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# Problem 153

Investigating Gaussian Integers

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 $\frac{5}{1+2i}$ 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: $\sum_{n=1}^{5}s(n)=35$.

For 1 ≤ n ≤ 105, ∑ s(n)=17924657155.

What is ∑ s(n) for 1 ≤ n ≤ 108?

x=3+2i和x=3-2i互称为共轭复数。

n 实数部分是正数的高斯整数约数 约数的和s(n)
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