Prove (or disprove) that there are infinitely many natural numbers of the form: $$n=\dfrac{25p^2-124q^2}{100q^2}$$ where $p$ and $q$ are positive integers with $q\neq 0$.
Now a more general question can be proving (or disproving) the existence of infinitely many natural numbers of the form $n=\dfrac{ap^2-bq^2}{cq^2}$, but I don't have any idea how to prove it, or start.
Can anybody please help?
THANKS
Hint
With $q^2(100n + 124) = 25p^2$, let $r = (q/5)$.
Then $q^2[100(n+1) + 24] = 25p^2 \implies r^2[100(n+1) + 24] = p^2.$
Therefore, I would explore $100(n+1) + 24$ being a perfect square.
If $(10a + s)^2 \equiv 24 \pmod{100}$, then so is $[100(b) + 10(a) + s]^2 ~:~ a,s \leq 9.$
What values for $a,s$ will result in $(10a + s)^2 \equiv 24 \pmod{100}$?
Consider $r^2 \times (100b + 10a + s)^2 = p^2.$