Exercise 1.5.2

Proof. As in (1.18) and (1.19), the equation

is equivalent to each equation

The first gives $y^2\le \lfloor \frac{199\times 12}{59}\rfloor =40$, thus $|y|\le 6$, and the second gives $x^2\le \lfloor \frac{199\times 28}{59}\rfloor =94$, thus $|x|\le 9$.

We obtain all the solutions with the python code

l = []
for x in range(-6, 7):
    for y in range(-9, 10):
        if 3*x**2 + 5 * x * y + 7 * y**2 == 199:
            l.append((x,y))
print(l)

All representations of $199$ by the form $3x^2+5\mathit{xy}+7y^2$ are $(3,4)$ and $(-3,-4)$. □