Question 29
Let $a$ be a positive integer which is not a square number. Denote by $A$ the set of all positive integers $k$ such that
$$
k=\frac{x^{2}-a}{x^{2}-y^{2}}
$$
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 $A=B$.
$$
k=\frac{x^{2}-a}{x^{2}-y^{2}}
$$
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 $A=B$.