Modulo-thirty-two obstruction for a congruent-number descent class
= Modulo-thirty-two obstruction for a congruent-number descent class
{title2=$N^2=2(U^4+p^2V^4)$}
If $p\equiv3\pmod8$, this <quartic covering in a two-isogeny descent> has no primitive integral solution. Opposite parities of $U,V$ give $N^2\equiv2\pmod8$. If both are odd, their fourth powers are one modulo sixteen and $p^2\equiv9\pmod{16}$, giving $N^2\equiv20\pmod{32}$. Neither residue is a square. This excludes the class $2$ on $Y^2=X^3+4p^2X$.