Problem 153: Investigating Gaussian Integers

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As we all know the equation x2=-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. x=3+2i and x=3-2i 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 .

So 1+2i is a divisor of 5.

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

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 partSum s(n) of these

divisors111 21, 1+i, 1-i, 25 31, 34 41, 1+i, 1-i, 2, 2+2i, 2-2i,413 51, 1+2i, 1-2i, 2+i, 2-i, 512 For divisors with positive real parts, then, we have: . For 1 ≤ n ≤ 105, ∑ s(n)=17924657155. What is ∑ s(n) for 1 ≤ n ≤ 108?