Number theory problem No.7

We need to factorize $441000 = 2^3 \times 3^2 \times 5^3 \times 7^2$ over the Gaussian integers $\mathbb{Z}[i]$ . Now $3,7$ are both irreducible in the Gaussian integers, while $2 = (1 + i)(1 - i)$ and $5 = (2 + i)(2 - i)$ . The units of $\mathbb{Z}[i]$ are $1,-1,i,-i$ , and we note that $1+i$ and $1-i$ are associate to each other (one is the other times a unit). Since we want to factorize $441000 = x^2 + y^2 = (x + iy)(x - iy)$ , we must have $x + iy \; = \; u(1+i)^3 \times 3 \times (2+i)^a(2-i)^{3-a} \times 7$ for some unit $u$ and some $0 \le i \le 3$ . This gives us $4$ essentially different solutions for integer coordinates $(x,y)$ such that $x^2 + y^2 = 441000$ (one for each value of $a$ ). For each such solution $(x,y)$ , there are four variations of that set of numbers, namely $(x,y)$ , $(-y,x)$ , $(-x,-y)$ and $(y,-x)$ - corresponding to the four possible choices of unit. Thus there are $4 \times 4 = \boxed{16}$ points with integer coordinates on this circle.