Do it like Gauss, Part 2

Algebra Level 5

A Gaussian Integer is a complex number of the form $z=a+bi$ , where $a$ and $b$ are integers. How many Gaussian Integers $z$ divide 1000, in the sense that $1000=z\times{w}$ for some Gaussian Integer $w$ ?

The answer is 448.

1 solution

Brian Charlesworth
May 16, 2015

The Gaussian prime factorization of $10$ is $(1 - i)^{2}(1 + 2i)(2 + i),$ which is unique up to multiplication by the units $\pm 1, \pm i.$

Thus the prime factorization of $1000 = 10^{3}$ is $(1 - i)^{6}(1 + 2i)^{3}(2 + i)^{3},$ again unique up to multiplication by the $4$ units. Thus just as we would calculate the number of divisors of a "regular" integer by way of its prime factorization, with the additional multiplicative term of $4$ to account for the four units, we find that the number of Gaussian divisors of $1000$ is

$4 \times (6 + 1) \times (3 + 1) \times (3 + 1) = \boxed{448}.$

Another beautiful and perfect solution; thank you! As I said earlier: You do it better than Gauss... you write like Euler.

Just to get to the bottom of things, can you please tell us a bit more about the uniqueness of the prime factorization?

Otto Bretscher - 6 years ago

Thanks! I believe that Gauss addressed the matter of "uniqueness" in his second monograph, (1832). An informative discussion is given here . I'm still trying to digest all this information myself, but it should prove to be a good starting point.

Brian Charlesworth - 6 years ago

I made mistakes, but here is a solution to young people: As z.w = 100 it foloows that | z| divide 1000;

Carlos Eddy Esaguy Nehab - 6 years ago

I used a brute force method based on the fact that 1000/(a+bi)=1000(a-bi)/(a^2+b^2). It checks every integer combination of (a,b) from -1000 to 1000 but I keep coming up with 208 every time! Your answer makes sense to me, but what combinations could my program be missing?

Garrett Clarke - 6 years ago

Do it like Gauss, Part 2

The answer is 448.

1 solution

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