Determine the integers $x$ and $y$ such that $x^2 + y^2 = 178$.
Prime factorisation gives us $178 = 2 \times 89$. By the Sum of Two Squares Theorem (8.5), factors 2 and $89 \equiv 1 \pmod 4$ tells us 178 can be written as a sum of two squares.
Experimenting with small numbers immediately gives us
$$ 178 = 3^2 + 13^2 $$