<s>[INST] on the quadratic form $5x^2-y^2$ Consider the following subsets of $\mathbb{Z}$:
$A=\{\frac{5p^2-y^2}{4}\, | \, p, y \,\,\text{odd positive integers}, p\,\, \text{prime}\}$ and
$B=\{\frac{5x^2-y^2}{4}\, | \, x, y \,\,\text{odd positive integers}\}$.
Are there elements in $B$ which are not in $A$?
 [/INST] Note that $\frac{5\cdot9^2-3^2}{4}=99$, and if $\frac{5p^2-y^2}{4}=99$ for some prime number $p$, then
$$5p^2-y^2=396\equiv0\pmod{3},$$
from which it follows that $p\equiv y\equiv0\pmod{3}$. Because $p$ is prime it follows that $p=3$, and hence
$$y^2=5p^2-396=-351,$$
a contradiction. So $99\in B$ but $99\notin A$.
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