Chapter 6.2, Problem 69E

Let G be the set of all integers x that can be written asthe sum of the squares of two integers, $x=a^2+b^2$ . Forexample, $13=3^2+2^2$ is in G, while 7 fails to be in G.
a. Listall integers $x\le 10$ thatare in G.
b. Show that G is closed under multiplication: If $x=a^2+b^2$ and $y=c^2+d^2$ are in G. then so is their
productxy. Hint: Consider the matrices $\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right],\left[\begin{array}{cc}c& -d\\ d& c\end{array}\right]$ , their product, and their determinants.
c. Given that $2642={31}^2+{41}^2$ and $3218={37}^2+{43}^2$ ,write $8,501,956=2642\cdot 3218$ as the sum ofthe squares of two positive integers. You may usetechnology.