Question 57
Let $a$ be a positive integer which is not a square number. Denote by $A$ the set of all positive integers $k$ such that
$$
k = \frac{x^{2} - a}{x^{2} - y^{2}} \tag{1}
$$
for some integers $x$ and $y$ with $x \gt \sqrt{a}$. Denote by $B$ the set of all positive integers $k$ such that (1) is satisfied for some integers $x$ and $y$ with $0 \leqslant x \lt \sqrt{a}$. Prove that $A = B$.
$$
k = \frac{x^{2} - a}{x^{2} - y^{2}} \tag{1}
$$
for some integers $x$ and $y$ with $x \gt \sqrt{a}$. Denote by $B$ the set of all positive integers $k$ such that (1) is satisfied for some integers $x$ and $y$ with $0 \leqslant x \lt \sqrt{a}$. Prove that $A = B$.