= Noncongruent primes congruent to three modulo eight
{title2=$p\equiv3\pmod8\Longrightarrow p\text{ is not congruent}$}
For a <prime> $p\equiv3\pmod8$, apply <two-isogeny descent> to the <congruent number elliptic curve> $y^2=x^3-p^2x$ and its companion $y^2=x^3+4p^2x$. The first square-class image is exactly $\{1,-1,p,-p\}$, already supplied by <2-torsion>. The second has only positive candidates $1,2,p,2p$. The <quartic covering in a two-isogeny descent> equations for $p$ and $2p$ are impossible modulo $p$, because $-1$ is not a <quadratic residue>. For $2$ the equation is $W^2=2U^4+2p^2V^4$. Coprimality forces $U,V$ odd and $W=2Z$; modulo sixteen this gives $2Z^2=1+9=10$, which is impossible. Thus the second image is trivial and the <square-class index formula for two-isogeny descent> gives rank zero. The same prime-reduction argument bounds its rational torsion by four, and it already has four rational <2-torsion> points, so it has no <rational point> corresponding to a <right triangle> of area $p$.
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