strictEqual(euler153(), 17971254122360636);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function euler153() {
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return true;
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}
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euler153();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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