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id	title	challengeType	forumTopicId	dashedName
5900f4051000cf542c50ff18	Problem 153: Investigating Gaussian Integers	5	301784	problem-153-investigating-gaussian-integers

--description--

As we all know the equation x^2 = -1 has no solutions for real x.

If we however introduce the imaginary number i this equation has two solutions: x = i and x = -i.

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.

Numbers of the form a + bi are called complex numbers.

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.

The regular integers are also Gaussian integers (with b = 0).

To distinguish them from Gaussian integers with b ≠ 0 we call such integers "rational integers."

A Gaussian integer is called a divisor of a rational integer n if the result is also a Gaussian integer.

If for example we divide 5 by 1 + 2i we can simplify in the following manner:

Multiply numerator and denominator by the complex conjugate of 1 + 2i: 1 − 2i.

The result is:

\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

So 1 + 2i is a divisor of 5.

Note that 1 + i is not a divisor of 5 because:

\frac{5}{1 + i} = \frac{5}{2} - \frac{5}{2}i

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}.

The following is a table of all of the divisors for the first five positive rational integers:

n	Gaussian integer divisors with positive real part	Sum s(n) of these divisors
1	1	1
2	1, 1 + i, 1 - i, 2	5
3	1, 3	4
4	1, 1 + i, 1 - i, 2, 2 + 2i, 2 - 2i, 4	13
5	1, 1 + 2i, 1 - 2i, 2 + i, 2 - i, 5	12

For divisors with positive real parts, then, we have: \displaystyle\sum_{n=1}^5 s(n) = 35.

For 1 ≤ n ≤ {10}^5, \displaystyle\sum_{n = 1}^{{10}^5} s(n) = 17924657155.

What is \displaystyle\sum_{n=1}^{{10}^8} s(n)?

--hints--

sumGaussianIntegers() should return 17971254122360636.

assert.strictEqual(sumGaussianIntegers(), 17971254122360636);

--seed--

--seed-contents--

function sumGaussianIntegers() {

  return true;
}

sumGaussianIntegers();

--solutions--

// solution required

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