on the quadratic form $5x^2-y^2$

Question

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 $A$?

Answer 1

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 $99\notin A$.